Integrable generalization of the Toda law to the square lattice

We generalize the Toda lattice (or Toda chain) equation to the square lattice, i.e., we construct an integrable nonlinear equation for a scalar field taking values on the square lattice and depending on a continuous (time) variable, characterized by an exponential law of interaction in both discrete directions of the square lattice. We construct the Darboux-Bäcklund transformations for such lattice, and the corresponding formulas describing their superposition. We finally use these Darboux-Bäcklund transformations to generate examples of explicit solutions of exponential and rational type. The exponential solutions describe the evolution of one and two smooth two-dimensional shock waves on the square lattice.

Figure 1

(Color online) If $\rho =0$ and $\alpha ,\beta ∊\mathbb{R}(F>4)$, the solution $Q_{m,n}+FCt$ in Eq. (31) describes a smooth 2D shock wave propagating with velocity ${\mathbf{v}}^{+}=-({\omega }^{+}∕2)\sin \left(2\theta \right)(1∕\beta ,1∕\alpha )$. The shock front is a straight line forming the angle $-\theta ,\theta ={\tan }^{-1}(\alpha ∕\beta )$ with the $m$ axis. In this figure $\alpha =5,\beta =4(F=101),{\delta }^{\pm }=1,C=1,\rho =0$.

Figure 2

(Color online) For $\alpha ,\beta ∊\mathbb{R}(F>4)$ and $\rho =O\left(1\right)$, the solution $Q_{m,n}+FCt$ in Eq. (31) describes two shock waves with fronts parallel to the $m$ and $n$ axes. The intersection point $P$ of these two fronts travels with velocity ${\mathbf{v}}_P=-C\mathbf{(}\left(\sinh \alpha \right)∕\alpha ,\left(\sinh \beta \right)∕\beta \mathbf{)}$. In this figure $\alpha =5,\beta =4(F=101),{\delta }^{\pm }=1,C=1,\rho =1$.

Figure 3

(Color online) A view from the top of the solution for $\rho ={10}^{-7}$. In the central (finite) region, the single shock prevails; this single shock matches with the two orthogonal shocks, which prevail instead in the outer region. In this figure $\alpha =5,\beta =4(F=101),{\delta }^{\pm }=1,C=1,\rho ={10}^{-7}$.

Figure 4

(Color online) A generic view of the solution for $\rho ={10}^{-7}$. In the central (finite) region, the single shock prevails; this single shock matches with the two orthogonal shocks, which prevail instead in the outer region. In this figure $\alpha =5,\beta =4(F=101),{\delta }^{\pm }=1,C=1,\rho ={10}^{-7}$.