Factorization of a class of 2 × 2 matrix symbols by reduction to a scalar factorization

Factorization of a class of 2 × 2 matrix symbols by reduction to a scalar factorization Abstract Using the reduction of a vector Riemann–Hilbert problem on the unit circle to a scalar problem on a contour in a Riemann surface, a factorization method for a class of symbols is described. The class of symbols involves outer functions and rational functions of the square root of a quotient of first degree polynomials. An application to a problem in the field of integrable systems of infinite dimension is presented. 1. Introduction Riemann–Hilbert problems appear in many areas of applications, in particular diffraction theory and integrable systems, as tools to formulate and obtain solutions to problems in those areas. In the case of vector Riemann–Hilbert problems, in particular with matrix-valued 2 × 2 symbols, no general method of solution exists. But known methods exist for classes of symbols (Bastos et al., 1995; Câmara et al., 2008; dos Santos & dos Santos, 2014; Kiyasov, 2012; Litvinchuk & Spitkovsy, 1987; Mishuris & Rogosin, 2016; Speck, 2017), which lead to classes of problems in the above mentioned areas that can be completely analysed and solved. Explicit solutions are often of great value as they give insight into the physics behind the problems and may also be useful as test solutions in dealing with more general problems by means of numerical methods. Finding new classes of Riemann–Hilbert problems that can be completely studied means that new problems can be solved in those applications. The aim of the present paper is to enlarge the classes of Riemann–Hilbert problems that can be studied and solved explicitly. In particular, the method proposed in this paper enlarges the class of problems dealt with in dos Santos & dos Santos (2014). The Riemann–Hilbert problems considered in the following sections may be written as  $$ G\phi^{+}=\phi^{-},\quad\quad\phi^{\pm}\in\left[C_{\alpha}^{\pm}(\mathbb{T})\right]^{2} $$ (1.1) where $$C_{\alpha }^{\pm }$$ are subspaces, defined below, of the space of Hölder functions on the unit circle in the complex plane, $$\mathbb {T}$$, $$C_{\alpha }(\mathbb {T})$$ and the symbol G is of the form  $$ G=\left[\begin{array}{cc} 1 & g_{1}h^{+}\\g_{2}(h^{+})^{-1} & 1\end{array}\right] $$ (1.2) with h+ an outer function such that $$(h^{+})^{\pm 1}\in C_{\alpha }^{+}(\mathbb {T})$$ and $$g_{j}\in L_{\infty }(\mathbb {T}), j=1,2$$ such that  $$ g_{1}(\xi)=q\left(\rho^{-}(\xi)\right)\,\,\,\textrm{and}\,\,\,g_{2}(\xi)=q^{-1}\left(-\rho^{-}(\xi)\right), \,\,\,\xi\in \mathbb{T}, $$ (1.3) with $$\rho ^{-}(\xi )=\left (\frac {\xi -a}{\xi +a}\right )^{\frac {1}{2}}$$, $$a\in \mathbb {D}$$, $$\mathbb {D}$$ the open unit disc, such that $$\operatorname {Re}\left [\rho ^{-}(\xi )\right ]\geqslant 0$$, q a rational function on $$L_{\infty }(\Gamma )$$, $$\Gamma =\rho ^{-}(\mathbb {T})\cup \left (-\rho ^{-}(\mathbb {T})\right )$$, and q−1 is the algebraic inverse of q. For a particular choice of g1 and g2 the above problems correspond to symbols in the area of integrable systems, as can be seen in the application presented in Section 5. It is the authors’ conviction (cf. Speck, 2017) that the above symbols may be related to diffraction problems associated with boundary conditions more general than Dirichlet or Neumann conditions, such as oblique derivative conditions (Castro & Moura Santos, 2004). Those symbols fall outside the Danielle–Khrapkov class and, in particular, the Rawlins and Williams subclass. It is worth noting that the method proposed in Rawlins & Williams (1981) is not applicable to cases which involve an essential singularity at infinity in the factorization procedure. The paper is organized as follows: in Section 2 some known results, which are needed in what follows, are recalled. In Section 3 the main results are presented, namely a factorization theorem for the class in study. This theorem establishes sufficient conditions for the existence of canonical factorization and explicit formulas for the factors subject to the solution to an algebraic system. The fourth section is devoted to the application of the obtained results to a subclass of matrix functions which in turn is related to an integrable system studied in the fifth and last section. To be more explicit, this last section gives an example of a solution to the Schröinger equation together with the corresponding potential function. For an appropriate modification of symbol G (cf. (5.2)) the potential function gives a solution to the KdV equation. 2. Preliminaries. Wiener–Hopf factorization and Riemann–Hilbert problems Throughout this paper $$\mathbb {T}$$ will denote the unit circle in the complex plane and $$\mathbb {D}$$ the open unit disc. If 0 < α ≤ 1, a function $$f:\mathbb {T}\rightarrow \mathbb {C}$$ is said to be Hölder continuous of index α if and only if, for some $$M\in \mathbb {R}^{+}$$,  $$ |\,f(x)-f(y)|<M|x-y|^{\alpha} $$ (2.1) for every $$x,y\in \mathbb {T}$$. $$C_{\alpha }(\mathbb {T})$$ shall denote the space of Hölder functions of index α in $$\mathbb {T}$$. This space admits the direct-sum decomposition  $$ C_{\alpha}(\mathbb{T})=C_{\alpha}^{+}(\mathbb{T})\oplus C_{\alpha}^{0-}(\mathbb{T}) $$ (2.2) where $$C_{\alpha }^{+}(\mathbb {T})$$ is the subspace of analytic functions in the unit disc $$\mathbb {D}$$ and $$C_{\alpha }^{0-}(\mathbb {T})$$ is the subspace of analytic functions in $$\mathbb {C}\setminus \overline {\mathbb {D}}$$ that vanish at infinity. Definition 2.1 A non-singular matrix-valued function $$G\in \left [C_{\alpha }(\mathbb {T})\right ]^{2\times 2}$$ is said to have a canonical right Wiener–Hopf factorization if and only if it admits a representation of the form  $$ G=G^{-}G^{+} $$ (2.3) with $$\left (G^{+}\right )^{\pm 1}\in C_{\alpha }^{+}(\mathbb {T})$$ and $$\left (G^{-}\right )^{\pm 1}\in C_{\alpha }^{-}(\mathbb {T})=C^{0-}_{\alpha }(\mathbb {T})\oplus \mathbb {C}$$. The Toeplitz operator $$T_{G}=P^{+}G\left |_{\operatorname {Im} P^{+}}\right .$$, where P± are the complementary projections associated to decomposition (2.2) and ImP+ is the image of P+, is invertible if and only if G has a canonical right Wiener–Hopf factorization. If G admits a canonical right factorization then $$G\left (G^{+}\right )^{-1}=G^{-}$$ and the factors $$G^{\pm }\in C_{\alpha }^{\pm }(\mathbb {T})$$ can be determined, cf. Clancey & Gohberg (1981) and Böttcher & Silbermann (1990), by solving the vector Riemann–Hilbert problem in $$C_{\alpha }(\mathbb {T})$$  $$ G\phi^{+}=\phi^{-}, $$ (2.4) whose solutions $$\phi _{j}^{\pm }\in C_{\alpha }^{\pm }(\mathbb {T}), j=1,2$$, with appropriate normalizing conditions, give the columns of the factors. 3. The factorization method This paper presents a method for finding a factorization for the class of non-singular symbols $$G\in \left [C_{\alpha }(\mathbb {T})\right ]_{2\times 2}$$ defined by  $$ G=\left[\begin{array}{cc}1 & h^{+}g_{1} \\ (h^{+})^{-1}g_{2} & 1\end{array}\right] $$ (3.1) where h+ is an outer function such that $$(h^+)^{\pm 1}\in C_\alpha ^+(\mathbb {T})$$ and $$g_j\in L_\infty (\mathbb {T}), j=1,2$$ are such that  $$ g_{1}(\xi)=q\left(\rho^{-}(\xi)\right)\,\,\,\textrm{and}\,\,\,g_{2}(\xi)=q^{-1}\left(-\rho^{-}(\xi)\right), \,\,\,\xi\in \mathbb{T}, $$ (3.2) with q a rational function such that $$g_1,g_2\in L_\infty (\mathbb {T})$$ and $$\rho ^-\in L_\infty ^-(\mathbb {T})$$ defined by $$\rho ^-(\xi )=\left (\frac {\xi -a}{\xi +a}\right )^{\frac {1}{2}}\!\!, a\in \mathbb {D}$$, such that $$\operatorname {Re}\left [\rho ^-(\xi )\right ]\geqslant 0$$. The factorization method here presented is based on the idea of reducing a vector Riemann–Hilbert problem on the unit circle to a scalar problem on a contour in the Riemann surface defined by the algebraic relation $$w^2=\frac {\xi -a}{\xi +a}$$, in this case the Riemann sphere (cf. dos Santos & dos Santos, 2014). The images of the unit circle through the transformations $$\xi \rightarrow \pm \rho ^-(\xi )$$ can be seen as contours in this Riemann surface and are represented by $$\Gamma _j=(-1)^{j+1}\rho ^-(\mathbb {T})$$, j = 1, 2. Using the same identification process one can define Γ = Γ1 ∪Γ2, Ω+ and $$\Omega ^-=\Omega _1^-\cup \Omega _2^-$$ where $$\Omega ^+=\bigcup _{j=1}^2(-1)^{j+1}\rho ^-(\mathbb {C}\setminus \mathbb {D})$$ and $$\Omega _j^-=(-1)^{j+1}\rho ^-(\mathbb {D})$$, j = 1, 2. Before presenting the main result we start by establishing an equivalence between the existence of solutions for the vector Riemann–Hilbert problem (2.4) and a scalar Riemann–Hilbert type problem on Γ. Since Ω− is the union of the two connected components $$\Omega _1^-$$ and $$\Omega _2^-$$, let X− be the set of all functions in Cα−(Γ) that vanish at the points 1 and −1. It follows that $$C_\alpha (\Gamma )=C^+_\alpha (\Gamma )\oplus X^-$$ and that the following holds: Proposition 1 Let G be as defined in (3.1), (3.2). Then the system of equations  $$ \left\{\begin{array}{l}\phi_{1}^{+}+g_{1}h^{+}\phi_{2}^{+}=\phi_{1}^{-}+f_{1}^{+}\\[3mm]g_{2}(h^{+})^{-1}\phi_{1}^{+}+\phi_{2}^{+}=\phi_{2}^{-}+f_{2}^{+}\end{array}\right. $$ (3.3) has a unique solution $$(\phi _1^+,\phi _2^+)\in \left [C^+_\alpha (\mathbb {T})\right ]^2$$, $$(\phi _1^-,\phi _2^-)\in \left [C^{0-}_\alpha (\mathbb {T})\right ]^2$$ for every $$(f_1^+,f_2^+)\in \left [C^+_\alpha (\mathbb {T})\right ]^2$$ if and only if equation  $$ \psi_{1}^{+}+q\widetilde{h}^{+}\psi_{2}^{+}=H(\psi^{-}+f^{+}) $$ (3.4) has a unique solution $$(\psi _1^+,\psi _2^+)\in \left [C_\alpha ^+(\Gamma )\right ]^2$$, ψ−∈ X− with $$\psi _j^+, j=1,2$$, even functions, for every $$f^+\in C^+_\alpha (\Gamma )$$, where  $$ \widetilde{h}^{+}(w)=h^{+}\left(a\frac{1+w^{2}}{1-w^{2}}\right)\quad\textrm{and}\quad H(w)=\left\{\begin{array}{ll} 1 & \textrm{if }w\in\Gamma_{1}\\[5mm]q(w)\widetilde{h}^{+}(w) & \textrm{if }w\in\Gamma_{2}\end{array}\right.. $$ (3.5) Fig. 1. View largeDownload slide Composed contour Γ. Fig. 1. View largeDownload slide Composed contour Γ. Proof. Suppose the system of equations (3.3) has a solution $$(\phi _1^+,\phi _2^+)\in \left [C^+_\alpha (\mathbb {T})\right ]^2$$, $$(\phi _1^-,\phi _2^-)\in \left [C^{0-}_\alpha (\mathbb {T})\right ]^2$$. Let w = ρ−(ξ). Through the transformation $$\xi \rightarrow w$$ the complex plane is transformed onto the right half-plane and $$\mathbb {C}\setminus \overline {\mathbb {D}}$$ is transformed onto the bounded region $$\Omega _1^-$$ whose boundary is the closed line Γ1 (cf. Fig. 1). Writing ξ in terms of w yields  $$ \xi=a\frac{1+w^{2}}{1-w^{2}} $$ (3.6) and defining functions $$\psi _j^\pm (w)=\phi ^\pm _j(a\frac {1+w^2}{1-w^2})$$, j = 1, 2, which are plus/minus functions with respect to the contour Γ1, taking in account its orientation, the first equation in (3.3) can be written as  $$ \psi_{1}^{+}(w)+q(w)\widetilde{h}^{+}(w)\psi_{2}^{+}(w)=\psi_{1}^{-}(w)+\widetilde{f}_{1}^{+}(w), w\in\Gamma_{1}\,, $$ (3.7) where $$\widetilde {h}$$ is defined in (3.5) and $$\widetilde {f}_1^+(w)=f^+_1(a\frac {1+w^2}{1-w^2})$$. On the other hand, putting w = −ρ−(ξ) the complex plane is transformed onto the left half-plane and $$\mathbb {C}\setminus \overline {\mathbb {D}}$$ is transformed onto the bounded region $$\Omega _2^-$$ whose boundary is the closed line Γ2 (again, cf. Fig. 1). Considering the above functions ψi± and $$\widetilde {h}^+$$ extended to the composed contour, and keeping in mind that they are even functions, the second equation in (3.3) can be written, using the mapping onto the left half-plane, as  $$ \psi_{1}^{+}(w)+q(w)\widetilde{h}^{+}(w)\psi_{2}^{+}(w)=q(w)\widetilde{h}^{+}(w)\left[\psi_{2}^{-}(w)+\widetilde{f}_{2}^{+}(w)\right], w\in\Gamma_{2} $$ (3.8) where $$\widetilde {f}_2^+(w)=f^+_2(a\frac {1+w^2}{1-w^2})$$. Defining H as in (3.5),  $$ \psi^{-}(w)=\left\{\begin{array}{ll} \phi_{1}^{-}\left(a\frac{1+w^{2}}{1-w^{2}}\right) & \textrm{if }w\in\Gamma_{1}\\[5mm]\phi_{2}^{-}\left(a\frac{1+w^{2}}{1-w^{2}}\right) & \textrm{if }w\in\Gamma_{2}\end{array}\right. \textrm{and}\ \ \ f^{+}(w)=\left\{\begin{array}{ll} \widetilde{f}_{1}^{+}(w) & \textrm{if }w\in\Gamma_{1}\\[5mm]\widetilde{f}_{2}^{+}(w) & \textrm{if }w\in\Gamma_{2}\end{array}\right. $$ (3.9) one gets (3.4). Since $$\psi _1^+,\psi _2^+$$ are even functions analytic in both left and right half-planes, except $$\Omega ^-_1$$ and $$\Omega _2^-$$ respectively, $$\psi _1^+,\psi _2^+\in C_\alpha ^+(\Gamma _1)\cap C_\alpha ^+(\Gamma _2)$$ thus $$\psi _1^+,\psi _2^+\in C_\alpha ^+(\Gamma )$$ and are even functions. Also it is easy to check that ψ−∈ Cα−(Γ), by computing the corresponding projection PΓ+ψ−, and that ψ−(1) = ψ−(−1) = 0. It follows that $$(\psi _1^+,\psi _2^+)\in \left [C_\alpha ^+(\Gamma )\right ]^2, \psi ^-\in X^-$$ is a solution of (3.4). Conversely, if $$(\psi _1^+,\psi _2^+)\in \left [C_\alpha ^+(\Gamma )\right ]^2, \psi ^-\in X^-$$ is a solution of (3.4), then, since $$\psi _1^+, \psi _2^+$$ are even, $$\phi ^+_j(\xi )=\psi _j^+(\rho ^-(\xi ))\in C_\alpha ^+(\mathbb {T})$$ and $$\phi _j^-(\xi )=\psi ^-\left |_{\Gamma _j}\right .((-1)^{j+1}\rho ^-(\xi ))\in C_\alpha ^{0-}(\mathbb {T})$$, j = 1, 2 satisfy (3.3). Proving the uniqueness is equivalent to showing that the homogeneous system (3.3) has only the trivial solution if and only if the same holds for equation (3.4). Let $$f_1^+,f_2^+=0$$. Then, from (3.9), f = 0 and the condition of unique solvability of (3.4) given in the statement of the proposition now implies that (3.4) has only the trivial solution, as required. The argument can be reversed to show that f = 0 in (3.4) implies that the system (3.3) has only the trivial solution. □ Having established an equivalence between Riemann–Hilbert problem (3.3) on the contour $$\mathbb {T}$$ and scalar Riemann–Hilbert type problem (3.4) on contour Γ it is time to describe the factorization method, starting by establishing sufficient conditions for the homogeneous scalar problem in Cα±(Γ) corresponding to (3.4) to have a unique solution and find explicit solutions for that problem. For every rational function q let $$Q_q(w)=\prod _{i=1}^N(w-z_i)^{m_i}$$ where N is the number of poles of q in $$\Omega ^+\setminus \{\infty \}$$, zi are those poles, mi their multiplicity and let Sq(w) = q(w)Qq(w). Then  $$ q(w)=\frac{S_{q}(w)}{Q_{q}(w)}, w\in\Gamma, $$ (3.10) Sq is a rational function with no poles in $$\Omega ^+\setminus \{\infty \}$$ and Qq is a polynomial whose zeros lie in Ω+ and are not zeros of Sq(Lang, 1999). For every such function define functions $$\widetilde {q}$$, $$\widetilde {S}_q$$ and $$\widetilde {Q}_q$$ by  $$ \widetilde{q}(w)=q(-w),\quad \widetilde{S}_{q}(w)=S_{q}(-w),\quad \widetilde{Q}_{q}(w)=Q_{q}(-w). $$ (3.11) Furthermore, for every F1, F2 ∈ Cα(Γ), let $$T_{F_1F_2}$$ be the function defined by  $$ T_{F_{1},F_{2}}=F_{1}\widetilde{F}_{2}-\widetilde{F}_{1}F_{2} $$ (3.12) where $$\widetilde {F}_j(w)=F_j(-w), j=1,2$$, let Dq be the rational function defined by  $$ D_{q}=T_{Q_{q},S_{q}}{,} $$ (3.13) nZ be the number of zeros of Dq in $$\Omega ^+\setminus \{\infty \}$$, wk, k = 1...nZ be those zeros, nk their multiplicity, $$n_D=\sum _{k=1}^{n_Z}n_k$$ and let $${\mathscr R}_q$$ be the set of all rational functions R of the form  $$ R(w)=\frac{P_{R}(w)}{Q_{q}(w)} $$ (3.14) where Qq is defined in (3.10) and PR is any polynomial of degree nP equal to the degree of Qq plus m0, $$n_P=m_0+\sum _{k=1}^{N}m_k$$, with m0 the order of the pole at infinity as a pole of q. Let, now, for every $$R\in {\mathscr R}_q$$  $$ \begin{array}{r@{\ }c@{\ }l} A_{R,k,j}&=&\displaystyle\frac{\partial^{j} T_{P_{R}H^{+},Q_{q}}}{\partial w^{j}}(w_{k}), 1\leqslant k\leqslant n_{Z}, 0\leqslant j\leqslant n_{k}-1\\[4mm] B_{R,k,j}&=&\displaystyle\frac{\partial^{j} T_{P_{R}H^{+},S_{q}}}{\partial w^{j}}(w_{k}), 1\leqslant k\leqslant n_{Z}, 0\leqslant j\leqslant n_{k}-1\\[4mm] C_{R,1}&=&\displaystyle\lim_{w\rightarrow\infty}\displaystyle\frac{T_{P_{R}H^{+},Q_{q}}(w)}{w^{n_{D}}}\\[4mm] C_{R,2}&=&\displaystyle\lim_{w\rightarrow\infty}\displaystyle\frac{T_{P_{R}H^{+},S_{q}}(w)}{w^{n_{D}}} \end{array}. $$ (3.15) Note that there is a bijection between the set $${\mathscr R}_q$$ and $$\mathbb {R}^{n_P}$$ since the nP coefficients of PR(w) determine uniquely $$R\in {\mathscr R}_q$$. Note, also, that, for a given q, AR, k, j, BR, k, j, CR, 1 and CR, 2 depend only on the coefficients of the polynomial PR(w). Proposition 2 Let $$\widetilde {h}^+$$ be an even function such that $$(\widetilde {h}^+)^{\pm 1}\in C_\alpha ^+(\Gamma )$$, q be a non-even rational function with no poles on Γ, represented in the form (3.10), such that Dq defined by (3.13) has no zeros on Γ and H be the scalar function defined by  $$ H(w)=\left\{\begin{array}{ll} 1 & w\in\Gamma_{1}\\[3mm]q(w)\widetilde{h}^{+}(w) & w\in\Gamma_{2}\end{array}\right..\nonumber $$ If H has a canonical right Wiener–Hopf factorization in Cα(Γ), H = H−H+, and there exists a unique rational function $$R\in { \mathscr R}_q$$, represented in the form (3.14), such that CR, 1, CR, 2 as defined in (3.15) are finite and the algebraic system in the nP coefficients of PR(w),  $$ \left\{\begin{array}{l}P_{R}(1)=c_{1}H^{-}(1)Q_{q}(1)\\[3mm]P_{R}(-1)=c_{2}H^{-}(-1)Q_{q}(-1)\\[3mm]A_{R,k,j}=0, \quad1\leqslant k\leqslant n_{Z}, 0\leqslant j\leqslant n_{k}-1\\[3mm] B_{R,k,j}=0, \quad1\leqslant k\leqslant n_{Z}, 0\leqslant j\leqslant n_{k}-1 \end{array}\right. $$ (3.16) where c1, c2 are arbitrary complex numbers and AR, k, j, BR, k, j are defined in (3.15), is uniquely solvable, then equation  $$ \psi_{1}^{+}+q\widetilde{h}^{+}\psi_{2}^{+}=H\psi^{-} $$ (3.17) with normalizing conditions ψ−((−1)j+1) = cj, j = 1, 2 has a unique solution $$\psi ^+_1,\psi ^+_2\in C^+_\alpha (\Gamma )$$, $$\psi ^-\in C^-_\alpha (\Gamma )$$ such that $$\psi ^+_1,\psi ^+_2$$ are even. That solution is given by  $$ \left\{\begin{array}{l} \psi_{1}^{+}=D_{q}^{-1}T_{P_{R}H^{+},S_{q}}\\[3mm] \psi_{2}^{+}=-(\widetilde{h}^{+})^{-1}D_{q}^{-1}T_{P_{R}H^{+},Q_{q}} \end{array}\right.. $$ (3.18) Proof. Since H = H−H+ with $$\left (H^+\right )^{\pm 1}\in C_\alpha ^+(\Gamma )$$ and $$\left (H^-\right )^{\pm 1}\in C_\alpha ^-(\Gamma )$$ then, by (3.17), $$\left (H^+\right )^{-1}\left (\psi _1^++q\widetilde {h}^+\psi _2^+\right )=H^-\psi ^-\in C_\alpha ^-(\Gamma )$$. On the other hand, since the only singularities of that function in Ω+ are the poles of q, it is a meromorphic function in $$\mathbb {C}$$ whose poles are in Ω+ and coincide with the poles of q in Ω+. Then there exists $$R\in {\mathscr R}_q$$ such that  $$ \left(H^{+}\right)^{-1}\left(\psi_{1}^{+}+q\widetilde{h}^{+}\psi_{2}^{+}\right)=H^{-}\psi^{-}=R. $$ (3.19) It suffices now to write the same equation in −w, keeping in mind that $$\widetilde {h}^+$$, $$\psi _1^+$$ and $$\psi _2^+$$ are even, by hypothesis, to write the system  $$ \left\{\begin{array}{l}\psi_{1}^{+}+q\,\widetilde{h}^{+}\psi_{2}^{+}=RH^{+}\\[3mm]\psi_{1}^{+}+\widetilde{q}\,\widetilde{h}^{+}\psi_{2}^{+}=\widetilde{R}\widetilde{H}^{+}\end{array}\right., $$ (3.20) defining $$\widetilde {R}(w)=R(-w)$$, $$\widetilde {q}(w)=q(-w)$$ and $$\widetilde {H}^+=H^+(-w)$$. Since Dq has no zeros on Γ, which means that $$q-\widetilde {q}$$ has no zeros on Γ, this system has a unique algebraic solution given by  $$ \left\{\begin{array}{l} \psi_{1}^{+}=\displaystyle\frac{\widetilde{q}\,R\,H^{+}-q\,\widetilde{R}\,\widetilde{H}^{+}}{\widetilde{q}-q}\\[3mm] \psi_{2}^{+}=(\widetilde{h}^{+})^{-1}\displaystyle\frac{R\,H^{+}-\widetilde{R}\,\widetilde{H}^{+}}{q-\widetilde{q}} \end{array}\right. $$ (3.21) which using (3.10), (3.11), (3.14) and (3.12) yields (3.18). In order for $$\psi _1^+$$ and $$\psi _2^+$$ to be solutions of Riemann–Hilbert problem (3.19), $$R\in {\mathscr R}_q$$ must be given by (3.14) with the coefficients of PR determined by the system of equations (3.16) where the first two equations of (3.16) correspond to the normalizing conditions ψ−((−1)j+1) = cj, j = 1, 2, which, in turn, correspond to the normalizing conditions at infinity for $$\phi _1^-$$ and $$\phi _2^-$$ in the initial homogeneous Riemann–Hilbert problem on $$\mathbb {T}$$, obtained by (3.3), and the remaining conditions are the natural conditions to ensure the cancellation of all singularities in Ω+ and, thus, the analiticity of $$\psi _1^+$$ and $$\psi _2^+$$ in Ω+. It remains to prove that these are even functions which can be easily checked computing $$\psi ^+_j(-w), j=1,2$$ and that they are unique which is a consequence of the uniqueness of the solution of that same system of equations and of the fact that any solution must satisfy (3.16). □ Remark 3 The canonical right Wiener–Hopf factorization of the non-singular scalar function H on the composed contour Γ can be obtained, if it exists, by  $$ \begin{array}{l} H^{+}(w)=\exp\left(P_{\Gamma}^{+}\log H\right)(w)\\[3mm] H^{-}(w)=\exp\left(P_{\Gamma}^{-}\log H\right)(w) \end{array} $$ (3.22) where PΓ± are complementary projections defined by  $$ \left(P_{\Gamma}^{\pm} \log H\right)(w)=\mp\frac{1}{2\pi\operatorname{i}}\int_{\Gamma}\frac{\log H(\xi)}{\xi-w}\;\textrm{d}\xi,\,\,w\in\Omega^{\pm} .$$ (3.23) A result similar to the previous proposition is easily obtained for the initial Riemann–Hilbert problem (3.3). Making use of the previous results it is now possible to state the main result in this section, which gives a method to get the factors of matrix G. Theorem 4 Let G be as defined in (3.1), (3.2), $$\widetilde {h}^+(w)=h^+(a\frac {1+w^2}{1-w^2})$$, Dq defined by (3.10), (3.13), Δ(ξ) = Dq(ρ−(ξ)) and  $$ H(w)=\left\{\begin{array}{ll} 1 & w\in\Gamma_{1}\\[3mm]q(w)\widetilde{h}^{+}(w) & w\in\Gamma_{2}\end{array}\right. .$$ (3.24) If H has a canonical Wiener–Hopf factorization H = H−H+, Δ has no zeros on $$\mathbb {T}$$ and, for every $$({c_1,c_2})\in \mathbb {C}^2$$, (3.16) is uniquely solvable then G has a canonical Wiener–Hopf factorization and the corresponding factors are given by $$(G^\pm )^{\mp 1}=\left [\phi _{jk}^\pm \right ]_{j,k=1,2}$$, where  $$ \begin{array}{l} \phi_{1,k}^{+}(\xi)=[\Delta(\xi)]^{-1}\left[T_{P_{R_{k}}H^{+},S_{q}}(\rho^{-}(\xi))\right]\\[3mm]\phi_{2,k}^{+}(\xi)=\left[h^{+}(\xi)\Delta(\xi)\right]^{-1}\left[T_{P_{R_{k}}H^{+},Q_{q}}(\rho^{-}(\xi))\right] \\[3mm] \phi_{1,k}^{-}(\xi)=R_{k}(\rho^{-}(\xi))H^{-}(\rho^{-}(\xi)) \\[3mm] \phi_{2,k}^{-}(\xi)=R_{k}(-\rho^{-}(\xi))H^{-}(-\rho^{-}(\xi)) \end{array} $$ (3.25) with $$R_k\in {\mathscr R}_q, k=1,2$$ are the unique solutions of (3.16) for cj = |k − j|, j = 1, 2, $$T_{P_{R_k}H^+,S_q}$$ and $$T_{P_{R_k}H^+,Q_q}$$ defined in (3.12). Proof. Under the assumptions of the theorem, by Proposition 2, it is possible to obtain a unique solution to the problem on Γ, for every set of normalizing conditions $$\psi ^-_j((-1)^{j+1})={c}_j, j=1,2$$ and thus, by Proposition 1, the initial Riemann–Hilbert problem (2.4) has a unique solution in $$C_\alpha ^\pm (\mathbb {T})$$ for every set of normalizing conditions $$\phi ^-_j(\infty )={c_j}, j=1,2$$ and it is possible to obtain the corresponding solutions on $$\mathbb {T}$$, keeping in mind that the solutions obtained through Proposition 2 are even. The fact that these solutions are even functions in $$C^+_\alpha (\Gamma )$$ enough to ensure that the solutions obtained for the initial problem (3.3) are in $$C^+_\alpha (\mathbb {T})$$, even after composing with ρ−, since this means that they are functions of w2. Choosing c1 = 0 and c2 = 1 for k = 1 and c1 = 1 and c2 = 0 for k = 2 one obtains the two independent solutions (3.25). □ Remark 5 Note that a similar result can be easily achieved assuming that instead of q being a rational function, q ∈ Cα+(Γ). Also, similar results can be obtained considering (2.4) on the real axis instead of $$\mathbb {T}$$. Theorem 4 is the basis of the factorization method for matrix symbol (3.1) presented in this paper. It gives sufficient conditions for the existence of canonical factorization of G and, under these conditions, presents formulas for the factors  $$ (G^{\pm})^{\mp1}=\left[\phi_{jk}^{\pm}\right]_{j,k=1,2} $$ (3.26) where functions ϕjk± are defined in (3.25). Given functions g1, g2 and h+ such that G is in the class studied in this paper, the algebraic system (3.16) can be written and if uniquely solvable, functions Rk can be explicitly written and thus formulas (3.25) explicitly give the factors of G. 4. Application to a factorization problem We now apply the theory presented in the previous section to a particular factorization problem. This particular problem is related to the integrable system considered in the next section. Since a particular choice of g1, g2 and h+ is given, which falls under the class to each the theory in the previous section applies, Theorem 4 allows for explicit formulas for the factors to be written. Let us consider the case where  $$ G=\left[\begin{array}{cc}1 & h^{+} g_{1} \\ (h^{+})^{-1}g_{2} & 1\end{array}\right] $$ (4.1) with $$g_{1}(\xi )=\operatorname {i} b^{-1}\displaystyle \frac {\rho ^{-}(\xi )-\operatorname {i} b}{\rho ^{-}(\xi )-\operatorname {i} b^{-1}}\rho ^{-}(\xi )$$, $$g_{2}(\xi )=\operatorname {i} b\displaystyle \frac {\rho ^{-}(\xi )+\operatorname {i} b^{-1}}{(\rho ^{-}(\xi )+\operatorname {i} b)\rho ^{-}(\xi )}$$, $$h^{+}(\xi )=\exp (2x\xi )$$ and $$\rho ^{-}(\xi )=\left (\frac {\xi -a}{\xi +a}\right )^{\frac {1}{2}}$$ with $$a,b,x\in \mathbb {C}$$ such that b = ρ−(b0) and $$b_{0}\in \mathbb {D}$$. These choices correspond to putting  $$ q(w)=\frac{\operatorname{i} w+b}{-\operatorname{i} w^{-1}+b},\quad w\in\Gamma,\quad b\in\mathbb{C}\setminus\{0\} .$$ (4.2) To obtain a factorization using the results of Section 3, in particular Theorem 4, we note that Dq, as defined in (3.13), depends on the location of the zeros and poles of q, the same happening to the factorization of H. Since b = ρ−(b0) with $$b_{0}\in \mathbb {D}$$, b ∈Ω+ and, making use of the transformation ξ↦ρ−(ξ) and its inverse, it is easy to check that ib−1 also lies in Ω+ thus q is a minus factor. Then,  $$ n_{0}=1,\quad Q_{q}(w)=w-\operatorname{i} b^{-1},\quad S_{q}(w)=\operatorname{i} b^{-1}w(w-\operatorname{i} b) $$ (4.3) with n0, Qq and Sq as defined in (3.10) and, since  $$ \widetilde{h}^{+}(w)=\exp\left(2ax\frac{1+w^{2}}{1-w^{2}}\right)=\exp(-2ax)\exp\left(\frac{-2ax}{w-1}\right)\exp\left(\frac{2ax}{w+1}\right), $$ (4.4) H can be canonically factorized, using (3.22), as  $$ H^{+}(w)=\exp\left(\frac{2ax}{w+1}\right) \,\,\, \textrm{and} \,\,\, H^{-}(w)=\left\{\begin{array}{ll}\exp\left(-\frac{2ax}{w+1}\right) & w\in\Gamma_{1}\\[5mm]q(w)\exp(-2ax)\exp\left(\frac{-2ax}{w-1}\right) & w\in\Gamma_{2} \end{array}\right.. $$ (4.5) It is a matter of a simple calculation to conclude that, by (3.11), (3.13),  $$ D_{q}(w)=2\operatorname{i} b^{-1}w(w^{2}+1) $$ (4.6) and, thus, Dq has no zeros on Γ, Δ has no zeros on $$\mathbb {T}$$ and there are three first order zeros of Dq in Ω+: w1 = 0, w2 = i and w3 = −i. Since q has two poles in Ω+, ib−1 and $$\infty $$, $${\mathscr R}_{q}$$, as defined in (3.14), is the set  $$ {\mathscr R}_{q}=\left\{\frac{\gamma_{0}+\gamma_{1}w+\gamma_{2}w^{2}}{w-\operatorname{i} b^{-1}}:\gamma_{0},\gamma_{1},\gamma_{2}\in\mathbb{C}\right\} .$$ (4.7) Then the third and fourth conditions on (3.16) are always satisfied for w1 = 0 and result in equivalent equations for w2 = i and w3 = −i, while the last condition on (3.16) is always fulfilled. Letting 2Aj = H−((−1)j+1)Qq((−1)j+1), j = 1, 2, (3.16) can be rewritten in the form  $$ \left\{\begin{array}{l}\gamma_{1}=A_{1}{c}_{1}-A_{2}{c}_{2}\\\gamma_{2}+\gamma_{0}=A_{1}{c}_{1}+A_{2}{c}_{2}\\ (\gamma_{2}-\gamma_{0})(b+1+(b-1)\exp(\operatorname{i} 2ax))=(A_{1}{c}_{1}-A_{2}{c}_{2})\operatorname{i}((b-1)\exp(\operatorname{i} 2ax)-b-1)\end{array}\right. $$ (4.8) Thus, if $$b+1+(b-1)\exp (\operatorname {i} 2ax)\neq 0$$ the system is uniquely solvable and there is a canonical factorization of G. On the other hand if that condition is not fulfilled then the system has more than one solution if b = −1 and no solutions if b ≠ −1 and in either case there is no canonical factorization for G. The previous results lead to the following: Proposition 6 Let $$b\in \mathbb {C}\setminus \{0\}$$ be such that ib ∈Ω+, G be as defined in (3.1) with  $$ g_{1}(\xi)=g_{2}(-\xi)=\frac{\operatorname{i} \rho^{-}(\xi)+b}{-\operatorname{i} (\rho^{-}(\xi))^{-1}+b} $$ (4.9) and $$\rho ^{-}(\xi )=\left (\frac {\xi -a}{\xi +a}\right )^{\frac {1}{2}}$$, $$a\in \mathbb {D}$$. Then if x is such that  $$ (1-b)\exp(\operatorname{i} 2ax)\neq b+1 $$ (4.10) G has a canonical factorization given by  $$ G^{+}=\left[\begin{array}{cc} \phi_{11}^{+} & \phi_{12}^{+}\\[3mm] \phi_{21}^{+} & \phi_{22}^{+}\end{array}\right]^{-1}\quad\quad G^{-}=\left[\begin{array}{cc} \phi_{11}^{-} & \phi_{12}^{-}\\[3mm]\phi_{21}^{-} & \phi_{22}^{-}\end{array}\right] $$ (4.11) where  $$ \begin{array}{l} \phi_{11}^{+}(\xi)=\frac{1}{4}(1+a\xi^{-1})\left[(\rho^{-}(\xi)+\operatorname{i} b)\varphi_{1}(\xi)-(\rho^{-}(\xi)-\operatorname{i} b)\widetilde{\varphi}_{1}(\xi)\right] \\[5mm] \phi_{12}^{+}(\xi)=\frac{1}{4}(1+a\xi^{-1})\left[(\rho^{-}(\xi)+\operatorname{i} b)\varphi_{2}(\xi)-(\rho^{-}(\xi)-\operatorname{i} b)\widetilde{\varphi}_{2}(\xi)\right]\\[5mm] \phi_{21}^{+}(\xi)=-\frac{\operatorname{i}}{4}b(1+a\xi^{-1})\left[\rho^{-}(\xi)\right]^{-1}\left[(\rho^{-}(\xi)+\operatorname{i} b^{-1})\varphi_{1}(\xi)+(\rho^{-}(\xi)-\operatorname{i} b^{-1})\widetilde{\varphi}_{1}(\xi)\right]\exp(-2\xi