New exact solutions for the KdV equation with higher order nonlinearity by using the variational method

The Korteweg–de Vries (KdV) equation with higher order nonlinearity models the wave propagation in one-dimensional nonlinear lattice. A higher-order extension of the familiar KdV equation is produced for internal solitary waves in a density and current stratified shear flow with a free surface. The variational approximation method is applied to obtain the solutions for the well-known KdV equation. Explicit solutions are presented and compared with the exact solutions. Very good agreement is achieved, demonstrating the high efficiency of variational approximation method. The existence of a Lagrangian and the invariant variational principle for the higher order KdV equation are discussed. The simplest version of the variational approximation, based on trial functions with two free parameters is demonstrated. The jost functions by quadratic, cubic and fourth order polynomials are approximated. Also, we choose the trial jost functions in the form of exponential and sinh solutions. All solutions are exact and stable, and have applications in physics.