Asymptotic solitons for a higher-order modified Korteweg–de Vries equation

Solitary wave interaction for a higher-order modified Korteweg–de Vries (mKdV) equation is examined. The higher-order mKdV equation can be asymptotically transformed to the mKdV equation, if the higher-order coefficients satisfy a certain algebraic relationship. The transformation is used to derive the higher-order two-soliton solution and it is shown that the interaction is asymptotically elastic. Moreover, the higher-order phase shifts are derived using the asymptotic theory. Numerical simulations of the interaction of two higher-order solitary waves are also performed. Two examples are considered, one satisfies the algebraic relationship derived from the asymptotic theory, and the other does not. For the example which satisfies the algebraic relationship the numerical results confirm that the collision is elastic. The numerical and theoretical predictions for the higher-order phase shifts are also in strong agreement. For the example which does not satisfy the algebraic relationship, the numerical results show that the collision is inelastic; an oscillatory wavetrain is produced by the interacting solitary waves. Also, the higher-order phase shifts for this inelastic example are tabulated, for a range of solitary wave amplitudes. An asymptotic mass-conservation law is derived and used to test the finite-difference scheme for the numerical solutions. It is shown that, in general, mass is not conserved by the higher-order mKdV equation, but varies during the interaction of the solitary waves.