An effective approximation to the dispersive soliton solutions of the coupled KdV equation via combination of two efficient methods

Abstract

The numerical solutions of the coupled Korteweg-de Vries equation is investigated via combination of two efficient methods that includes Crank–Nicolson scheme for time integration and quintic B-spline based differential quadrature method for space integration. The differential quadrature method has the advantage of using less number of grid points. And the advantage of the Crank–Nicolson scheme is the prevention of long and tedious algebraic computations for time integration. Those advantages come together and produce better results. To display the accuracy and efficiency of the present hybrid method three well-known test problems, namely single soliton, interaction of two soliton and birth of solitons are solved and the error norms $$L_{2}$$ and $$ L _{\infty }$$ are computed and compared with earlier works. Present hybrid method obtained superior results than earlier works by using the same parameters and less number of grid points. This situation is shown by comparison of the earlier works. At the same time, two lowest invariants and numerical and analytical values of the amplitudes of the solitons during the simulations are computed and tabulated. Besides those, relative changes of invariants are computed. Properties of solitons observed clearly at the all of the test problems and figures of the all of the simulations are given.