Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 32D (Integrable Systems)

The transformations obey $g_1^s{g}_1^r=g_1^{s+r}$ and $g_1^0$ is the identity. If $\tilde{x}(\tilde{t})=x(t)+st$, then $d^2\tilde{x}/d{\tilde{t}}^2=d^{2}x/d{t}^2$. Thus solutions of the free-particle equation are mapped to solutions, so this is a one-parameter group of Lie symmetries.

The flow equations of the displayed vector field are ${\dot{\psi }}_1=-{\psi }_2$ and ${\dot{\psi }}_2={\psi }_1$. Hence $d\psi /ds=i\psi$, whose solution is $\psi (s)=e^{is}\psi (0)$. The vector field therefore generates the stated phase rotation.

Direct substitution gives $g^s{g}^r=g^{s+r}$. Differentiating at $s=0$ gives $t{\partial }_x+ix(\psi {\partial }_{\psi }-{\psi }^{\ast }{\partial }_{{\psi }^{\ast }})$.

Writing the transformed field in transformed coordinates givesConsequentlywhere $\Phi =s\tilde{x}-s^{2}t/2$. These formulas, together with their complex conjugates and the analogous mixed derivatives, give the second prolongation. Differentiating at zero yields

Substituting the prolonged formulas showsso the nonlinear Schrödinger equation is invariant. Applying the symmetry to the given solitary wave produceswhich travels with speed $s$.