The -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 -breather solution in the determinant form for the -dimensional elliptic Toda equation via Bäcklund transformation and nonlinear superposition formulae. The lump solutions of the -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 -soliton solution, it is found that the -dimensional elliptic Toda equation possesses line soliton molecules, breather-soliton molecules, and breather molecules. Based on the -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 -dimensional elliptic Toda equation—exhibits line soliton molecules.

中文翻译：

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-维椭圆 Toda 方程的呼吸器、团块和孤子分子

这 -维椭圆 Toda 方程是 Toda 格的高维推广，也是 Kadomtsev-Petviashvili-1 (KP1) 方程的离散版本。在本文中，我们以行列式形式推导出-呼吸器解-通过 Bäcklund 变换和非线性叠加公式的维度椭圆 Toda 方程。团块解决方案-维椭圆 Toda 方程是通过退化过程从呼吸器解导出的。构建了由两个线孤子和一个呼吸器/块组成的混合解决方案。通过将速度共振引入-孤子解，发现-维度椭圆 Toda 方程包含线孤子分子、呼吸孤子分子和呼吸分子。基于-孤子解决方案，我们还演示了孤子/呼吸孤子分子与团块之间的相互作用以及孤子分子与呼吸器之间的相互作用。有趣的是发现 KP1 方程不具有线孤子分子，而是它的离散形式——-维度椭圆 Toda 方程——展示线孤子分子。