Integrable spatiotemporally varying KdV and MKdV equations: Generalized Lax pairs and an extended Estabrook-Wahlquist method

This paper develops two approaches to Lax-integrable 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 KdV and MKdV equations. 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 Painlevé 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. Hence we attempt to apply the Estabrook-Wahlquist (EW) prolongation technique, a relatively self-consistent procedure requiring little prior information. However, this immediately requires that the technique be significantly generalized or broadened in several different ways, including solving matrix partial differential equations instead of algebraic ones as the structure of the Lax Pair is deduced systematically following the standard Lie-algebraic procedure. The same is true while finding the explicit forms for the various ‘coefficient’ matrices which occur in the procedure and which must satisfy the various constraint equations which result at various stages of the calculation.

The new and extended EW technique which results is illustrated by algorithmically deriving generalized Lax-integrable versions of the generalized fifth-order KdV and MKdV equations.