Soliton solutions with generalized Kudryashov method and study of variational integrators with Lagrangian to shallow water wave equation

In this paper, we study variational integrators (VIs) with the help of projection technique for Korteweg–de Vries (KdV) equation. First, we use forward, backward and central difference schemes. After that, we use Lagrangian, Euler–Lagrange equation and discrete Euler–Lagrange equation to find numerical solution to KdV equation. Finally, we obtain soliton solutions like W-shape, bright and kink soliton with the help of generalized Kudryashov method (GKM). These solitons are used in optical communication, Bose–Einstein condensate, plasma physics, fiber optics sensors and so on.