• # Paper 1 , Section II, 33D

(a) Let $U(z, \bar{z}, \lambda)$ and $V(z, \bar{z}, \lambda)$ be matrix-valued functions, whilst $\psi(z, \bar{z}, \lambda)$ is a vector-valued function. Show that the linear system

$\partial_{z} \psi=U \psi, \quad \partial_{\bar{z}} \psi=V \psi$

is over-determined and derive a consistency condition on $U, V$ that is necessary for there to be non-trivial solutions.

(b) Suppose that

$U=\frac{1}{2 \lambda}\left(\begin{array}{cc} \lambda \partial_{z} u & e^{-u} \\ e^{u} & -\lambda \partial_{z} u \end{array}\right) \quad \text { and } \quad V=\frac{1}{2}\left(\begin{array}{cc} -\partial_{\bar{z}} u & \lambda e^{u} \\ \lambda e^{-u} & \partial_{\bar{z}} u \end{array}\right)$

where $u(z, \bar{z})$ is a scalar function. Obtain a partial differential equation for $u$ that is equivalent to your consistency condition from part (a).

(c) Now let $z=x+i y$ and suppose $u$ is independent of $y$. Show that the trace of $(U-V)^{n}$ is constant for all positive integers $n$. Hence, or otherwise, construct a non-trivial first integral of the equation

$\frac{d^{2} \phi}{d x^{2}}=4 \sinh \phi, \quad \text { where } \quad \phi=\phi(x)$

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• # Paper 2, Section II, 34D

(a) Explain briefly how the linear operators $L=-\partial_{x}^{2}+u(x, t)$ and $A=4 \partial_{x}^{3}-3 u \partial_{x}-$ $3 \partial_{x} u$ can be used to give a Lax-pair formulation of the $\mathrm{KdV}$ equation $u_{t}+u_{x x x}-6 u u_{x}=0$.

(b) Give a brief definition of the scattering data

$\mathcal{S}_{u(t)}=\left\{\{R(k, t)\}_{k \in \mathbb{R}},\left\{-\kappa_{n}(t)^{2}, c_{n}(t)\right\}_{n=1}^{N}\right\}$

attached to a smooth solution $u=u(x, t)$ of the KdV equation at time $t$. [You may assume $u(x, t)$ to be rapidly decreasing in $x$.] State the time dependence of $\kappa_{n}(t)$ and $c_{n}(t)$, and derive the time dependence of $R(k, t)$ from the Lax-pair formulation.

(c) Show that

$F(x, t)=\sum_{n=1}^{N} c_{n}(t)^{2} e^{-\kappa_{n}(t) x}+\frac{1}{2 \pi} \int_{-\infty}^{\infty} R(k, t) e^{i k x} d k$

satisfies $\partial_{t} F+8 \partial_{x}^{3} F=0$. Now let $K(x, y, t)$ be the solution of the equation

$K(x, y, t)+F(x+y, t)+\int_{x}^{\infty} K(x, z, t) F(z+y, t) d z=0$

and let $u(x, t)=-2 \partial_{x} \phi(x, t)$, where $\phi(x, t)=K(x, x, t)$. Defining $G(x, y, t)$ by $G=$ $\left(\partial_{x}^{2}-\partial_{y}^{2}-u(x, t)\right) K(x, y, t)$, show that

$G(x, y, t)+\int_{x}^{\infty} G(x, z, t) F(z+y, t) d z=0$

(d) Given that $K(x, y, t)$ obeys the equations

\begin{aligned} \left(\partial_{x}^{2}-\partial_{y}^{2}\right) K-u K &=0 \\ \left(\partial_{t}+4 \partial_{x}^{3}+4 \partial_{y}^{3}\right) K-3\left(\partial_{x} u\right) K-6 u \partial_{x} K &=0 \end{aligned}

where $u=u(x, t)$, deduce that

$\partial_{t} K+\left(\partial_{x}+\partial_{y}\right)^{3} K-3 u\left(\partial_{x}+\partial_{y}\right) K=0$

and hence that $u$ solves the $\mathrm{KdV}$ equation.

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• # Paper 3, Section II, 32D

(a) Consider the group of transformations of $\mathbb{R}^{2}$ given by $g_{1}^{s}:(t, x) \mapsto(\tilde{t}, \tilde{x})=$ $(t, x+s t)$, where $s \in \mathbb{R}$. Show that this acts as a group of Lie symmetries for the equation $d^{2} x / d t^{2}=0$.

