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

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 all increasing values of n is an element of 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 Painleve Test since the resultant third-order nonlinear ordinary differential equation passes the Painleve Test. This family does, in fact, have a similar form of leading order, resonances and truncated series of solution too.