Lax pairs and isospectral flows

Let $L$ and $K$ be two $n \times n$ matrices. Assume that the entries depend on a parameter $t$ and are differentiable with respect to $t$. Assume that $K^{-1}(t)$ exists for all $t$. Assume that the time-evolution of $L$ is given by

Consider the linear operators $L$ and $M$ defined by

Find the condition on $L$ and $M$ such that $[L,M] = 0$, where $[\, , \, ]$ denotes the commutator. The potentials $U(x,t,\lambda )$ and $V(x,t,\lambda )$ (square matrices) are typically chosen as elements of some semisimple Lie algebra.

Consider the differential operators

From the condition $[L,B]\phi (x) = L\phi (x)$ find the ordinary differential equation that the function $f$ obeys.

Consider the linear equations

where $U, V$ are functions of $u, v, \lambda$. The linear operators

are the Lax pair. The integrability condition for $\phi$ is the zero curvature condition on the Lax pair

Let $g(u,v)$ be an invertible $2 \times 2$ matrix and let

Find the nonlinear differential equation for $[L,M] = 0$.

Consider the $n \times n$ tridiagonal matrices

where the entries $a_j$ and $b_j$ are real analytic functions of $t$. Thus $A$ is symmetric over $\mathbb {R}$ and $B$ is skew-symmetric over $\mathbb {R}$.

Many soliton equations in $1+1$ dimensions can be written as an integrability condition for a set of linear equations called the linear scattering problem

where $\Omega$ is an $N \times N$ matrix consisting of a one-parameter $\zeta$ (spectral parameter) family of differential one-forms. The integrability condition is $\Theta = 0$ (zero curvature equation), where

$d$ is the exterior derivative and $\wedge$ is the exterior product. Assume that

where $\zeta$ is the complex spectral parameter, $D$ is a diagonal $N \times N$ matrix with $d_{jj}$ complex and pairwise distinct. The $N \times N$ matrix $P(x,t)$ is off-diagonal and it is assumed to satisfy the boundary conditions

A number of Hamilton systems can be written in the form

where $L$ and $A$ are time-dependent $n \times n$ matrices. This is called the Lax representation of the Hamilton system. We find that

and the that $\tr [L^k]$ ($k=1,2,\ldots$) are first integrals. Consider the Hamilton function (Toda lattice)

Introducing the quantities

and cyclic boundary conditions (i.e., $q_4 \equiv q_1$) we find that the Hamilton equations of motion take the form (with $b_3 = 0$)

where $j=1,2$. Introducing the matrices (Lax pair)

the equations of motion can be written as Lax representation. From $L$ we find the first integral as $\tr [L^n]$, where $n = 1,2,\ldots$, where $\tr$ denotes the trace. We obtain

Write a SymbolicC++ program lax.cpp that finds $[L,A]$ and shows that $\tr (L)$, $\tr (L^2)$ and the determinant of $L$ are first integrals.

Let $\partial := \partial /\partial x$. The two pseudodifferential operators

are called the Lax operator and gauge operator, respectively. With $\partial^{-1}\partial = \partial \partial^{-1} = 1$ the generalized Leibniz rule is satisfied

for any $k \in \mathbb {Z}$ and $L = W\partial W^{-1}$. The KP hierarchy is given by the Lax equations

where $\partial_n := \partial /\partial t_n$ and $B_n := L_+^n$ is the differential part of $L^n = L_+^n+L_-^n$ with

Derive the first three equations of the KP hierarchy.

Let $\overrightarrow {v}(x,t) = (v_1(x,t),v_2(x,t))^T$. Consider the linear equations

Show that the compatibility condition of these two equations with $\partial \zeta /\partial t = 0$ provides the unstable nonlinear Schrodinger equation