Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 2 / 8A (Differential Equations)

Write $\psi =a{e}^{iS}$ with $a=\sqrt{\rho }$ and $v=S_x$. ThenwhileEquating imaginary parts givesand equating real parts and differentiating in $x$ givesThusThis is the hydrodynamic form of the Defocusing nonlinear Schrödinger equation.

For a stationary gas with constant phase, $\psi =a(x)$ is real and time independent. The field equation reduces toA positive constant solution obeys $d(1-d^2)=0$, so

The travelling-wave substitution $\xi =x-Ut$ givesWriteIts real part satisfiesFor $U>0$, the imaginary part also givesWith $\alpha =\sqrt{1-2U^2}$ and a translation chosen so that the notch is centred at zero,Its limiting modulus is $\sqrt{{\alpha }^2+2U^2}=1=d$, as required. The same expression at $U=0$ is obtained directly from the second-order equation.

The dark soliton density isFor $U=0$,a stationary notch reaching zero at its centre. For $U=1/2$,a shallower notch with minimum $1/2$. Since $\xi =x-Ut$, the first profile remains fixed and the second translates rigidly to the right at speed $1/2$.