Interaction of solitons and the effect of radiation for the generalized KdV equation

NUMERICAL AND ASYMPTOTIC DESCRIPTION OF SOLITONS FOR GKdV EQUATIONS WITH NONHOMOGENEOUS NONLINEARITIES

“…3 and 2 = κ Almost the same is true for nonintegrable homogeneous case: the solitary waves interact elastically in the principal term in an asymptotic sense, whereas the nonintegrability implies the appearance of small radiation-type corrections [3][4][5][6], [9,10]). At the same time, the existence of travelling wave solutions and the character of the solitary wave collision remains unknown for arbitrary nonlinearity.…”

“…So, we will construct a weak asymptotic solution. The weak asymptotics method (see, e.g., [1][2][3][4][5][6][7], [9,10] and references therein) takes into account the fact that soliton-type solutions which are smooth for 0 > ε become non-smooth in the limit as . 0 → ε Thus, it is possible to treat such solutions as a mapping…”

A Perturbation Theory for Nonintegrable Equations with Small Dispersion

“…It is obvious that the existence of the weak asymptotics (13) with the properties (14) and (15) implies that the solitary waves interact like the KdV solitons at least in the leading term. Numerical simulations ( [14,15,17]) confirm the traced analysis, see Figure 1. Note that a small oscillating tail appears after the soliton collision, see [15] for detail.…”

“…The weak asymptotics method has been proposed at first for shock wave type solutions [8] and for soliton-type solutions [9] many years ago. Further generalizations, modifications, and adaptations to other problems can be found in publications by M. Colombeau, Danilov, Mitrovic, Omel'yanov, Shelkovich, and others, see, for example, [10][11][12][13][14][15][16][17][18][19][20] and references therein.…”

Collision of solitons for a nonhomogeneous version of the KdV equation

“…then the perturbed soliton generates a rapidly oscillating tail of the amplitude o(1) (the so called "radiation") instead of the smooth tail εu − (x, t) (see [18] for the perturbed KdV equation and numerical results in [9]). However, εu − (x, t) describes sufficiently well the tendency of the radiation amplitude behavior.…”