Weakly non-integrable solitary wave collisions in the modified Benjamin–Bona–Mahony equation

Wave Motion. 2027. Vol. 148. Article 103867 .

Flamarion M. V., Pelinovsky E.

We consider the interaction of two solitary waves in the modified Benjamin–Bona–Mahony
(mBBM) equation in a regime where a small parameter measures the departure from the classical
modified Korteweg–de Vries (mKdV) equation. A regular perturbation method is used to
analyze the modification of the two-soliton wave field induced by the weak non-integrability of
the mBBM equation. We show that, after the interaction, the solitary wave with the larger amplitude
increases in modulus, whereas the one with the smaller amplitude decreases in modulus.
More precisely, the amplitude loss in modulus of the smaller wave exceeds the corresponding
gain of the larger wave, with the excess transferred to dispersive radiation. These analytical predictions
are confirmed by direct pseudospectral simulations of the mBBM equation, in which the
outgoing amplitudes are measured by windowed projection onto the exact solitary-wave family
— a diagnostic that suppresses contamination by the dispersive wake exponentially and resolves
amplitude changes several orders of magnitude smaller than the waves themselves.

Keywords: солитоны, solitons

Publication based on the results of: Differential-geometric properties of dynamical systems and mathematical modeling of processes (2025)