x)\\[5mm] \phi_{22}^{+}(\xi)=-\frac{\operatorname{i}}{4}b(1+a\xi^{-1})\left[\rho^{-}(\xi)\right]^{-1}\left[(\rho^{-}(\xi)+\operatorname{i} b^{-1})\varphi_{2}(\xi)+(\rho^{-}(\xi)-\operatorname{i} b^{-1})\widetilde{\varphi}_{2}(\xi)\right]\exp(-2\xi x)\\[5mm] \phi_{11}^{-}(\xi)=\displaystyle\frac{\varphi_{1}(\xi)}{\rho^{-}(\xi)-\operatorname{i} b^{-1}}\\[5mm] \phi_{12}^{-}(\xi)=\displaystyle\frac{\varphi_{2}(\xi)}{\rho^{-}(\xi)-\operatorname{i} b^{-1}}\\[5mm] \phi_{21}^{-}(\xi)=\displaystyle\frac{\widetilde{\varphi}_{1}(\xi)\exp(-2x\xi)}{\operatorname{i} b^{-1}(\rho^{-}(\xi)+\operatorname{i} b)\rho^{-}(\xi)}\\[5mm] \phi_{22}^{-}(\xi)=\displaystyle\frac{\widetilde{\varphi}_{2}(\xi)\exp(-2x\xi)}{\operatorname{i} b^{-1}(\rho^{-}(\xi)+\operatorname{i} b)\rho^{-}(\xi)} \end{array} $$ with  $$ \begin{array}{l} \varphi_{i}(\xi)=\left(\gamma_{0i}+\gamma_{1i}\rho^{-}(\xi)+\gamma_{2i}(\rho^{-}(\xi))^{2}\right)\exp\frac{2ax}{\rho^{-}(\xi)+1}),\\[5mm] \widetilde{\varphi}_{i}(\xi)=\left(\gamma_{0i}-\gamma_{1i}\rho^{-}(\xi)+\gamma_{2i}(\rho^{-}(\xi))^{2}\right)\exp\frac{2ax}{-\rho^{-}(\xi)+1}), \end{array} $$   $$ \begin{array}{lll} \gamma_{01}=\frac{1}{2}A_{1}(1+A_{3}), & \gamma_{11}=A_{1}, & \gamma_{21}=\frac{1}{2}A_{1}(1-A_{3}))\\[5mm]\gamma_{02}=\frac{1}{2}A_{2}(1-A_{3}), & \gamma_{12}=-A_{2}, & \gamma_{22}=\frac{1}{2}A_{2}(1+A_{3}) \end{array} $$ and  $$ \begin{array}{l@{,\quad }l@{,\quad }l} A_{1}=\frac{1}{2}\exp(-ax)(1-\operatorname{i} b^{-1}) & A_{2}=\frac{1}{2}\exp(-ax)(1+\operatorname{i} b^{-1}) & A_{3}=\operatorname{i}\frac{(b-1)\exp(\operatorname{i} 2ax)-(b+1)}{(b-1)\exp(\operatorname{i} 2ax)+(b+1)}. \end{array} $$ Proof. It is enough to use the results of the last theorem of the previous section. Note that the functions between square brackets in the expression given for $$\phi _{1k}^{+}$$ are even functions in ρ− that vanish at 0 and those in the expression of $$\phi _{2k}^{+}$$ are odd, so $$\phi _{1k}^{+}$$ are indeed plus functions. Also note that although $$\exp (2x\xi )$$ is not a minus function, when multiplied by $$\widetilde {\varphi }_{j}$$ it is indeed a minus function. □ 5. Application to an integrable systems problem In this section we consider an application of the theory of Section 3 and results of Section 4 to a problem in the field of integrable systems of infinite dimension, specifically to a problem related to the KdV equation. As shown in dos Santos & dos Santos (2016) families of solutions for the KdV equation and for the linear Schrödinger equation can be obtained by solving a Riemann–Hilbert problem with shift on the unit circle, $$\mathbb {T}$$, of the form  $$ E(\gamma_{a}\phi^{+}+\gamma_{b}\phi^{+}(-))+\phi^{-}=0, $$ (5.1) subject to $$\phi ^{-}(\infty )=1$$, where E is an outer function of the form given below, $$\gamma _{a}, \gamma _{b}:\mathbb {T}\to \mathbb {C}$$ are scalar functions assumed to be in $$C_{\alpha }(\mathbb {T})$$, 0 < α < 1. In (5.1), ϕ+(−)(ξ) = ϕ+(−ξ) and ϕ± belong to the subspaces of $$C_{\alpha }(\mathbb {T})$$ of functions analytic on the unit disc and on $$\mathbb {C}\setminus \overline {\mathbb {D}}$$, respectively. The above problem is equivalent to the standard vector Riemann–Hilbert problem  $$ G\phi^{+}=\phi^{-}, \phi^{\pm}\in\left[C_{\alpha}^{\pm}(\mathbb{T})\right]^{2} $$ (5.2) with symbol G of the form  $$ G=\left[\begin{array}{cc}\gamma_{a}&\gamma_{b}E^{-2}\\\gamma_{b}(-)E^{2} & \gamma_{a}(-)\end{array}\right] $$ (5.3) where $$E(\xi )=\exp (\xi x), x\in \mathbb {R}$$ for the Schrödinger equation and $$E(\xi )=\exp (\xi x-\xi ^{3}t), x,t\in \mathbb {R}$$ for the KdV equation. Function ϕ− in (5.1) gives the solution to the Schrödinger equation with potential function u given by   $$ u(x)=-2\frac{\;\textrm{d}\phi_{1}}{\;\textrm{d} x}(x) $$ (5.4) where ϕ1 is the coefficient of λ−1 in the Laurent expansion of ϕ− in $$\mathbb {C}\setminus \overline {\mathbb {D}}$$ (cf. dos Santos & dos Santos, 2016). For $$E(\xi )=\exp (\xi x-\xi ^{3} t)$$, the potential function u is a solution to the KdV equation. For simplicity, in the example below we take, in (3.1) $$h^{+}(\xi )=\exp (\xi x), x\in \mathbb {R}$$ and in (3.2)  $$ q(w)=\frac{\operatorname{i} w+b}{-\operatorname{i} w^{-1}+b}, b\in\mathbb{C}\setminus\{0\}, w\in\Gamma .$$ (5.5) These choices yield g2(ξ) = g1(−ξ), with g1, g2 defined by (3.2), which means that, for γb = q(ρ−), γb(−) = q−1(−ρ−), γa = 1 and $$E(\xi )=\exp (\xi x)$$, symbol G belongs to the class considered in Section 3 which is not included in the class studied in dos Santos & dos Santos (2014, 2016). Before proceeding with the explicit solution of Riemann–Hilbert problem (5.2) we note that all functions involved in the analysis depend on the parameter x. In order to find the solution to the Schrödinger equation one has to find the first coordinate of the $$C_{\alpha }^{-}(\mathbb {T})$$ solution of Riemann–Hilbert problem (5.2) for normalizing conditions $$\phi ^{-}_{1}(\infty )=\phi ^{-}_{2}(\infty )=1$$. Since the choices made for g1, g2 and h+ lead to the same Riemann–Hilbert problem dealt with in the previous section, one can use the same computations but with the new normalizing conditions. Then the solution $$\phi ^{-}_{1}$$ to the Riemann–Hilbert problem is given by  $$ \phi^{-}_{1}=\frac{\gamma_{2}(\rho^{-})^{2}+\gamma_{1}\rho^{-}+\gamma_{0}}{\rho^{-}-\operatorname{i} b^{-1}}\exp\left(\frac{2ax}{\rho^{-}+1}\right) $$ (5.6) with $$\gamma _{2}=\operatorname {i} C_{1}\exp (-ax)\left [1-\frac {2}{1+C_{2}\exp (\operatorname {i} 2ax)}\right ]=-\gamma _{0}$$, $$\gamma _{1}=2C_{1}\exp (-ax)$$, C1 = 1 − ib−1 and $$C_{2}=\frac {b-1}{b+1}$$. This leads to $$u(x)=\frac {-16a^{2}C_{2}\exp (2\operatorname {i} ax)}{(1+C_{2}\exp (2\operatorname {i} ax))^{2}}$$. Rearranging this expression and putting $$\exp (2\operatorname {i}\theta _{0})=\frac {b-1}{b+1}$$ one gets  $$ u(x)=\frac{-4a^{2}C_{2}\exp(-2\operatorname{i}\theta_{0})}{\cos^{2}(ax+\theta_{0})}\,. $$ (5.7) We note that the above expression corresponds to a soliton solution (see e.g. dos Santos & dos Santos, 2016, Section 3.1), for ai and θi real. Writing a = iα, θ = iφ we get  $$ u(x)=\frac{A}{\cosh^{2}(\alpha x+\varphi)} $$ (5.8) where $$A=-4a^{2}C_{2}\exp (-2\varphi )$$. Actually, expression (5.8) is the potential function for which the solution to the linear Schrödinger equation is given by (5.6). As noted above to obtain the solution for the KdV equation we have to replace the exponential in (5.3) by $$E(\xi )=\exp (\xi x-\xi ^{3}t)$$. Proposition 7 The Riemann–Hilbert problem with shift (5.3) with γa = 1 and γb = q(ρ−) yields, for the particular choice (5.5) of q, the soliton type potential (5.7) for which the solution to the linear Schrodinger equation is given by (5.6). An interesting question to be dealt with in a future paper is which subclasses of the class of symbols considered in Section 3 lead to interesting new solutions to either the KdV or the Schrödinger equations, in particular which subclasses lead to almost periodic solutions. References Bastos, M. A., Câmara, M. C. & dos Santos, A. F. ( 1995) Generalized factorization for Daniele–Khrapkov matrix functions—explicit formulas. J. Math. Anal. Appl. , 190, 295-- 328. Google Scholar CrossRef Search ADS   Böttcher, A. & Silbermann, B. ( 1990) Analysis of Toeplitz Operators . Berlin: Springer. Google Scholar CrossRef Search ADS   Câmara, M. C., dos Santos, A. F. & dos Santos, P. F. ( 2008) Matrix Riemann–Hilbert problems and factorization on Riemann surfaces. J. Funct. Anal. , 255, 228-- 254. Google Scholar CrossRef Search ADS   Castro, L. P. & Moura Santos, A. ( 2004) An operator approach for an oblique derivative boundary-transmission problem. Math. Meth Appl. Sci. , 27, 1469-- 1491. Google Scholar CrossRef Search ADS   Clancey, K. & Gohberg, I. ( 1981) Factorization of Matrix Functions and Singular Integral Operators . Basel: Birkhäuser. Google Scholar CrossRef Search ADS   dos Santos, A. F. & dos Santos, P. F. ( 2014) Factorization of a class of symbols with outer functions. J. Math. Anal. Appl. , 413, 185-- 194. Google Scholar CrossRef Search ADS   dos Santos, A. F. & dos Santos, P. F. ( 2016) Segal–Wilson approach to integrable systems and Riemann–Hilbert problems. J. Math. Anal. Appl. , 443, 797-- 816. Google Scholar CrossRef Search ADS   Kiyasov, S. N. ( 2012) Some cases of efficient factorization of second-order matrix functions. Russ. Mathematics , 56, 30-- 36. Google Scholar CrossRef Search ADS   Lang, S. ( 1999) Complex Analysis . New York: Springer. Google Scholar CrossRef Search ADS   Litvinchuk, G. S. & Spitkovsy, I. ( 1987) The Factorization of Matrix Valued Functions . Basel-Boston: Birkhäuser. Google Scholar CrossRef Search ADS   Mishuris, G. & Rogosin, S. ( 2016) Constructive methods for factorization of matrix functions. IMA J. Appl. Math. , 81, 365-- 391. Google Scholar CrossRef Search ADS   Rawlins, A. D. & Williams, W. E. ( 1981) Matrix Wiener–Hopf factorisation. Q.J. Mech. Appl. Math. , 34, 1-- 8. Google Scholar CrossRef Search ADS   Speck, F.O. ( 2017) A class of interface problems for the Helmholtz equation in $$\mathbb {R}^n$$. Math. Methods Appl. Sci. , 40, 391-- 402. Google Scholar CrossRef Search ADS   © The Author(s) 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Applied Mathematics Oxford University Press

Factorization of a class of 2 × 2 matrix symbols by reduction to a scalar factorization

