PT-Symmetric Extension of the Korteweg-de Vries Equation

TL;DR

This paper explores PT-symmetric extensions of the Korteweg-de Vries equation into the complex domain, analyzing their properties, conservation laws, and solitary wave solutions, with a focus on the case where epsilon equals 3.

Contribution

It introduces a family of complex nonlinear wave equations extending the KdV equation while preserving PT symmetry, and investigates their mathematical properties and solutions.

Findings

Conservation laws are derived for the epsilon=3 case.

Solitary wave solutions are analyzed for the extended equations.

The equations exhibit interesting symmetry and integrability features.

Abstract

The Korteweg-de Vries equation u_t+uu_x+u_{xxx}=0 is PT symmetric (invariant under space-time reflection). Therefore, it can be generalized and extended into the complex domain in such a way as to preserve the PT symmetry. The result is the family of complex nonlinear wave equations u_t-iu(i u_x)^epsilon+u_{xxx}=0, where epsilon is real. The features of these equations are discussed. Special attention is given to the epsilon=3 equation, for which conservation laws are derived and solitary waves are investigated.