Algebraic Properties for Certain Form of the Members of Sequence on Generalized Modified Camassa-Holm Equation

TL;DR

This paper investigates the symmetry, integrability, and algebraic properties of a generalized modified Camassa-Holm equation, revealing its linearizability and Painlevé integrability through symmetry analysis and series solutions.

Contribution

It introduces a family of equations with increasing nonlinearity, analyzes their symmetries, and demonstrates their linearizability and Painlevé property, advancing understanding of their algebraic structure.

Findings

The second-order ODE derived from GMCH has eight-dimensional symmetries.

The equation is linearizable due to its symmetry properties.

It passes the Painlevé test indicating integrability.

Abstract

We study the symmetry and integrability of a Generalized Modified Camassa-Holm Equation (GMCH) of the form $$u_{t}-u_{xxt}+2nu_{x}(u^2-u_{x}^2)^{n-1}(u-u_{xx})^2+(u^2-u_{x}^2)^{n}(u_{x}-u_{xxx})=0.$$ We observe that for increasing values of $n\in \mathbb{N}$, $\mathbb{N}$ denotes natural number, the above equation gives a family of equations in which nonlinearity is rapidly increasing as $n$ increases. However, this family has similar form of symmetries except the values of $n$. Interestingly the resultant second-order nonlinear ODE which is to be obtained from GMCH equation has eight dimensional symmetries. Hence the second-order nonlinear ODE is linearizable. Finally we conclude that the resultant second-order nonlinear ordinary differential equation which is obtained from the family of GMCH passes the Painlev\'e Test also it posses the similar form of leading order, resonances and truncated series solution too.