(b) Let $\left(\psi_{1}, \psi_{2}\right) \in \mathbb{R}^{2}$ and define $\psi=\psi_{1}+i \psi_{2}$. Show that the vector field $\psi_{1} \partial_{\psi_{2}}-\psi_{2} \partial_{\psi_{1}}$ generates the group of phase rotations $g_{2}^{s}: \psi \rightarrow e^{i s} \psi$.

(c) Show that the transformations of $\mathbb{R}^{2} \times \mathbb{C}$ defined by

$g^{s}:(t, x, \psi) \mapsto(\tilde{t}, \tilde{x}, \tilde{\psi})=\left(t, x+s t, \psi e^{i s x+i s^{2} t / 2}\right)$

form a one-parameter group generated by the vector field

$V=t \partial_{x}+x\left(\psi_{1} \partial_{\psi_{2}}-\psi_{2} \partial_{\psi_{1}}\right)=t \partial_{x}+i x\left(\psi \partial_{\psi}-\psi^{*} \partial_{\psi^{*}}\right)$

and find the second prolongation $\operatorname{Pr}^{(2)} g^{s}$ of the action of $\left\{g^{s}\right\}$. Hence find the coefficients $\eta^{0}$ and $\eta^{11}$ in the second prolongation of $V$,

$\mathrm{pr}^{(2)} V=t \partial_{x}+\left(i x \psi \partial_{\psi}+\eta^{0} \partial_{\psi_{t}}+\eta^{1} \partial_{\psi_{x}}+\eta^{00} \partial_{\psi_{t t}}+\eta^{01} \partial_{\psi_{x t}}+\eta^{11} \partial_{\psi_{x x}}+\right.$ complex conjugate $)$.

(d) Show that the group $\left\{g^{s}\right\}$ of transformations in part (c) acts as a group of Lie symmetries for the nonlinear Schrödinger equation $i \partial_{t} \psi+\frac{1}{2} \partial_{x}^{2} \psi+|\psi|^{2} \psi=0$. Given that $a e^{i a^{2} t / 2} \operatorname{sech}(a x)$ solves the nonlinear Schrödinger equation for any $a \in \mathbb{R}$, find a solution which describes a solitary wave travelling at arbitrary speed $s \in \mathbb{R}$.

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• # Paper 1, Section II, 33C

(a) Show that if $L$ is a symmetric matrix $\left(L=L^{T}\right)$ and $B$ is skew-symmetric $\left(B=-B^{T}\right)$ then $[B, L]=B L-L B$ is symmetric.

(b) Consider the real $n \times n$ symmetric matrix

$L=\left(\begin{array}{cccccccc} 0 & a_{1} & 0 & 0 & \cdots & \cdots & \cdots & 0 \\ a_{1} & 0 & a_{2} & 0 & \cdots & \cdots & \cdots & 0 \\ 0 & a_{2} & 0 & a_{3} & \cdots & \cdots & \cdots & 0 \\ 0 & 0 & a_{3} & \cdots & \cdots & \cdots & \cdots & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & \cdots & \cdots & \cdots & \cdots & \cdots & a_{n-2} & 0 \\ 0 & \cdots & \cdots & \cdots & \cdots & a_{n-2} & 0 & a_{n-1} \\ 0 & \cdots & \cdots & \cdots & \cdots & 0 & a_{n-1} & 0 \end{array}\right)$

(i.e. $L_{i, i+1}=L_{i+1, i}=a_{i}$ for $1 \leqslant i \leqslant n-1$, all other entries being zero) and the real $n \times n$ skew-symmetric matrix

$B=\left(\begin{array}{cccccccc} 0 & 0 & a_{1} a_{2} & 0 & \cdots & \cdots & \cdots & 0 \\ 0 & 0 & 0 & a_{2} a_{3} & \cdots & \ldots & \ldots & 0 \\ -a_{1} a_{2} & 0 & 0 & 0 & \ldots & \ldots & \ldots & 0 \\ 0 & -a_{2} a_{3} & 0 & \ldots & \cdots & \ldots & \ldots & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & \ldots & \ldots & \ldots & \ldots & \ldots & 0 & a_{n-2} a_{n-1} \\ 0 & \ldots & \cdots & \cdots & \cdots & 0 & 0 & 0 \\ 0 & \ldots & \cdots & \cdots & \ldots & -a_{n-2} a_{n-1} & 0 & 0 \end{array}\right)$

(i.e. $B_{i, i+2}=-B_{i+2, i}=a_{i} a_{i+1}$ for $1 \leqslant i \leqslant n-2$, all other entries being zero).

