Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 33A (Integrable Systems)

Lax's equation in the convention used here isLet $\mathcal{L}\psi =\lambda \psi$ with $(\psi ,\psi )=1$. Since $\mathcal{L}$ is self-adjoint and $\mathcal{A}$ is anti-self-adjoint, differentiating $\lambda =(\psi ,\mathcal{L}\psi )$ givesThe first and third terms equal $\lambda ,d(\psi ,\psi )/dt=0$. For the middle term,Thus $\dot{\lambda }=0$, proving the Isospectral Lax equation property: every eigenvalue is independent of $t$.

Write $D=d/dx$. For rapidly decaying smooth functions, integration by parts gives $D^{\ast }=-D$, while multiplication by the real functions $p$ and $q$ is self-adjoint. Thereforeand henceLikewise $D^2$ and multiplication by $q$ are self-adjoint, so their purely imaginary multiples in $A=3i{D}^2-4iq$ are anti-self-adjoint:

If $L\psi =\lambda \psi$ with $\psi \ne 0$, self-adjointness givesConsequently every eigenvalue of $L$ is real.

The supplied identity saysExpanding it and differentiating $L$ with respect to $t$ givesEquating the coefficient of ${\psi }_x$ in Lax's equation $L_t=[L,A]$ yieldsThe imaginary scalar terms then cancel because $q_{tx}=3p_{xx}$, leavingDifferentiate the first equation in $t$ and the second in $x$ to eliminate $p$. The third-order Boussinesq Lax pair therefore producesThus the requested Boussinesq equation has