<math xmlns='http://www.w3.org/1998/Math/MathML' id='M1'> <mi>M</mi> </math>-Breather, Lumps, and Soliton Molecules for the <math xmlns='http://www.w3.org/1998/Math/MathML' id='M2'> <mfenced open='(' close=')'> <mrow> <mn>2</mn> <mo>+</mo> <mn>1</mn> </mrow> </mfenced> </math>-Dimensional Elliptic Toda Equation

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 B&#228;cklund 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.