Abstract :
The collision of solitary waves for a higher-order modified Korteweg-de Vries (mKdV) equation is examined. In particular, the collision between solitary waves with sech-type and algebraic (which only exist on a non-zero mean level) profiles is considered. An asymptotic transformation, valid if the higher-order coefficients satisfy a certain algebraic relationship, is used to transform the higher-order mKdV equation to an integrable member of the mKdV hierarchy. The transformation is used to show that the higher-order collision is asymptotically elastic and to derive the higher-order phase shifts.
Numerical simulations of both elastic and inelastic collisions are performed. For the example covered by the asymptotic theory the numerical results confirm that the collision is elastic and the theoretical predictions for the higher-order phase shifts. For the example not covered by the asymptotic theory the numerical results show that the collision is inelastic; an oscillatory wavetrain is produced by the colliding solitary waves. The higher-order phase shift for the faster (sech-type) solitary wave is found. For the slower (algebraic) wave, however, the shed radiation never completely separates from the solitary wave. Interaction with the shed radiation causes the phase shift to evolve long after collision is over, with no final higher-order phase shift able to be determined.
An asymptotic mass-conservation law is to test the accuracy of 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.