(i) Compute $[B, L]$.

(ii) Assume that the $a_{j}$ are smooth functions of time $t$ so the matrix $L=L(t)$ also depends smoothly on $t$. Show that the equation $\frac{d L}{d t}=[B, L]$ implies that

$\frac{d a_{j}}{d t}=f\left(a_{j-1}, a_{j}, a_{j+1}\right)$

for some function $f$ which you should find explicitly.

(iii) Using the transformation $a_{j}=\frac{1}{2} \exp \left[\frac{1}{2} u_{j}\right]$ show that

$\frac{d u_{j}}{d t}=\frac{1}{2}\left(e^{u_{j+1}}-e^{u_{j-1}}\right)$

for $j=1, \ldots n-1$. [Use the convention $u_{0}=-\infty, a_{0}=0, u_{n}=-\infty, a_{n}=0 .$ ]

(iv) Deduce that given a solution of equation ( $\dagger$, there exist matrices $\{U(t)\}_{t \in \mathbb{R}}$ depending on time such that $L(t)=U(t) L(0) U(t)^{-1}$, and explain how to obtain first integrals for $(t)$ from this.

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• # Paper 2, Section II, 33C

(i) Explain how the inverse scattering method can be used to solve the initial value problem for the $\mathrm{KdV}$ equation

$u_{t}+u_{x x x}-6 u u_{x}=0, \quad u(x, 0)=u_{0}(x)$

including a description of the scattering data associated to the operator $L_{u}=-\partial_{x}^{2}+u(x, t)$, its time dependence, and the reconstruction of $u$ via the inverse scattering problem.

(ii) Solve the inverse scattering problem for the reflectionless case, in which the reflection coefficient $R(k)$ is identically zero and the discrete scattering data consists of a single bound state, and hence derive the 1-soliton solution of $\mathrm{KdV}$.

(iii) Consider the direct and inverse scattering problems in the case of a small potential $u(x)=\epsilon q(x)$, with $\epsilon$ arbitrarily small: $0<\epsilon \ll 1$. Show that the reflection coefficient is given by

$R(k)=\epsilon \int_{-\infty}^{\infty} \frac{e^{-2 i k z}}{2 i k} q(z) d z+O\left(\epsilon^{2}\right)$

and verify that the solution of the inverse scattering problem applied to this reflection coefficient does indeed lead back to the potential $u=\epsilon q$ when calculated to first order in

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• # Paper 3, Section II, 32C

(a) Given a smooth vector field

$V=V_{1}(x, u) \frac{\partial}{\partial x}+\phi(x, u) \frac{\partial}{\partial u}$

on $\mathbb{R}^{2}$ define the prolongation of $V$ of arbitrary order $N$.

Calculate the prolongation of order two for the group $S O(2)$ of transformations of $\mathbb{R}^{2}$ given for $s \in \mathbb{R}$ by

$g^{s}\left(\begin{array}{l} u \\ x \end{array}\right)=\left(\begin{array}{l} u \cos s-x \sin s \\ u \sin s+x \cos s \end{array}\right)$

and hence, or otherwise, calculate the prolongation of order two of the vector field $V=-x \partial_{u}+u \partial_{x}$. Show that both of the equations $u_{x x}=0$ and $u_{x x}=\left(1+u_{x}^{2}\right)^{\frac{3}{2}}$ are invariant under this action of $S O(2)$, and interpret this geometrically.

(b) Show that the sine-Gordon equation

$\frac{\partial^{2} u}{\partial X \partial T}=\sin u$

admits the group $\left\{g^{s}\right\}_{s \in \mathbb{R}}$, where

$g^{s}:\left(\begin{array}{c} X \\ T \\ u \end{array}\right) \mapsto\left(\begin{array}{c} e^{s} X \\ e^{-s} T \\ u \end{array}\right)$

as a group of Lie point symmetries. Show that there is a group invariant solution of the form $u(X, T)=F(z)$ where $z$ is an invariant formed from the independent variables, and hence obtain a second order equation for $w=w(z)$ where $\exp [i F]=w$.

