Tag Matthew Russo

Integrable Spatiotemporally Varying NLS, PT-Symmetric NLS, and DNLS Equations: Generalized Lax Pairs and Lie Algebras

Matthew Russo, S. Roy Choudhury

This paper develops two approaches to Lax-integrbale systems with spatiotemporally varying coefficients. A technique based on extended Lax Pairs is first considered to derive variable-coefficient generalizations of various Lax-integrable NLPDE hierarchies recently introduced in the literature. As illustrative examples, we consider generalizations of the NLS and DNLS equations, as well as a PT-symmetric version of the NLS equation. It is demonstrated that the techniques yield Lax- or S-integrable NLPDEs with both time- AND space-dependent coefficients which are thus more general than almost all cases considered earlier via other methods such as the Painleve Test, Bell Polynomials, and various similarity methods. However, this technique, although operationally effective, has the significant disadvantage that, for any integrable system with spatiotemporally varying coefficients, one must ‘guess’ a generalization of the structure of the known Lax Pair for the corresponding system with constant coefficients. Motivated by the somewhat arbitrary nature of the above procedure, we therefore next attempt to systematize the derivation of Lax-integrable sytems with variable coefficients. We attempt to apply the Estabrook- Wahlquist (EW) prolongation technique, a relatively self-consistent procedure requiring little prior infomation. However, this immediately requires that the technique be significantly generalized or broadened in several different ways. The new and extended EW technique which results is illustrated by algorithmically deriving generalized Lax-integrable versions of NLS, PT-symmetric NLS, and DNLS equations.

http://arxiv.org/abs/1410.0645
Analysis of PDEs (math.AP); Exactly Solvable and Integrable Systems (nlin.SI)

Integrable Generalized KdV, MKdV, and Nonlocal PT-Symmetric NLS Equations with Spatiotemporally Varying Coefficients

Matthew Russo, S. Roy Choudhury

We present a technique based on extended Lax Pairs to derive variable-coefficient generalizations of various Lax-integrable NLPDE hierarchies. As illustrative examples, we consider generalizations of KdV equations, three variants of generalized MKdV equations, and a recently-considered nonlocal PT-symmetric NLS equation. It is demonstrated that the technique yields Lax- or S-integrable NLPDEs with both time- AND space-dependent coefficients which are thus more general than almost all cases considered earlier via other methods such as the Painleve Test, Bell Polynomials, and various similarity methods. Employing the Painleve singular manifold method, some solutions are also presented for the generalized variable-coefficient integrable KdV and MKdV equations derived here. Current and future work is centered on generalizing other integrable hierarchies of NLPDEs similarly, and deriving various integrability properties such as solutions, Backlund Transformations, and hierarchies of conservation laws for these new integrable systems with variable coefficients.

http://arxiv.org/abs/1404.4602

Mathematical Physics (math-ph)