# Trying to solve 2D Toda Lattice Equation with Lax Pair Approach

I am working on this Hamiltonian: $$H = \frac{p_1^2 + p_2^2}{2m} + e^{q_2-q_1} + e^{q_2} + e^{-q_1} -3$$ Thank you for the hint that it is a modification of the Toda Lattice Equation. Let me sketch what I tried until now and why it is not working: Analogous to the mentioned publications I introduced $$b_n:= \frac{1}{2}Exp{(\frac{q_n-q_{n+1}}{2})}\\ a_n:= -\frac{p_n}{2}$$ where it follows directly with $\frac{\partial H}{\partial q_i}=-p_i$ and $\frac{\partial H}{\partial p_i}=q_i$: $$\dot{b_n} = (a_{n+1} - a_n)b_n \\ \dot{a_n} = 2 (b_{n}^2 - b_{n-1}^2)$$ When now using the Lax Pair $L$,$B$: $$L f_n = b_n f_{n+1} +b_{n-1} f_{n-1} + a_n f_{n}$$

$$B f_n = b_n f_{n+1} - b_{n-1} f_{n-1}$$ it can be shown that $\partial_t L=[B,L]$. My problem arises in defining the border conditions of my couple $q_1$ and $q_2$ in the 2d lattice above, since one needs to shift to the 3d representation $\{b_0,b_1,b_2\}$ in order to satisfy the periodic conditions (One mutual coordinate $q_3 = 0$ coupled to the others). Since it can be shown easily that $\dot{\lambda} = 0$ (where $\lambda$ is an eigenvalue $Lv=\lambda v$) the constants of motion reduce to the calculation of the eigenvalues. But in this case the eigenvalues of $L$ dont seem to simplify, in fact it doesnt seem to be a solution, which was my inital goal.

In general this approach seems to be at overkill for the 2d problem since it solves the n-dimensional Toda lattice.

1. Anyone knows of an easier approach to the 2d problem?
2. The Matrix $L$ seems to yield the wrong solution: $$L = \begin{pmatrix} a_0 & b_0 & 0 \\ b_0 & a_1 & b_1\\ 0 & b_1 & a_2 \end{pmatrix}$$ The Matrix does neigher solve $\partial_t L = [B,L]$ (with $B=L_+ - L_-$) nor are eigenvalues constants of motion. Was has gone wrong?

3. Since the inverse scattering method can be applied here, I tried to get the scattering data, but actually I was not able to do the task. Any literature?