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© The Author(s) 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
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Abstract

Abstract Using the reduction of a vector Riemann–Hilbert problem on the unit circle to a scalar problem on a contour in a Riemann surface, a factorization method for a class of symbols is described. The class of symbols involves outer functions and rational functions of the square root of a quotient of first degree polynomials. An application to a problem in the field of integrable systems of infinite dimension is presented. 1. Introduction Riemann–Hilbert problems appear in many areas of applications, in particular diffraction theory and integrable systems, as tools to formulate and obtain solutions to problems in those areas. In the case of vector Riemann–Hilbert problems, in particular with matrix-valued 2 × 2 symbols, no general method of solution exists. But known methods exist for classes of symbols (Bastos et al., 1995; Câmara et al., 2008; dos Santos & dos Santos, 2014; Kiyasov, 2012; Litvinchuk & Spitkovsy, 1987; Mishuris & Rogosin, 2016; Speck, 2017), which lead to classes of problems in the above mentioned areas that can be completely analysed and solved. Explicit solutions are often of great value as they give insight into the physics behind the problems and may also be useful as test solutions in dealing with more general problems by means of numerical methods. Finding new classes of Riemann–Hilbert problems that can be completely studied means that new problems can be solved in those applications. The aim of the present paper is to enlarge the classes of Riemann–Hilbert problems that can be studied and solved explicitly. In particular, the method proposed in this paper enlarges the class of problems dealt with in dos Santos & dos Santos (2014). The Riemann–Hilbert problems considered in the following sections may be written as  $$ G\phi^{+}=\phi^{-},\quad\quad\phi^{\pm}\in\left[C_{\alpha}^{\pm}(\mathbb{T})\right]^{2} $$ (1.1) where $$C_{\alpha }^{\pm }$$ are subspaces, defined below, of the space of Hölder functions on the unit circle in the complex plane, $$\mathbb {T}$$, $$C_{\alpha }(\mathbb {T})$$ and the symbol G is of the form  $$ G=\left[\begin{array}{cc} 1 & g_{1}h^{+}\\g_{2}(h^{+})^{-1} & 1\end{array}\right] $$ (1.2) with h+ an outer function such that $$(h^{+})^{\pm 1}\in C_{\alpha }^{+}(\mathbb {T})$$ and $$g_{j}\in L_{\infty }(\mathbb {T}), j=1,2$$ such that  $$ g_{1}(\xi)=q\left(\rho^{-}(\xi)\right)\,\,\,\textrm{and}\,\,\,g_{2}(\xi)=q^{-1}\left(-\rho^{-}(\xi)\right), \,\,\,\xi\in \mathbb{T}, $$ (1.3) with $$\rho ^{-}(\xi )=\left (\frac {\xi -a}{\xi +a}\right )^{\frac {1}{2}}$$, $$a\in \mathbb {D}$$, $$\mathbb {D}$$ the open unit disc, such that $$\operatorname {Re}\left [\rho ^{-}(\xi )\right ]\geqslant 0$$, q a rational function on $$L_{\infty }(\Gamma )$$, $$\Gamma =\rho ^{-}(\mathbb {T})\cup \left (-\rho ^{-}(\mathbb {T})\right )$$, and q−1 is the algebraic inverse of q. For a particular choice of g1 and g2 the above problems correspond to symbols in the area of integrable systems, as can be seen in the application presented in Section 5. It is the authors’ conviction (cf. Speck, 2017) that the above symbols may be related to diffraction problems associated with boundary conditions more general than Dirichlet or Neumann conditions, such as oblique derivative conditions (Castro & Moura Santos, 2004). Those symbols fall outside the Danielle–Khrapkov class and, in particular, the Rawlins and Williams subclass. It is worth noting that the method proposed in Rawlins & Williams (1981) is not applicable to cases which involve an essential singularity at infinity in the factorization procedure. The paper is organized as follows: in Section 2 some known results, which are needed in what follows, are recalled. In Section 3 the main results are presented, namely a factorization theorem for the class in study. This theorem establishes sufficient conditions for the existence of canonical factorization and explicit formulas for the factors subject to the solution to an algebraic system. The fourth section is devoted to the application of the obtained results to a subclass of matrix functions which in turn is related to an integrable system studied in the fifth and last section. To be more explicit, this last section gives an example of a solution to the Schröinger equation together with the corresponding potential function. For an appropriate modification of symbol G (cf. (5.2)) the potential function gives a solution to the KdV equation. 2. Preliminaries. Wiener–Hopf factorization and Riemann–Hilbert problems Throughout this paper $$\mathbb {T}$$ will denote the unit circle in the complex plane and $$\mathbb {D}$$ the open unit disc. If 0 < α ≤ 1, a function $$f:\mathbb {T}\rightarrow \mathbb {C}$$ is said to be Hölder continuous of index α if and only if, for some $$M\in \mathbb {R}^{+}$$,  $$ |\,f(x)-f(y)|<M|x-y|^{\alpha} $$ (2.1) for every $$x,y\in \mathbb {T}$$. $$C_{\alpha }(\mathbb {T})$$ shall denote the space of Hölder functions of index α in $$\mathbb {T}$$. This space admits the direct-sum decomposition  $$ C_{\alpha}(\mathbb{T})=C_{\alpha}^{+}(\mathbb{T})\oplus C_{\alpha}^{0-}(\mathbb{T}) $$ (2.2) where $$C_{\alpha }^{+}(\mathbb {T})$$ is the subspace of analytic functions in the unit disc $$\mathbb {D}$$ and $$C_{\alpha }^{0-}(\mathbb {T})$$ is the subspace of analytic functions in $$\mathbb {C}\setminus \overline {\mathbb {D}}$$ that vanish at infinity. Definition 2.1 A non-singular matrix-valued function $$G\in \left [C_{\alpha }(\mathbb {T})\right ]^{2\times 2}$$ is said to have a canonical right Wiener–Hopf factorization if and only if it admits a representation of the form  $$ G=G^{-}G^{+} $$ (2.3) with $$\left (G^{+}\right )^{\pm 1}\in C_{\alpha }^{+}(\mathbb {T})$$ and $$\left (G^{-}\right )^{\pm 1}\in C_{\alpha }^{-}(\mathbb {T})=C^{0-}_{\alpha }(\mathbb {T})\oplus \mathbb {C}$$. The Toeplitz operator $$T_{G}=P^{+}G\left |_{\operatorname {Im} P^{+}}\right .$$, where P± are the complementary projections associated to decomposition (2.2) and ImP+ is the image of P+, is invertible if and only if G has a canonical right Wiener–Hopf factorization. If G admits a canonical right factorization then $$G\left (G^{+}\right )^{-1}=G^{-}$$ and the factors $$G^{\pm }\in C_{\alpha }^{\pm }(\mathbb {T})$$ can be determined, cf. Clancey & Gohberg (1981) and Böttcher & Silbermann (1990), by solving the vector Riemann–Hilbert problem in $$C_{\alpha }(\mathbb {T})$$  $$ G\phi^{+}=\phi^{-}, $$ (2.4) whose solutions $$\phi _{j}^{\pm }\in C_{\alpha }^{\pm }(\mathbb {T}), j=1,2$$, with appropriate normalizing conditions, give the columns of the factors. 3. The factorization method This paper presents a method for finding a factorization for the class of non-singular symbols $$G\in \left [C_{\alpha }(\mathbb {T})\right ]_{2\times 2}$$ defined by  $$ G=\left[\begin{array}{cc}1 & h^{+}g_{1} \\ (h^{+})^{-1}g_{2} & 1\end{array}\right] $$ (3.1) where h+ is an outer function such that $$(h^+)^{\pm 1}\in C_\alpha ^+(\mathbb {T})$$ and $$g_j\in L_\infty (\mathbb {T}), j=1,2$$ are such that  $$ g_{1}(\xi)=q\left(\rho^{-}(\xi)\right)\,\,\,\textrm{and}\,\,\,g_{2}(\xi)=q^{-1}\left(-\rho^{-}(\xi)\right), \,\,\,\xi\in \mathbb{T}, $$ (3.2) with q a rational function such that $$g_1,g_2\in L_\infty (\mathbb {T})$$ and $$\rho ^-\in L_\infty ^-(\mathbb {T})$$ defined by $$\rho ^-(\xi )=\left (\frac {\xi -a}{\xi +a}\right )^{\frac {1}{2}}\!\!, a\in \mathbb {D}$$, such that $$\operatorname {Re}\left [\rho ^-(\xi )\right ]\geqslant 0$$. The factorization method here presented is based on the idea of reducing a vector Riemann–Hilbert problem on the unit circle to a scalar problem on a contour in the Riemann surface defined by the algebraic relation $$w^2=\frac {\xi -a}{\xi +a}$$, in this case the Riemann sphere (cf. dos Santos & dos Santos, 2014). The images of the unit circle through the transformations $$\xi \rightarrow \pm \rho ^-(\xi )$$ can be seen as contours in this Riemann surface and are represented by $$\Gamma _j=(-1)^{j+1}\rho ^-(\mathbb {T})$$, j = 1, 2. Using the same identification process one can define Γ = Γ1 ∪Γ2, Ω+ and $$\Omega ^-=\Omega _1^-\cup \Omega _2^-$$ where $$\Omega ^+=\bigcup _{j=1}^2(-1)^{j+1}\rho ^-(\mathbb {C}\setminus \mathbb {D})$$ and $$\Omega _j^-=(-1)^{j+1}\rho ^-(\mathbb {D})$$, j = 1, 2. Before presenting the main result we start by establishing an equivalence between the existence of solutions for the vector Riemann–Hilbert problem (2.4) and a scalar Riemann–Hilbert type problem on Γ. Since Ω− is the union of the two connected components $$\Omega _1^-$$ and $$\Omega _2^-$$, let X− be the set of all functions in Cα−(Γ) that vanish at the points 1 and −1. It follows that $$C_\alpha (\Gamma )=C^+_\alpha (\Gamma )\oplus X^-$$ and that the following holds: Proposition 1 Let G be as defined in (3.1), (3.2). Then the system of equations  $$ \left\{\begin{array}{l}\phi_{1}^{+}+g_{1}h^{+}\phi_{2}^{+}=\phi_{1}^{-}+f_{1}^{+}\\[3mm]g_{2}(h^{+})^{-1}\phi_{1}^{+}+\phi_{2}^{+}=\phi_{2}^{-}+f_{2}^{+}\end{array}\right. $$ (3.3) has a unique solution $$(\phi _1^+,\phi _2^+)\in \left [C^+_\alpha (\mathbb {T})\right ]^2$$, $$(\phi _1^-,\phi _2^-)\in \left [C^{0-}_\alpha (\mathbb {T})\right ]^2$$ for every $$(f_1^+,f_2^+)\in \left [C^+_\alpha (\mathbb {T})\right ]^2$$ if and only if equation  $$ \psi_{1}^{+}+q\widetilde{h}^{+}\psi_{2}^{+}=H(\psi^{-}+f^{+}) $$ (3.4) has a unique solution $$(\psi _1^+,\psi _2^+)\in \left [C_\alpha ^+(\Gamma )\right ]^2$$, ψ−∈ X− with $$\psi _j^+, j=1,2$$, even functions, for every $$f^+\in C^+_\alpha (\Gamma )$$, where  $$ \widetilde{h}^{+}(w)=h^{+}\left(a\frac{1+w^{2}}{1-w^{2}}\right)\quad\textrm{and}\quad H(w)=\left\{\begin{array}{ll} 1 & \textrm{if }w\in\Gamma_{1}\\[5mm]q(w)\widetilde{h}^{+}(w) & \textrm{if }w\in\Gamma_{2}\end{array}\right.. $$ (3.5) Fig. 1. View largeDownload slide Composed contour Γ. Fig. 1. View largeDownload slide Composed contour Γ. Proof. Suppose the system of equations (3.3) has a solution $$(\phi _1^+,\phi _2^+)\in \left [C^+_\alpha (\mathbb {T})\right ]^2$$, $$(\phi _1^-,\phi _2^-)\in \left [C^{0-}_\alpha (\mathbb {T})\right ]^2$$. Let w = ρ−(ξ). Through the transformation $$\xi \rightarrow w$$ the complex plane is transformed onto the right half-plane and $$\mathbb {C}\setminus \overline {\mathbb {D}}$$ is transformed onto the bounded region $$\Omega _1^-$$ whose boundary is the closed line Γ1 (cf. Fig. 1). Writing ξ in terms of w yields  $$ \xi=a\frac{1+w^{2}}{1-w^{2}} $$ (3.6) and defining functions $$\psi _j^\pm (w)=\phi ^\pm _j(a\frac {1+w^2}{1-w^2})$$, j = 1, 2, which are plus/minus functions with respect to the contour Γ1, taking in account its orientation, the first equation in (3.3) can be written as  $$ \psi_{1}^{+}(w)+q(w)\widetilde{h}^{+}(w)\psi_{2}^{+}(w)=\psi_{1}^{-}(w)+\widetilde{f}_{1}^{+}(w), w\in\Gamma_{1}\,, $$ (3.7) where $$\widetilde {h}$$ is defined in (3.5) and $$\widetilde {f}_1^+(w)=f^+_1(a\frac {1+w^2}{1-w^2})$$. On the other hand, putting w = −ρ−(ξ) the complex plane is transformed onto the left half-plane and $$\mathbb {C}\setminus \overline {\mathbb {D}}$$ is transformed onto the bounded region $$\Omega _2^-$$ whose boundary is the closed line Γ2 (again, cf. Fig. 1). Considering the above functions ψi± and $$\widetilde {h}^+$$ extended to the composed contour, and keeping in mind that they are even functions, the second equation in (3.3) can be written, using the mapping onto the left half-plane, as  $$ \psi_{1}^{+}(w)+q(w)\widetilde{h}^{+}(w)\psi_{2}^{+}(w)=q(w)\widetilde{h}^{+}(w)\left[\psi_{2}^{-}(w)+\widetilde{f}_{2}^{+}(w)\right], w\in\Gamma_{2} $$ (3.8) where $$\widetilde {f}_2^+(w)=f^+_2(a\frac {1+w^2}{1-w^2})$$. Defining H as in (3.5),  $$ \psi^{-}(w)=\left\{\begin{array}{ll} \phi_{1}^{-}\left(a\frac{1+w^{2}}{1-w^{2}}\right) & \textrm{if }w\in\Gamma_{1}\\[5mm]\phi_{2}^{-}\left(a\frac{1+w^{2}}{1-w^{2}}\right) & \textrm{if }w\in\Gamma_{2}\end{array}\right. \textrm{and}\ \ \ f^{+}(w)=\left\{\begin{array}{ll} \widetilde{f}_{1}^{+}(w) & \textrm{if }w\in\Gamma_{1}\\[5mm]\widetilde{f}_{2}^{+}(w) & \textrm{if }w\in\Gamma_{2}\end{array}\right. $$ (3.9) one gets (3.4). Since $$\psi _1^+,\psi _2^+$$ are even functions analytic in both left and right half-planes, except $$\Omega ^-_1$$ and $$\Omega _2^-$$ respectively, $$\psi _1^+,\psi _2^+\in C_\alpha ^+(\Gamma _1)\cap C_\alpha ^+(\Gamma _2)$$ thus $$\psi _1^+,\psi _2^+\in C_\alpha ^+(\Gamma )$$ and are even functions. Also it is easy to check that ψ−∈ Cα−(Γ), by computing the corresponding projection PΓ+ψ−, and that ψ−(1) = ψ−(−1) = 0. It follows that $$(\psi _1^+,\psi _2^+)\in \left [C_\alpha ^+(\Gamma )\right ]^2, \psi ^-\in X^-$$ is a solution of (3.4). Conversely, if $$(\psi _1^+,\psi _2^+)\in \left [C_\alpha ^+(\Gamma )\right ]^2, \psi ^-\in X^-$$ is a solution of (3.4), then, since $$\psi _1^+, \psi _2^+$$ are even, $$\phi ^+_j(\xi )=\psi _j^+(\rho ^-(\xi ))\in C_\alpha ^+(\mathbb {T})$$ and $$\phi _j^-(\xi )=\psi ^-\left |_{\Gamma _j}\right .((-1)^{j+1}\rho ^-(\xi ))\in C_\alpha ^{0-}(\mathbb {T})$$, j = 1, 2 satisfy (3.3). Proving the uniqueness is equivalent to showing that the homogeneous system (3.3) has only the trivial solution if and only if the same holds for equation (3.4). Let $$f_1^+,f_2^+=0$$. Then, from (3.9), f = 0 and the condition of unique solvability of (3.4) given in the statement of the proposition now implies that (3.4) has only the trivial solution, as required. The argument can be reversed to show that f = 0 in (3.4) implies that the system (3.3) has only the trivial solution. □ Having established an equivalence between Riemann–Hilbert problem (3.3) on the contour $$\mathbb {T}$$ and scalar Riemann–Hilbert type problem (3.4) on contour Γ it is time to describe the factorization method, starting by establishing sufficient conditions for the homogeneous scalar problem in Cα±(Γ) corresponding to (3.4) to have a unique solution and find explicit solutions for that problem. For every rational function q let $$Q_q(w)=\prod _{i=1}^N(w-z_i)^{m_i}$$ where N is the number of poles of q in $$\Omega ^+\setminus \{\infty \}$$, zi are those poles, mi their multiplicity and let Sq(w) = q(w)Qq(w). Then  $$ q(w)=\frac{S_{q}(w)}{Q_{q}(w)}, w\in\Gamma, $$ (3.10) Sq is a rational function with no poles in $$\Omega ^+\setminus \{\infty \}$$ and Qq is a polynomial whose zeros lie in Ω+ and are not zeros of Sq(Lang, 1999). For every such function define functions $$\widetilde {q}$$, $$\widetilde {S}_q$$ and $$\widetilde {Q}_q$$ by  $$ \widetilde{q}(w)=q(-w),\quad \widetilde{S}_{q}(w)=S_{q}(-w),\quad \widetilde{Q}_{q}(w)=Q_{q}(-w). $$ (3.11) Furthermore, for every F1, F2 ∈ Cα(Γ), let $$T_{F_1F_2}$$ be the function defined by  $$ T_{F_{1},F_{2}}=F_{1}\widetilde{F}_{2}-\widetilde{F}_{1}F_{2} $$ (3.12) where $$\widetilde {F}_j(w)=F_j(-w), j=1,2$$, let Dq be the rational function defined by  $$ D_{q}=T_{Q_{q},S_{q}}{,} $$ (3.13) nZ be the number of zeros of Dq in $$\Omega ^+\setminus \{\infty \}$$, wk, k = 1...nZ be those zeros, nk their multiplicity, $$n_D=\sum _{k=1}^{n_Z}n_k$$ and let $${\mathscr R}_q$$ be the set of all rational functions R of the form  $$ R(w)=\frac{P_{R}(w)}{Q_{q}(w)} $$ (3.14) where Qq is defined in (3.10) and PR is any polynomial of degree nP equal to the degree of Qq plus m0, $$n_P=m_0+\sum _{k=1}^{N}m_k$$, with m0 the order of the pole at infinity as a pole of q. Let, now, for every $$R\in {\mathscr R}_q$$  $$ \begin{array}{r@{\ }c@{\ }l} A_{R,k,j}&=&\displaystyle\frac{\partial^{j} T_{P_{R}H^{+},Q_{q}}}{\partial w^{j}}(w_{k}), 1\leqslant k\leqslant n_{Z}, 0\leqslant j\leqslant n_{k}-1\\[4mm] B_{R,k,j}&=&\displaystyle\frac{\partial^{j} T_{P_{R}H^{+},S_{q}}}{\partial w^{j}}(w_{k}), 1\leqslant k\leqslant n_{Z}, 0\leqslant j\leqslant n_{k}-1\\[4mm] C_{R,1}&=&\displaystyle\lim_{w\rightarrow\infty}\displaystyle\frac{T_{P_{R}H^{+},Q_{q}}(w)}{w^{n_{D}}}\\[4mm] C_{R,2}&=&\displaystyle\lim_{w\rightarrow\infty}\displaystyle\frac{T_{P_{R}H^{+},S_{q}}(w)}{w^{n_{D}}} \end{array}. $$ (3.15) Note that there is a bijection between the set $${\mathscr R}_q$$ and $$\mathbb {R}^{n_P}$$ since the nP coefficients of PR(w) determine uniquely $$R\in {\mathscr R}_q$$. Note, also, that, for a given q, AR, k, j, BR, k, j, CR, 1 and CR, 2 depend only on the coefficients of the polynomial PR(w). Proposition 2 Let $$\widetilde {h}^+$$ be an even function such that $$(\widetilde {h}^+)^{\pm 1}\in C_\alpha ^+(\Gamma )$$, q be a non-even rational function with no poles on Γ, represented in the form (3.10), such that Dq defined by (3.13) has no zeros on Γ and H be the scalar function defined by  $$ H(w)=\left\{\begin{array}{ll} 1 & w\in\Gamma_{1}\\[3mm]q(w)\widetilde{h}^{+}(w) & w\in\Gamma_{2}\end{array}\right..\nonumber $$ If H has a canonical right Wiener–Hopf factorization in Cα(Γ), H = H−H+, and there exists a unique rational function $$R\in { \mathscr R}_q$$, represented in the form (3.14), such that CR, 1, CR, 2 as defined in (3.15) are finite and the algebraic system in the nP coefficients of PR(w),  $$ \left\{\begin{array}{l}P_{R}(1)=c_{1}H^{-}(1)Q_{q}(1)\\[3mm]P_{R}(-1)=c_{2}H^{-}(-1)Q_{q}(-1)\\[3mm]A_{R,k,j}=0, \quad1\leqslant k\leqslant n_{Z}, 0\leqslant j\leqslant n_{k}-1\\[3mm] B_{R,k,j}=0, \quad1\leqslant k\leqslant n_{Z}, 0\leqslant j\leqslant n_{k}-1 \end{array}\right. $$ (3.16) where c1, c2 are arbitrary complex numbers and AR, k, j, BR, k, j are defined in (3.15), is uniquely solvable, then equation  $$ \psi_{1}^{+}+q\widetilde{h}^{+}\psi_{2}^{+}=H\psi^{-} $$ (3.17) with normalizing conditions ψ−((−1)j+1) = cj, j = 1, 2 has a unique solution $$\psi ^+_1,\psi ^+_2\in C^+_\alpha (\Gamma )$$, $$\psi ^-\in C^-_\alpha (\Gamma )$$ such that $$\psi ^+_1,\psi ^+_2$$ are even. That solution is given by  $$ \left\{\begin{array}{l} \psi_{1}^{+}=D_{q}^{-1}T_{P_{R}H^{+},S_{q}}\\[3mm] \psi_{2}^{+}=-(\widetilde{h}^{+})^{-1}D_{q}^{-1}T_{P_{R}H^{+},Q_{q}} \end{array}\right.. $$ (3.18) Proof. Since H = H−H+ with $$\left (H^+\right )^{\pm 1}\in C_\alpha ^+(\Gamma )$$ and $$\left (H^-\right )^{\pm 1}\in C_\alpha ^-(\Gamma )$$ then, by (3.17), $$\left (H^+\right )^{-1}\left (\psi _1^++q\widetilde {h}^+\psi _2^+\right )=H^-\psi ^-\in C_\alpha ^-(\Gamma )$$. On the other hand, since the only singularities of that function in Ω+ are the poles of q, it is a meromorphic function in $$\mathbb {C}$$ whose poles are in Ω+ and coincide with the poles of q in Ω+. Then there exists $$R\in {\mathscr R}_q$$ such that  $$ \left(H^{+}\right)^{-1}\left(\psi_{1}^{+}+q\widetilde{h}^{+}\psi_{2}^{+}\right)=H^{-}\psi^{-}=R. $$ (3.19) It suffices now to write the same equation in −w, keeping in mind that $$\widetilde {h}^+$$, $$\psi _1^+$$ and $$\psi _2^+$$ are even, by hypothesis, to write the system  $$ \left\{\begin{array}{l}\psi_{1}^{+}+q\,\widetilde{h}^{+}\psi_{2}^{+}=RH^{+}\\[3mm]\psi_{1}^{+}+\widetilde{q}\,\widetilde{h}^{+}\psi_{2}^{+}=\widetilde{R}\widetilde{H}^{+}\end{array}\right., $$ (3.20) defining $$\widetilde {R}(w)=R(-w)$$, $$\widetilde {q}(w)=q(-w)$$ and $$\widetilde {H}^+=H^+(-w)$$. Since Dq has no zeros on Γ, which means that $$q-\widetilde {q}$$ has no zeros on Γ, this system has a unique algebraic solution given by  $$ \left\{\begin{array}{l} \psi_{1}^{+}=\displaystyle\frac{\widetilde{q}\,R\,H^{+}-q\,\widetilde{R}\,\widetilde{H}^{+}}{\widetilde{q}-q}\\[3mm] \psi_{2}^{+}=(\widetilde{h}^{+})^{-1}\displaystyle\frac{R\,H^{+}-\widetilde{R}\,\widetilde{H}^{+}}{q-\widetilde{q}} \end{array}\right. $$ (3.21) which using (3.10), (3.11), (3.14) and (3.12) yields (3.18). In order for $$\psi _1^+$$ and $$\psi _2^+$$ to be solutions of Riemann–Hilbert problem (3.19), $$R\in {\mathscr R}_q$$ must be given by (3.14) with the coefficients of PR determined by the system of equations (3.16) where the first two equations of (3.16) correspond to the normalizing conditions ψ−((−1)j+1) = cj, j = 1, 2, which, in turn, correspond to the normalizing conditions at infinity for $$\phi _1^-$$ and $$\phi _2^-$$ in the initial homogeneous Riemann–Hilbert problem on $$\mathbb {T}$$, obtained by (3.3), and the remaining conditions are the natural conditions to ensure the cancellation of all singularities in Ω+ and, thus, the analiticity of $$\psi _1^+$$ and $$\psi _2^+$$ in Ω+. It remains to prove that these are even functions which can be easily checked computing $$\psi ^+_j(-w), j=1,2$$ and that they are unique which is a consequence of the uniqueness of the solution of that same system of equations and of the fact that any solution must satisfy (3.16). □ Remark 3 The canonical right Wiener–Hopf factorization of the non-singular scalar function H on the composed contour Γ can be obtained, if it exists, by  $$ \begin{array}{l} H^{+}(w)=\exp\left(P_{\Gamma}^{+}\log H\right)(w)\\[3mm] H^{-}(w)=\exp\left(P_{\Gamma}^{-}\log H\right)(w) \end{array} $$ (3.22) where PΓ± are complementary projections defined by  $$ \left(P_{\Gamma}^{\pm} \log H\right)(w)=\mp\frac{1}{2\pi\operatorname{i}}\int_{\Gamma}\frac{\log H(\xi)}{\xi-w}\;\textrm{d}\xi,\,\,w\in\Omega^{\pm} .$$ (3.23) A result similar to the previous proposition is easily obtained for the initial Riemann–Hilbert problem (3.3). Making use of the previous results it is now possible to state the main result in this section, which gives a method to get the factors of matrix G. Theorem 4 Let G be as defined in (3.1), (3.2), $$\widetilde {h}^+(w)=h^+(a\frac {1+w^2}{1-w^2})$$, Dq defined by (3.10), (3.13), Δ(ξ) = Dq(ρ−(ξ)) and  $$ H(w)=\left\{\begin{array}{ll} 1 & w\in\Gamma_{1}\\[3mm]q(w)\widetilde{h}^{+}(w) & w\in\Gamma_{2}\end{array}\right. .$$ (3.24) If H has a canonical Wiener–Hopf factorization H = H−H+, Δ has no zeros on $$\mathbb {T}$$ and, for every $$({c_1,c_2})\in \mathbb {C}^2$$, (3.16) is uniquely solvable then G has a canonical Wiener–Hopf factorization and the corresponding factors are given by $$(G^\pm )^{\mp 1}=\left [\phi _{jk}^\pm \right ]_{j,k=1,2}$$, where  $$ \begin{array}{l} \phi_{1,k}^{+}(\xi)=[\Delta(\xi)]^{-1}\left[T_{P_{R_{k}}H^{+},S_{q}}(\rho^{-}(\xi))\right]\\[3mm]\phi_{2,k}^{+}(\xi)=\left[h^{+}(\xi)\Delta(\xi)\right]^{-1}\left[T_{P_{R_{k}}H^{+},Q_{q}}(\rho^{-}(\xi))\right] \\[3mm] \phi_{1,k}^{-}(\xi)=R_{k}(\rho^{-}(\xi))H^{-}(\rho^{-}(\xi)) \\[3mm] \phi_{2,k}^{-}(\xi)=R_{k}(-\rho^{-}(\xi))H^{-}(-\rho^{-}(\xi)) \end{array} $$ (3.25) with $$R_k\in {\mathscr R}_q, k=1,2$$ are the unique solutions of (3.16) for cj = |k − j|, j = 1, 2, $$T_{P_{R_k}H^+,S_q}$$ and $$T_{P_{R_k}H^+,Q_q}$$ defined in (3.12). Proof. Under the assumptions of the theorem, by Proposition 2, it is possible to obtain a unique solution to the problem on Γ, for every set of normalizing conditions $$\psi ^-_j((-1)^{j+1})={c}_j, j=1,2$$ and thus, by Proposition 1, the initial Riemann–Hilbert problem (2.4) has a unique solution in $$C_\alpha ^\pm (\mathbb {T})$$ for every set of normalizing conditions $$\phi ^-_j(\infty )={c_j}, j=1,2$$ and it is possible to obtain the corresponding solutions on $$\mathbb {T}$$, keeping in mind that the solutions obtained through Proposition 2 are even. The fact that these solutions are even functions in $$C^+_\alpha (\Gamma )$$ enough to ensure that the solutions obtained for the initial problem (3.3) are in $$C^+_\alpha (\mathbb {T})$$, even after composing with ρ−, since this means that they are functions of w2. Choosing c1 = 0 and c2 = 1 for k = 1 and c1 = 1 and c2 = 0 for k = 2 one obtains the two independent solutions (3.25). □ Remark 5 Note that a similar result can be easily achieved assuming that instead of q being a rational function, q ∈ Cα+(Γ). Also, similar results can be obtained considering (2.4) on the real axis instead of $$\mathbb {T}$$. Theorem 4 is the basis of the factorization method for matrix symbol (3.1) presented in this paper. It gives sufficient conditions for the existence of canonical factorization of G and, under these conditions, presents formulas for the factors  $$ (G^{\pm})^{\mp1}=\left[\phi_{jk}^{\pm}\right]_{j,k=1,2} $$ (3.26) where functions ϕjk± are defined in (3.25). Given functions g1, g2 and h+ such that G is in the class studied in this paper, the algebraic system (3.16) can be written and if uniquely solvable, functions Rk can be explicitly written and thus formulas (3.25) explicitly give the factors of G. 4. Application to a factorization problem We now apply the theory presented in the previous section to a particular factorization problem. This particular problem is related to the integrable system considered in the next section. Since a particular choice of g1, g2 and h+ is given, which falls under the class to each the theory in the previous section applies, Theorem 4 allows for explicit formulas for the factors to be written. Let us consider the case where  $$ G=\left[\begin{array}{cc}1 & h^{+} g_{1} \\ (h^{+})^{-1}g_{2} & 1\end{array}\right] $$ (4.1) with $$g_{1}(\xi )=\operatorname {i} b^{-1}\displaystyle \frac {\rho ^{-}(\xi )-\operatorname {i} b}{\rho ^{-}(\xi )-\operatorname {i} b^{-1}}\rho ^{-}(\xi )$$, $$g_{2}(\xi )=\operatorname {i} b\displaystyle \frac {\rho ^{-}(\xi )+\operatorname {i} b^{-1}}{(\rho ^{-}(\xi )+\operatorname {i} b)\rho ^{-}(\xi )}$$, $$h^{+}(\xi )=\exp (2x\xi )$$ and $$\rho ^{-}(\xi )=\left (\frac {\xi -a}{\xi +a}\right )^{\frac {1}{2}}$$ with $$a,b,x\in \mathbb {C}$$ such that b = ρ−(b0) and $$b_{0}\in \mathbb {D}$$. These choices correspond to putting  $$ q(w)=\frac{\operatorname{i} w+b}{-\operatorname{i} w^{-1}+b},\quad w\in\Gamma,\quad b\in\mathbb{C}\setminus\{0\} .$$ (4.2) To obtain a factorization using the results of Section 3, in particular Theorem 4, we note that Dq, as defined in (3.13), depends on the location of the zeros and poles of q, the same happening to the factorization of H. Since b = ρ−(b0) with $$b_{0}\in \mathbb {D}$$, b ∈Ω+ and, making use of the transformation ξ↦ρ−(ξ) and its inverse, it is easy to check that ib−1 also lies in Ω+ thus q is a minus factor. Then,  $$ n_{0}=1,\quad Q_{q}(w)=w-\operatorname{i} b^{-1},\quad S_{q}(w)=\operatorname{i} b^{-1}w(w-\operatorname{i} b) $$ (4.3) with n0, Qq and Sq as defined in (3.10) and, since  $$ \widetilde{h}^{+}(w)=\exp\left(2ax\frac{1+w^{2}}{1-w^{2}}\right)=\exp(-2ax)\exp\left(\frac{-2ax}{w-1}\right)\exp\left(\frac{2ax}{w+1}\right), $$ (4.4) H can be canonically factorized, using (3.22), as  $$ H^{+}(w)=\exp\left(\frac{2ax}{w+1}\right) \,\,\, \textrm{and} \,\,\, H^{-}(w)=\left\{\begin{array}{ll}\exp\left(-\frac{2ax}{w+1}\right) & w\in\Gamma_{1}\\[5mm]q(w)\exp(-2ax)\exp\left(\frac{-2ax}{w-1}\right) & w\in\Gamma_{2} \end{array}\right.. $$ (4.5) It is a matter of a simple calculation to conclude that, by (3.11), (3.13),  $$ D_{q}(w)=2\operatorname{i} b^{-1}w(w^{2}+1) $$ (4.6) and, thus, Dq has no zeros on Γ, Δ has no zeros on $$\mathbb {T}$$ and there are three first order zeros of Dq in Ω+: w1 = 0, w2 = i and w3 = −i. Since q has two poles in Ω+, ib−1 and $$\infty $$, $${\mathscr R}_{q}$$, as defined in (3.14), is the set  $$ {\mathscr R}_{q}=\left\{\frac{\gamma_{0}+\gamma_{1}w+\gamma_{2}w^{2}}{w-\operatorname{i} b^{-1}}:\gamma_{0},\gamma_{1},\gamma_{2}\in\mathbb{C}\right\} .$$ (4.7) Then the third and fourth conditions on (3.16) are always satisfied for w1 = 0 and result in equivalent equations for w2 = i and w3 = −i, while the last condition on (3.16) is always fulfilled. Letting 2Aj = H−((−1)j+1)Qq((−1)j+1), j = 1, 2, (3.16) can be rewritten in the form  $$ \left\{\begin{array}{l}\gamma_{1}=A_{1}{c}_{1}-A_{2}{c}_{2}\\\gamma_{2}+\gamma_{0}=A_{1}{c}_{1}+A_{2}{c}_{2}\\ (\gamma_{2}-\gamma_{0})(b+1+(b-1)\exp(\operatorname{i} 2ax))=(A_{1}{c}_{1}-A_{2}{c}_{2})\operatorname{i}((b-1)\exp(\operatorname{i} 2ax)-b-1)\end{array}\right. $$ (4.8) Thus, if $$b+1+(b-1)\exp (\operatorname {i} 2ax)\neq 0$$ the system is uniquely solvable and there is a canonical factorization of G. On the other hand if that condition is not fulfilled then the system has more than one solution if b = −1 and no solutions if b ≠ −1 and in either case there is no canonical factorization for G. The previous results lead to the following: Proposition 6 Let $$b\in \mathbb {C}\setminus \{0\}$$ be such that ib ∈Ω+, G be as defined in (3.1) with  $$ g_{1}(\xi)=g_{2}(-\xi)=\frac{\operatorname{i} \rho^{-}(\xi)+b}{-\operatorname{i} (\rho^{-}(\xi))^{-1}+b} $$ (4.9) and $$\rho ^{-}(\xi )=\left (\frac {\xi -a}{\xi +a}\right )^{\frac {1}{2}}$$, $$a\in \mathbb {D}$$. Then if x is such that  $$ (1-b)\exp(\operatorname{i} 2ax)\neq b+1 $$ (4.10) G has a canonical factorization given by  $$ G^{+}=\left[\begin{array}{cc} \phi_{11}^{+} & \phi_{12}^{+}\\[3mm] \phi_{21}^{+} & \phi_{22}^{+}\end{array}\right]^{-1}\quad\quad G^{-}=\left[\begin{array}{cc} \phi_{11}^{-} & \phi_{12}^{-}\\[3mm]\phi_{21}^{-} & \phi_{22}^{-}\end{array}\right] $$ (4.11) where  $$ \begin{array}{l} \phi_{11}^{+}(\xi)=\frac{1}{4}(1+a\xi^{-1})\left[(\rho^{-}(\xi)+\operatorname{i} b)\varphi_{1}(\xi)-(\rho^{-}(\xi)-\operatorname{i} b)\widetilde{\varphi}_{1}(\xi)\right] \\[5mm] \phi_{12}^{+}(\xi)=\frac{1}{4}(1+a\xi^{-1})\left[(\rho^{-}(\xi)+\operatorname{i} b)\varphi_{2}(\xi)-(\rho^{-}(\xi)-\operatorname{i} b)\widetilde{\varphi}_{2}(\xi)\right]\\[5mm] \phi_{21}^{+}(\xi)=-\frac{\operatorname{i}}{4}b(1+a\xi^{-1})\left[\rho^{-}(\xi)\right]^{-1}\left[(\rho^{-}(\xi)+\operatorname{i} b^{-1})\varphi_{1}(\xi)+(\rho^{-}(\xi)-\operatorname{i} b^{-1})\widetilde{\varphi}_{1}(\xi)\right]\exp(-2\xi