Similarity Analysis of Lax Pairs for a Class of Nonlinear Evolution Equations.

Abstract

The similarity transformation (ST) method is applied to reduce the Lax pair for some nonlinear partial differential equations (NLPDEs) into a system of ordinary differential equations (ODEs) to obtain its similarity solutions. Then the ODE system is considered to find the analytical solutions of the PDE by plotting the acquired similarity solutions. The method is applied to the three different equations named as; Modified Boussinesq (MBQ) equation, Kadomtsev–Petviashvili (KP) equation, and (2+1) - Korteweg-de-Vries breaking type (KDV-BS) equation. The Lie transformation method is utilized to convert the modified Boussinesq equation’s Lax Pair into a system of ordinary differential equations and obtain the analytical solutions of this equation. Likewise, this method is used for the KP and (2+1)-dimensional KdV Lax Pairs. The Lie vectors are optimized through the commutation operation. The reduction of the Lax pair instead of the original equation reveals a new solution. The applied method is effective in spreading the solution of NLPDEs.