Isospectral Lax equation

For $\dot{L}=[L,A]$, the operator ${\partial }_t+A$ maps each eigenspace of $L$ into itself, so the spectrum is time-independent.

The Lax equation $L_t=[L,A]$ for these operators is equivalent toEliminating $p$ gives the Boussinesq equation

Ifthen every positive power has constant trace:The final equality is the cyclic property of the matrix trace.

With boundary values $q_0=q_n=0$, the finite Volterra lattice isup to rescaling or reversal of time. Writing $q_j=a_j^2$ gives ${\dot{a}}_j=a_j(a_{j+1}^2-a_{j-1}^2)$.

Let $L$ be the symmetric zero-diagonal Jacobi matrix with $L_{j,j+1}=L_{j+1,j}=a_j$, and let the skew-symmetric matrix $B$ haveThen $\dot{L}=[B,L]$ is equivalent to the Finite Volterra lattice. The corresponding similarity flow makes the eigenvalues of $L$, the coefficients of its characteristic polynomial, and its trace invariants first integrals.

For real periodic KdV data, translation by one period acts on a conjugate basis of scattering solutions byLax compatibility gives $\dot{T}=[\Lambda ,T]$ for the matrix of ${\partial }_t+A$ on the eigenspace. Hence $\mathrm{tr}T=2\mathrm{Re}a$ is conserved.