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• # Paper 1, Section II, C

Let $M=\mathbb{R}^{2 n}=\left\{(\mathbf{q}, \mathbf{p}) \mid \mathbf{q}, \mathbf{p} \in \mathbb{R}^{n}\right\}$ be equipped with its standard Poisson bracket.

(a) Given a Hamiltonian function $H=H(\mathbf{q}, \mathbf{p})$, write down Hamilton's equations for $(M, H)$. Define a first integral of the system and state what it means that the system is integrable.

(b) Show that if $n=1$ then every Hamiltonian system is integrable whenever

$\left(\frac{\partial H}{\partial q}, \frac{\partial H}{\partial p}\right) \neq \mathbf{0}$

Let $\tilde{M}=\mathbb{R}^{2 m}=\left\{(\tilde{\mathbf{q}}, \tilde{\mathbf{p}}) \mid \tilde{\mathbf{q}}, \tilde{\mathbf{p}} \in \mathbb{R}^{m}\right\}$ be another phase space, equipped with its standard Poisson bracket. Suppose that $\tilde{H}=\tilde{H}(\tilde{\mathbf{q}}, \tilde{\mathbf{p}})$ is a Hamiltonian function for $\tilde{M}$. Define $\mathbf{Q}=\left(q_{1}, \ldots, q_{n}, \tilde{q}_{1}, \ldots, \tilde{q}_{m}\right), \mathbf{P}=\left(p_{1}, \ldots, p_{n}, \tilde{p}_{1}, \ldots, \tilde{p}_{m}\right)$ and let the combined phase space $\mathcal{M}=\mathbb{R}^{2(n+m)}=\{(\mathbf{Q}, \mathbf{P})\}$ be equipped with the standard Poisson bracket.

(c) Show that if $(M, H)$ and $(\tilde{M}, \tilde{H})$ are both integrable, then so is $(\mathcal{M}, \mathcal{H})$, where the combined Hamiltonian is given by:

$\mathcal{H}(\mathbf{Q}, \mathbf{P})=H(\mathbf{q}, \mathbf{p})+\tilde{H}(\tilde{\mathbf{q}}, \tilde{\mathbf{p}})$

(d) Consider the $n$-dimensional simple harmonic oscillator with phase space $M$ and Hamiltonian $H$ given by:

$H=\frac{1}{2} p_{1}^{2}+\ldots+\frac{1}{2} p_{n}^{2}+\frac{1}{2} \omega_{1}^{2} q_{1}^{2}+\ldots+\frac{1}{2} \omega_{n}^{2} q_{n}^{2}$

where $\omega_{i}>0$. Using the results above, or otherwise, show that $(M, H)$ is integrable for $(\mathbf{q}, \mathbf{p}) \neq \mathbf{0}$.

(e) Is it true that every bounded orbit of an integrable system is necessarily periodic? You should justify your answer.

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• # Paper 2, Section II, C

Suppose $p=p(x)$ is a smooth, real-valued, function of $x \in \mathbb{R}$ which satisfies $p(x)>0$ for all $x$ and $p(x) \rightarrow 1, p_{x}(x), p_{x x}(x) \rightarrow 0$ as $|x| \rightarrow \infty$. Consider the Sturm-Liouville operator:

$L \psi:=-\frac{d}{d x}\left(p^{2} \frac{d \psi}{d x}\right)$

which acts on smooth, complex-valued, functions $\psi=\psi(x)$. You may assume that for any $k>0$ there exists a unique function $\varphi_{k}(x)$ which satisfies:

$L \varphi_{k}=k^{2} \varphi_{k}$

and has the asymptotic behaviour:

$\varphi_{k}(x) \sim \begin{cases}e^{-i k x} & \text { as } x \rightarrow-\infty \\ a(k) e^{-i k x}+b(k) e^{i k x} & \text { as } x \rightarrow+\infty\end{cases}$

(a) By analogy with the standard Schrödinger scattering problem, define the reflection and transmission coefficients: $R(k), T(k)$. Show that $|R(k)|^{2}+|T(k)|^{2}=1$. [Hint: You may wish to consider $W(x)=p(x)^{2}\left[\psi_{1}(x) \psi_{2}^{\prime}(x)-\psi_{2}(x) \psi_{1}^{\prime}(x)\right]$ for suitable functions $\psi_{1}$ and $\left.\psi_{2} \cdot\right]$

(b) Show that, if $\kappa>0$, there exists no non-trivial normalizable solution $\psi$ to the equation

