Optimal system, similarity solution and Painlevé test on generalized modified Camassa-Holm equation

We study the symmetry and integrability of a Generalized Modified Camassa-Holm Equation (GMCH) of the form $$\begin{aligned} u_{t}-u_{xxt}+2nu_{x}(u^2-u_{x}^2)^{n-1}(u-u_{xx})^2+(u^2-u_{x}^2)^{n}(u_{x}-u_{xxx})=0. \end{aligned}$$ u t - u xxt + 2 n u 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 all increasing values of $$n\in {\mathbb {R}}$$ n ∈ R , $${\mathbb {R}}$$ R denotes the set of real 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, a commutator table, an adjoint representation, and a one-dimensional optimal system. Interestingly, we show that the resultant second-order nonlinear ODE generated from the GMCH equation is linearizable because it possesses maximal symmetries. Finally, we conclude that the GMCH family passes the Painlevé Test since the resultant third-order nonlinear ordinary differential equation passes the Painlevé Test. This family does, in fact, have a similar form of leading order, resonances and truncated series of solution too.