Asymptotic solitons for a third-order Korteweg–de Vries equation

Abstract

Abstract Solitary wave interaction for a higher-order version of the Korteweg–de Vries (KdV) equation is considered. The equation is obtained by retaining third-order terms in the perturbation expansion, where for the KdV equation only first-order terms are retained. The third-order KdV equation can be asymptotically transformed to the KdV equation, if the third-order coefficients satisfy a certain algebraic relationship. The third-order two-soliton solution is derived using the transformation. The third-order phase shift corrections are found and it is shown that the collision is asymptotically elastic. The interaction of two third-order solitary waves is also considered numerically. Examples of an elastic and an inelastic collision are both considered. For the elastic collision (which satisfies the algebraic relationship) the numerical results confirm the theoretical predictions, in particular there is good agreement found when comparing the third-order phase shift corrections. For the inelastic collision (which does not satisfy the algebraic relationship) an oscillatory wavetrain is produced by the interacting solitary waves. Also, the third-order phase shift corrections are found numerically for a range of solitary wave amplitudes. An asymptotic mass-conservation law is used to test the finite-difference scheme for the numerical solutions. In general, mass is not conserved by the third-order KdV equation, but varies during the interaction of the solitary waves.