Integrable Generalized KdV and MKdV Equations with Spatiotemporally Varying Coefficients

摘要: 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 KdV equations and three variants generalized MKdV equations. It demonstrated that techniques yield Lax- or S-integrable NLPDEs with both time- AND space-dependent coefficients which are thus more general than almost all cases earlier via other methods such as Painleve Test, Bell Polynomials, similarity methods. However, this technique, although operationally effective, has significant disadvantage that, for any integrable system spatiotemporally varying coefficients, one must guess a generalization structure known Pair corresponding constant coefficients. Motivated by somewhat arbitrary nature above procedure, embark paper an attempt systematize derivation sytems variable Hence apply Estabrook-Wahlquist (EW) prolongation relatively self-consistent procedure requiring little prior information. immediately requires be significantly broadened several different ways, including solving matrix partial differential instead algebraic ones. The new EW whch results illustrated algorithmically deriving versions fifth-order KdV,