Z. Angew. Math. Phys. (2017) 68:103
2017 Springer International Publishing AG
published online August 19, 2017
Zeitschrift f¨ur angewandte
Mathematik und Physik ZAMP
Double-wave solutions to quasilinear hyperbolic systems of ﬁrst-order PDEs
C. Curr´o and N. Manganaro
Abstract. A reduction procedure for determining double-wave exact solutions to ﬁrst-order hyperbolic systems of PDEs is
proposed. The basic idea is to reduce the integration of the governing hyperbolic set of N partial diﬀerential equations to
that of a 2 × 2 reduced hyperbolic model along with a further diﬀerential constraint. Therefore, the method of diﬀerential
constraints is used in order to solve the auxiliary 2 × 2 system. An example of interest to viscoelasticity is presented.
Mathematics Subject Classiﬁcation. 35L40, 35N10.
Keywords. Hyperbolic systems, Double-wave solutions, Diﬀerential constraints.
1. Introduction and general remarks
Many nonlinear wave propagation phenomena are described by quasilinear systems of ﬁrst-order PDEs
of the form
= B(U) (1.1)
where x and t denote, respectively, space and time coordinates, U(x, t) ∈ R
is a column vector repre-
senting the ﬁeld variables, A and B are matrix coeﬃcients and a subscript means for partial derivatives
with respect to the indicated variable. We assume system (1.1) to be hyperbolic with respect to t, so that
det (A−λ I) = 0 (1.2)
admits N real roots λ
to which there corresponds a complete set of left and right eigenvectors l
I) = 0, (A−λ
= 0, (i =1, .., N). (1.3)
Moreover, the vector B represents a source-like term and it can model a number of dissipative or dispersive
mechanisms which are present in the nonlinear wave dynamics of interest.
The particular case N = 2 is of great interest since such systems naturally arise in several contexts and
play a prominent role in several ﬁelds of application of hyperbolic waves theory. A striking feature of these
models is that, under the assumption of strict hyperbolicity, they can be recast into a form which expresses
the evolution of a privileged set of ﬁeld variables, the Riemann variables, along the related characteristic
curves, and in the homogeneous case (B = 0) through the classical hodograph transformation, system
(1.1) reduces to a linear form . Nevertheless, the solution, expressed in terms of the Riemann functions,
is of very limited use in describing one-dimensional wave processes so that several ad hoc approaches have
been worked out for integrating the associated hodograph system by means of closed-form or parametric
solutions [2,3]. If equations (1.1) are not homogeneous, then alternative linearizing approaches have been
proposed through the use of direct methods [4,5] and group methods [6–10].
This work was supported by the Italian National Group of Mathematical Physics (GNFM–INdAM).