8
$\begingroup$

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?

$\endgroup$
2
0
$\begingroup$

That's the Toda lattice, check, e.g., here. Also, Berry's paper has a discussion of the three-particle Toda lattice, which is what this is. The paper itself is a really good read, but that may give you a good start.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.