$M$-Breather, Lumps, and Soliton Molecules for the $\left(2+1\right)$-Dimen

Abstract

The 2 + 1 -dimensional elliptic Toda equation is a higher dimensional generalization of the Toda lattice and also a discrete version of the Kadomtsev-Petviashvili-1 (KP1) equation. In this paper, we derive the M -breather solution in the determinant form for the 2 + 1 -dimensional elliptic Toda equation via Bäcklund transformation and nonlinear superposition formulae. The lump solutions of the 2 + 1 -dimensional elliptic Toda equation are derived from the breather solutions through the degeneration process. Hybrid solutions composed of two line solitons and one breather/lump are constructed. By introducing the velocity resonance to the N -soliton solution, it is found that the 2 + 1 -dimensional elliptic Toda equation possesses line soliton molecules, breather-soliton molecules, and breather molecules. Based on the N -soliton solution, we also demonstrate the interactions between a soliton/breather-soliton molecule and a lump and the interaction between a soliton molecule and a breather. It is interesting to find that the KP1 equation does not possess a line soliton molecule, but its discrete version—the 2 + 1 -dimensional elliptic Toda equation—exhibits line soliton molecules.

Resonant interactions among two-dimensional nonlinear localized waves and lump molecules for the (2+1)-dimensional elliptic Toda equation

“…The N-soliton solution of the (2 + 1)-dimensional elliptic Toda equation has been derived by the Bäcklund transformation and nonlinear superposition formula [16]…”

“…where subscript i denotes the breather i (i = 1, 2). The nonsingular conditions of the breather 1 and the breather 2 are [16] gives that the phase shifts of two breathers before and after the interaction between two breathers are both Γ , where Γ = ln |L 1 L 2 |. When Γ → ∞, the interaction between two breathers becomes resonant interaction.…”

“…[15]. More recently, N-soliton, M-breather and lump solutions of the (2 + 1)-dimensional elliptic Toda equation have been investigated by the Bäcklund transformation and nonlinear superposition formula, [16] which provides the possibility to obtain more hybrid solutions.…”