A unified transform method for solving linear and certain nonlinear PDEs. (English) Zbl 0876.35102

Fokas, A. S.

Proc. R. Soc. Lond., Ser. A 453, No. 1962, 1411-1443 (1997).

Summary: A new transform method for solving initial boundary value problems for linear and for integrable nonlinear PDEs in two independent variables is introduced. This unified method is based on the fact that linear and integrable nonlinear equations have the distinguished property that they possess a Lax pair formulation. The implementation of this method involves performing a simultaneous spectral analysis of both parts of the Lax pair and solving a Riemann-Hilbert problem. In addition to a unification in the method of solution, there also exists a unification in the representation of the solution. The sine-Gordon equation in light-cone coordinates, the nonlinear Schrödinger equation, and their linearized versions are used as illustrative examples. It is also shown that appropriate deformations of the Lax pairs of linear equations can be used to construct Lax pairs for integrable nonlinear equations. As an example, a new Lax pair of the nonlinear Schrödinger equation is derived.

MSC:

35Q53	KdV equations (Korteweg-de Vries equations)
35A22	Transform methods (e.g., integral transforms) applied to PDEs
35Q15	Riemann-Hilbert problems in context of PDEs
37J35	Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10	Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)

Keywords:

transform method; Lax pair formulation; Riemann-Hilbert problem; sine-Gordon equation; nonlinear Schrödinger equation