Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 3 / 32A (Integrable Systems)

Differentiating the two equations for a fundamental matrix of solutions givesEquality for all independent initial vectors is equivalent to the zero-curvature conditionA single possibly zero solution vector alone would not imply a matrix identity; compatibility is understood for the complete linear system.

Write $q={\varphi }_{\rho }$, $s=\sin \varphi$, $c=\cos \varphi$ and use Pauli matrices:The matrix products givewhereas $V_{\rho }=-(iqc/(4\lambda )){\sigma }_2+(iqs/(4\lambda )){\sigma }_3$. Thus these last two components cancel, and the remaining curvature is $(i/2)({\varphi }_{\rho \tau }-\sin \varphi ){\sigma }_1$. Thereforethe Sine-Gordon equation in light-cone coordinates.

If $U,V$ depend only on $x=\rho +\tau$, then ${\partial }_{\rho }={\partial }_{\tau }={\partial }_x$ on them. Put $A=U-V$ and $B=U+V$. The curvature equation givesBy cyclicity of matrix trace,This is an Isospectral Lax equation expressed as a Lax equation, and it proves the assertion for every positive integer $p$.

For the given matrices and $\varphi =\varphi (x)$,Using $(a\cdot \sigma {)}^2=(a\cdot a)I$ givesSince $\lambda$ is fixed, constancy of this trace yields the first integralDifferentiating confirms $dC/dx={\varphi }^{\prime }({\varphi }^{\prime \prime }-\sin \varphi )=0$ for the reduced equation.