x)\\[5mm] \phi_{22}^{+}(\xi)=-\frac{\operatorname{i}}{4}b(1+a\xi^{-1})\left[\rho^{-}(\xi)\right]^{-1}\left[(\rho^{-}(\xi)+\operatorname{i} b^{-1})\varphi_{2}(\xi)+(\rho^{-}(\xi)-\operatorname{i} b^{-1})\widetilde{\varphi}_{2}(\xi)\right]\exp(-2\xi x)\\[5mm] \phi_{11}^{-}(\xi)=\displaystyle\frac{\varphi_{1}(\xi)}{\rho^{-}(\xi)-\operatorname{i} b^{-1}}\\[5mm] \phi_{12}^{-}(\xi)=\displaystyle\frac{\varphi_{2}(\xi)}{\rho^{-}(\xi)-\operatorname{i} b^{-1}}\\[5mm] \phi_{21}^{-}(\xi)=\displaystyle\frac{\widetilde{\varphi}_{1}(\xi)\exp(-2x\xi)}{\operatorname{i} b^{-1}(\rho^{-}(\xi)+\operatorname{i} b)\rho^{-}(\xi)}\\[5mm] \phi_{22}^{-}(\xi)=\displaystyle\frac{\widetilde{\varphi}_{2}(\xi)\exp(-2x\xi)}{\operatorname{i} b^{-1}(\rho^{-}(\xi)+\operatorname{i} b)\rho^{-}(\xi)} \end{array} $$ with  $$ \begin{array}{l} \varphi_{i}(\xi)=\left(\gamma_{0i}+\gamma_{1i}\rho^{-}(\xi)+\gamma_{2i}(\rho^{-}(\xi))^{2}\right)\exp\frac{2ax}{\rho^{-}(\xi)+1}),\\[5mm] \widetilde{\varphi}_{i}(\xi)=\left(\gamma_{0i}-\gamma_{1i}\rho^{-}(\xi)+\gamma_{2i}(\rho^{-}(\xi))^{2}\right)\exp\frac{2ax}{-\rho^{-}(\xi)+1}), \end{array} $$   $$ \begin{array}{lll} \gamma_{01}=\frac{1}{2}A_{1}(1+A_{3}), & \gamma_{11}=A_{1}, & \gamma_{21}=\frac{1}{2}A_{1}(1-A_{3}))\\[5mm]\gamma_{02}=\frac{1}{2}A_{2}(1-A_{3}), & \gamma_{12}=-A_{2}, & \gamma_{22}=\frac{1}{2}A_{2}(1+A_{3}) \end{array} $$ and  $$ \begin{array}{l@{,\quad }l@{,\quad }l} A_{1}=\frac{1}{2}\exp(-ax)(1-\operatorname{i} b^{-1}) & A_{2}=\frac{1}{2}\exp(-ax)(1+\operatorname{i} b^{-1}) & A_{3}=\operatorname{i}\frac{(b-1)\exp(\operatorname{i} 2ax)-(b+1)}{(b-1)\exp(\operatorname{i} 2ax)+(b+1)}. \end{array} $$ Proof. It is enough to use the results of the last theorem of the previous section. Note that the functions between square brackets in the expression given for $$\phi _{1k}^{+}$$ are even functions in ρ− that vanish at 0 and those in the expression of $$\phi _{2k}^{+}$$ are odd, so $$\phi _{1k}^{+}$$ are indeed plus functions. Also note that although $$\exp (2x\xi )$$ is not a minus function, when multiplied by $$\widetilde {\varphi }_{j}$$ it is indeed a minus function. □ 5. Application to an integrable systems problem In this section we consider an application of the theory of Section 3 and results of Section 4 to a problem in the field of integrable systems of infinite dimension, specifically to a problem related to the KdV equation. As shown in dos Santos & dos Santos (2016) families of solutions for the KdV equation and for the linear Schrödinger equation can be obtained by solving a Riemann–Hilbert problem with shift on the unit circle, $$\mathbb {T}$$, of the form  $$ E(\gamma_{a}\phi^{+}+\gamma_{b}\phi^{+}(-))+\phi^{-}=0, $$ (5.1) subject to $$\phi ^{-}(\infty )=1$$, where E is an outer function of the form given below, $$\gamma _{a}, \gamma _{b}:\mathbb {T}\to \mathbb {C}$$ are scalar functions assumed to be in $$C_{\alpha }(\mathbb {T})$$, 0 < α < 1. In (5.1), ϕ+(−)(ξ) = ϕ+(−ξ) and ϕ± belong to the subspaces of $$C_{\alpha }(\mathbb {T})$$ of functions analytic on the unit disc and on $$\mathbb {C}\setminus \overline {\mathbb {D}}$$, respectively. The above problem is equivalent to the standard vector Riemann–Hilbert problem  $$ G\phi^{+}=\phi^{-}, \phi^{\pm}\in\left[C_{\alpha}^{\pm}(\mathbb{T})\right]^{2} $$ (5.2) with symbol G of the form  $$ G=\left[\begin{array}{cc}\gamma_{a}&\gamma_{b}E^{-2}\\\gamma_{b}(-)E^{2} & \gamma_{a}(-)\end{array}\right] $$ (5.3) where $$E(\xi )=\exp (\xi x), x\in \mathbb {R}$$ for the Schrödinger equation and $$E(\xi )=\exp (\xi x-\xi ^{3}t), x,t\in \mathbb {R}$$ for the KdV equation. Function ϕ− in (5.1) gives the solution to the Schrödinger equation with potential function u given by   $$ u(x)=-2\frac{\;\textrm{d}\phi_{1}}{\;\textrm{d} x}(x) $$ (5.4) where ϕ1 is the coefficient of λ−1 in the Laurent expansion of ϕ− in $$\mathbb {C}\setminus \overline {\mathbb {D}}$$ (cf. dos Santos & dos Santos, 2016). For $$E(\xi )=\exp (\xi x-\xi ^{3} t)$$, the potential function u is a solution to the KdV equation. For simplicity, in the example below we take, in (3.1) $$h^{+}(\xi )=\exp (\xi x), x\in \mathbb {R}$$ and in (3.2)  $$ q(w)=\frac{\operatorname{i} w+b}{-\operatorname{i} w^{-1}+b}, b\in\mathbb{C}\setminus\{0\}, w\in\Gamma .$$ (5.5) These choices yield g2(ξ) = g1(−ξ), with g1, g2 defined by (3.2), which means that, for γb = q(ρ−), γb(−) = q−1(−ρ−), γa = 1 and $$E(\xi )=\exp (\xi x)$$, symbol G belongs to the class considered in Section 3 which is not included in the class studied in dos Santos & dos Santos (2014, 2016). Before proceeding with the explicit solution of Riemann–Hilbert problem (5.2) we note that all functions involved in the analysis depend on the parameter x. In order to find the solution to the Schrödinger equation one has to find the first coordinate of the $$C_{\alpha }^{-}(\mathbb {T})$$ solution of Riemann–Hilbert problem (5.2) for normalizing conditions $$\phi ^{-}_{1}(\infty )=\phi ^{-}_{2}(\infty )=1$$. Since the choices made for g1, g2 and h+ lead to the same Riemann–Hilbert problem dealt with in the previous section, one can use the same computations but with the new normalizing conditions. Then the solution $$\phi ^{-}_{1}$$ to the Riemann–Hilbert problem is given by  $$ \phi^{-}_{1}=\frac{\gamma_{2}(\rho^{-})^{2}+\gamma_{1}\rho^{-}+\gamma_{0}}{\rho^{-}-\operatorname{i} b^{-1}}\exp\left(\frac{2ax}{\rho^{-}+1}\right) $$ (5.6) with $$\gamma _{2}=\operatorname {i} C_{1}\exp (-ax)\left [1-\frac {2}{1+C_{2}\exp (\operatorname {i} 2ax)}\right ]=-\gamma _{0}$$, $$\gamma _{1}=2C_{1}\exp (-ax)$$, C1 = 1 − ib−1 and $$C_{2}=\frac {b-1}{b+1}$$. This leads to $$u(x)=\frac {-16a^{2}C_{2}\exp (2\operatorname {i} ax)}{(1+C_{2}\exp (2\operatorname {i} ax))^{2}}$$. Rearranging this expression and putting $$\exp (2\operatorname {i}\theta _{0})=\frac {b-1}{b+1}$$ one gets  $$ u(x)=\frac{-4a^{2}C_{2}\exp(-2\operatorname{i}\theta_{0})}{\cos^{2}(ax+\theta_{0})}\,. $$ (5.7) We note that the above expression corresponds to a soliton solution (see e.g. dos Santos & dos Santos, 2016, Section 3.1), for ai and θi real. Writing a = iα, θ = iφ we get  $$ u(x)=\frac{A}{\cosh^{2}(\alpha x+\varphi)} $$ (5.8) where $$A=-4a^{2}C_{2}\exp (-2\varphi )$$. Actually, expression (5.8) is the potential function for which the solution to the linear Schrödinger equation is given by (5.6). As noted above to obtain the solution for the KdV equation we have to replace the exponential in (5.3) by $$E(\xi )=\exp (\xi x-\xi ^{3}t)$$. Proposition 7 The Riemann–Hilbert problem with shift (5.3) with γa = 1 and γb = q(ρ−) yields, for the particular choice (5.5) of q, the soliton type potential (5.7) for which the solution to the linear Schrodinger equation is given by (5.6). An interesting question to be dealt with in a future paper is which subclasses of the class of symbols considered in Section 3 lead to interesting new solutions to either the KdV or the Schrödinger equations, in particular which subclasses lead to almost periodic solutions. References Bastos, M. A., Câmara, M. C. & dos Santos, A. F. ( 1995) Generalized factorization for Daniele–Khrapkov matrix functions—explicit formulas. J. Math. Anal. Appl. , 190, 295-- 328. Google Scholar CrossRef Search ADS   Böttcher, A. & Silbermann, B. ( 1990) Analysis of Toeplitz Operators . Berlin: Springer. Google Scholar CrossRef Search ADS   Câmara, M. C., dos Santos, A. F. & dos Santos, P. F. ( 2008) Matrix Riemann–Hilbert problems and factorization on Riemann surfaces. J. Funct. Anal. , 255, 228-- 254. Google Scholar CrossRef Search ADS   Castro, L. P. & Moura Santos, A. ( 2004) An operator approach for an oblique derivative boundary-transmission problem. Math. Meth Appl. Sci. , 27, 1469-- 1491. Google Scholar CrossRef Search ADS   Clancey, K. & Gohberg, I. ( 1981) Factorization of Matrix Functions and Singular Integral Operators . Basel: Birkhäuser. Google Scholar CrossRef Search ADS   dos Santos, A. F. & dos Santos, P. F. ( 2014) Factorization of a class of symbols with outer functions. J. Math. Anal. Appl. , 413, 185-- 194. Google Scholar CrossRef Search ADS   dos Santos, A. F. & dos Santos, P. F. ( 2016) Segal–Wilson approach to integrable systems and Riemann–Hilbert problems. J. Math. Anal. Appl. , 443, 797-- 816. Google Scholar CrossRef Search ADS   Kiyasov, S. N. ( 2012) Some cases of efficient factorization of second-order matrix functions. Russ. Mathematics , 56, 30-- 36. Google Scholar CrossRef Search ADS   Lang, S. ( 1999) Complex Analysis . New York: Springer. Google Scholar CrossRef Search ADS   Litvinchuk, G. S. & Spitkovsy, I. ( 1987) The Factorization of Matrix Valued Functions . Basel-Boston: Birkhäuser. Google Scholar CrossRef Search ADS   Mishuris, G. & Rogosin, S. ( 2016) Constructive methods for factorization of matrix functions. IMA J. Appl. Math. , 81, 365-- 391. Google Scholar CrossRef Search ADS   Rawlins, A. D. & Williams, W. E. ( 1981) Matrix Wiener–Hopf factorisation. Q.J. Mech. Appl. Math. , 34, 1-- 8. Google Scholar CrossRef Search ADS   Speck, F.O. ( 2017) A class of interface problems for the Helmholtz equation in $$\mathbb {R}^n$$. Math. Methods Appl. Sci. , 40, 391-- 402. Google Scholar CrossRef Search ADS   © The Author(s) 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

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IMA Journal of Applied MathematicsOxford University Press

Published: Feb 1, 2018

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