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.

Date-Jimbo-Kashiwara-Miwa (DJKM) equation is an integrable (2+1)- dimensional extension of the KdV equation and describes two-dimensional nonlinear dispersive waves. In this paper, we first derive the M-breather solution in terms of determinants for the DJKM equation applying the nonlinear superposition formula and analyze the dynamical properties of the breathers. The M-lump waves for the DJKM equation are obtained through the full degeneration of the breathers and hybrid solutions composed of line solitons, breathers and lumps are constructed. By using the velocity resonance mechanism, we also show that the DJKM equation possesses some resonant structures for breathers and solitons such as breather molecules and breather-soliton molecules. Furthermore, based on the N-soliton solution, the interactions between breather/breather-soliton molecules and breathers/lumps are investigated.

In this paper, we apply Hirota bilinear method and determinant technique to derive the Nth-order rational solution expressed compactly in terms of Matsuno determinants for the variable-coefficient extended modified Kadomtsev-Petviashvili (mKP) equation. As a special case, we obtain the M-lump solution expressed in terms of $2M \times 2M$ determinants for the mKP\uppercase\expandafter{\romannumeral1} equation and investigate the dynamical behaviors of 1- and 2-lump solutions. Furthermore, we present the Wronskian and Grammian solution for the variable-coefficient extended mKP equation. Based on the Grammian solution, we construct the line soliton, the line breather and the semi-rational solution on constant and periodic backgrounds for the mKP\uppercase\expandafter{\romannumeral1} equation. Through the asymptotic analysis, we show that the semi-rational solutions describe the fission and fusion of lumps and line solitons. In addition, we construct the variable-coefficient extended mKP equation with self-consistent sources via the source generation procedure and derive its N-soliton solution in the compact form of Grammian and Wronskian.

The (2+1)-dimensional elliptic Toda equation is a high-dimensional generalization of the Toda lattice and a semi-discrete Kadomtsev-Petviashvili I equation. This paper focuses on investigating the resonant interactions between two breathers, a breather/lump and line solitons as well as lump molecules for the (2+1)-dimensional elliptic Toda equation. Based on the N-soliton solution, we obtain the hybrid solutions consisting of line solitons, breathers and lumps. Through the asymptotic analysis of these hybrid solutions, we derive the phase shifts of the breather, lump and line solitons before and after the interaction between a breather/lump and line solitons. By making the phase shifts infinite, we obtain the resonant solution of two breathers and the resonant solutions of a breather/lump and line solitons. Through the asymptotic analysis of these resonant solutions, we demonstrate the resonant interactions exhibit the fusion, fission, time-localized breather and rogue lump phenomena. Utilizing the velocity resonance method, we obtain lump-soliton, lump-breather, lump-soliton-breather and lump-breather-breather molecules. The above works have not been reported in the (2+1)-dimensional discrete nonlinear wave equations.

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