Solitary wave solutions with non-monotone shapes for the modified Kawahara equation

The propagation of stationary solitary waves of the non-linear modified Kawahara equation is considered. The asymptotic boundary conditions admit the trivial solution along with the solitary wave type solution, which is a bifurcation problem. The bifurcation is treated by reformulating the problem into a problem for identification of an unknown coefficient from over-posed boundary data, in which the trivial solution is excluded. Making use of the method of variational imbedding, the inverse problem for the coefficient identification is reformulated as a higher-order boundary value problem. This approach to solving the fifth-order modified Kawahara equation is allowing identification of non-trivial solutions. The obtained boundary-value problem is solved by means of an iterative difference scheme, which is thoroughly validated. New traveling wave solutions with monotone and non-monotone shapes are obtained for different values of the problem parameters.