The Asymptotic Approach to the Description of Two-Dimensional Symmetric Soliton Patterns

Abstract

The asymptotic approach is suggested for the description of interacting surface and internal oceanic solitary waves. This approach allows one to describe stationary moving symmetric wave patterns consisting of two plane solitary waves of equal amplitudes moving at an angle to each other. The results obtained within the approximate asymptotic theory are validated by comparison with the exact two-soliton solution of the Kadomtsev–Petviashvili equation (KP2-equation). The suggested approach is equally applicable to a wide class of non-integrable equations too. As an example, the asymptotic theory is applied to the description of wave patterns in the 2D Benjamin–Ono equation describing internal waves in the infinitely deep ocean containing a relatively thin one of the layers.

(2+1)-dimensional KdV, fifth-order KdV, and Gardner equations derived from the ideal fluid model. Soliton, cnoidal and superposition solutions

“…Since in zeroth-order u t = −u x , u xxt = −u xxx , u yyt = −u xyy and u yyt dx = −u yy , then equation (31) does receive the same form as equation (30). Equation ( 30) takes correctly into account an uneven bottom, but only under assumption that h xx = 0.…”

“…Equation ( 30) takes correctly into account an uneven bottom, but only under assumption that h xx = 0. In the case when the bottom is even, δ = 0, and the last term in (30) vanishes. In such case…”

“…In [30] Y. Stepanyants presented 2-soliton solution to KP2 equation. This solution should be the solution to (2+1)-dimensional KdV equation (32), as well.…”

“…It turns out that in (2+1)-dimensional theory all these results hold. In section III we derived equation (30) which is (2+1)-dimensional KdV equation taking into account small bottom variations. Analogous procedures (not presented here) including δ = 0 for the case discussed in section IV lead to the equation…”

Lump interactions with plane solitons

“…Under certain conditions, two interacting lump chains can form only one lump chain or vice versa, one lump chain can experience a splitting onto two other lump chains moving in the space under an angle to each other. The entire pattern moves stationary similar to a plane soliton triad under the resonance interaction within the KP2 equation (see [16] and references therein).…”