An analytical study on the existence of solitary wave and double layer solution of the well-known energy integral at M= Mc

TL;DR

This paper provides a comprehensive analytical framework for the existence of solitary wave and double layer solutions at the critical Mach number in plasma systems, supported by rigorous theorems and applied to dust acoustic waves.

Contribution

It introduces ten theorems establishing conditions for solitary wave and double layer existence at M= Mc, advancing the theoretical understanding of nonlinear plasma structures.

Findings

Existence of solitary waves or double layers at M= Mc depends on the Sagdeev potential conditions.

Certain potential configurations preclude coexistence of positive and negative structures at M= Mc.

Application to dust acoustic waves reveals new insights into plasma nonlinear phenomena.

Abstract

A general theory for the existence of solitary wave and double layer at M= Mc has been discussed, where Mc is the lower bound of the Mach number M, i.e., solitary wave and/or double layer solutions of the well-known energy integral start to exist for M> Mc. Ten important theorems have been proved to confirm the existence of solitary wave and double layer at M = Mc. If V({\phi})({\equiv}V(M,{\phi})) denotes the Sagdeev potential with {\phi} is the perturbed field or perturbed dependent variable associated with the specific problem, V(M,{\phi}) is well defined as a real number for all M {\in} \mathcal{M} and for all {\phi} {\in} {\Phi}, and V(M,0)=V'(M,0)=V"(Mc,0)=0, V"'(Mc,0)<0 (V"'(Mc,0)>0), \deltaV/\deltaM < 0 for all M({\in} \mathcal{M}) > 0 and for all {\phi}({\in} {\Phi}) > 0 ({\phi}({\in}{\Phi}) < 0), where " '{\equiv} \delta/\delta{\phi} ", the main analytical results for the existence of solitary wave and double layer solution of the energy integral at M= Mc are as follows. Result-1: If there exists at least one value M0 of M such that the system supports positive (negative) potential solitary waves for all Mc<M<M0, then there exist either a positive (negative) potential solitary wave or a positive (negative) potential double layer at M= Mc. Result-2: If the system supports only negative (positive) potential solitary waves for M> Mc, then there does not exist positive (negative) potential solitary wave at M= Mc. Result-3: It is not possible to have coexistence of both positive and negative potential solitary structures (including double layers) at M= Mc. Apart from the conditions of Result-1, the double layer solution at M= Mc is possible only when there exists a double layer solution in any right neighborhood of Mc. Finally these analytical results have been applied to a specific problem on dust acoustic waves in nonthermal plasma in search of new results.