Studies in Applied Mathematics

Gerdjikov, V. S.; Yanovski, A. B.

doi: 10.1111/sapm.12065pmid: N/A

We consider the Caudrey–Beals–Coifman linear problem (CBC problem) and the theory of the Recursion Operators (Generating Operators) related to it in the presence of Zh‐reductions of Mikhailov type defined by Coxeter automorphism of the underlying algebra. We discuss mainly the spectral aspects of the theory of the Recursion Operators related to the expansions over the adjoint solutions of the CBC problem but pay attention also to the algebraic issues related to the recursion relations through which the hierarchies of the nonlinear evolution equations associated with the CBC auxiliary problem are obtained.

Bian, Dongfen; Guo, Boling; Ling, Liming

doi: 10.1111/sapm.12051pmid: N/A

The Landau–Lifshitz equation is analyzed via the inverse scattering method. First, we give the well‐posedness theory for Landau–Lifshitz equation with the frame of inverse scattering method. The generalized Darboux transformation is rigorous considered in the frame of inverse scattering transformation. Finally, we give the high‐order soliton solution formula of Landau–Lifshitz equation and vortex filament equation.

O'Malley, Robert E.; Kirkinis, Eleftherios

doi: 10.1111/sapm.12052pmid: N/A

This paper presents a straightforward procedure for using Renormalization Group methods to solve a significant variety of perturbation problems, including some that result from applying a nonlinear version of variation of parameters. A regular perturbation procedure typically provides asymptotic solutions valid for bounded t values as a positive parameter ε tends to zero. One can eliminate secular terms by introducing a slowly‐varying amplitude obtained as a solution of an amplitude equation on intervals where εt is bounded. With sufficient stability hypotheses, the results may even hold for all t≥0. These ideas are illustrated for a number of nontrivial problems involving ordinary differential equations.

Joshi, Nalini

doi: 10.1111/sapm.12066pmid: N/A

In this paper, we present new, unstable solutions, which we call quicksilver solutions, of a q‐difference Painlevé equation in the limit as the independent variable approaches infinity. The specific equation we consider in this paper is a discrete version of the first Painlevé equation (qPI), whose phase space (space of initial values) is a rational surface of type A7(1). We describe four families of almost stationary behaviors, but focus on the most complicated case, which is the vanishing solution. We derive this solution's formal power series expansion, describe the growth of its coefficients, and show that, while the series is divergent, there exist true analytic solutions asymptotic to such a series in a certain q‐domain. The method, while demonstrated for qPI, is also applicable to other q‐difference Painlevé equations.