$L \psi=-\kappa^{2} \psi$

Assume now that $p=p(x, t)$, such that $p(x, t)>0$ and $p(x, t) \rightarrow 1, p_{x}(x, t), p_{x x}(x, t) \rightarrow$ 0 as $|x| \rightarrow \infty$. You are given that the operator $A$ defined by:

$A \psi:=-4 p^{3} \frac{d^{3} \psi}{d x^{3}}-18 p^{2} p_{x} \frac{d^{2} \psi}{d x^{2}}-\left(12 p p_{x}^{2}+6 p^{2} p_{x x}\right) \frac{d \psi}{d x}$

satisfies:

$(L A-A L) \psi=-\frac{d}{d x}\left(2 p^{4} p_{x x x} \frac{d \psi}{d x}\right)$

(c) Show that $L, A$ form a Lax pair if the Harry Dym equation,

$p_{t}=p^{3} p_{x x x}$

is satisfied. [You may assume $L=L^{\dagger}, A=-A^{\dagger}$.]

(d) Assuming that $p$ solves the Harry Dym equation, find how the transmission and reflection amplitudes evolve as functions of $t$.

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• # Paper 3, Section II, C

Suppose $\psi^{s}:(x, u) \mapsto(\tilde{x}, \tilde{u})$ is a smooth one-parameter group of transformations acting on $\mathbb{R}^{2}$, with infinitesimal generator

$V=\xi(x, u) \frac{\partial}{\partial x}+\eta(x, u) \frac{\partial}{\partial u}$

(a) Define the $n^{\text {th }}$prolongation $\operatorname{Pr}^{(n)} V$ of $V$, and show that

$\operatorname{Pr}^{(n)} V=V+\sum_{i=1}^{n} \eta_{i} \frac{\partial}{\partial u^{(i)}}$

where you should give an explicit formula to determine the $\eta_{i}$ recursively in terms of $\xi$ and $\eta$.

(b) Find the $n^{t h}$ prolongation of each of the following generators:

$V_{1}=\frac{\partial}{\partial x}, \quad V_{2}=x \frac{\partial}{\partial x}, \quad V_{3}=x^{2} \frac{\partial}{\partial x}$

(c) Given a smooth, real-valued, function $u=u(x)$, the Schwarzian derivative is defined by,

$S=S[u]:=\frac{u_{x} u_{x x x}-\frac{3}{2} u_{x x}^{2}}{u_{x}^{2}}$

Show that,

$\operatorname{Pr}^{(3)} V_{i}(S)=c_{i} S,$

for $i=1,2,3$ where $c_{i}$ are real functions which you should determine. What can you deduce about the symmetries of the equations: (i) $S[u]=0$, (ii) $S[u]=1$, (iii) $S[u]=\frac{1}{x^{2}}$ ?

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• # Paper 1, Section II, A

Let $M=\mathbb{R}^{2 n}=\left\{(\mathbf{q}, \mathbf{p}) \mid \mathbf{q}, \mathbf{p} \in \mathbb{R}^{n}\right\}$ be equipped with the standard symplectic form so that the Poisson bracket is given by:

$\{f, g\}=\frac{\partial f}{\partial q_{j}} \frac{\partial g}{\partial p_{j}}-\frac{\partial f}{\partial p_{j}} \frac{\partial g}{\partial q_{j}}$

for $f, g$ real-valued functions on $M$. Let $H=H(\mathbf{q}, \mathbf{p})$ be a Hamiltonian function.

(a) Write down Hamilton's equations for $(M, H)$, define a first integral of the system and state what it means that the system is integrable.

(b) State the Arnol'd-Liouville theorem.

(c) Define complex coordinates $z_{j}$ by $z_{j}=q_{j}+i p_{j}$, and show that if $f, g$ are realvalued functions on $M$ then:

$\{f, g\}=-2 i \frac{\partial f}{\partial z_{j}} \frac{\partial g}{\partial \overline{z_{j}}}+2 i \frac{\partial g}{\partial z_{j}} \frac{\partial f}{\partial \bar{z}_{j}}$

(d) For an $n \times n$ anti-Hermitian matrix $A$ with components $A_{j k}$, let $I_{A}:=\frac{1}{2 i} \overline{z_{j}} A_{j k} z_{k}$. Show that:

$\left\{I_{A}, I_{B}\right\}=-I_{[A, B]},$

where $[A, B]=A B-B A$ is the usual matrix commutator.

(e) Consider the Hamiltonian:

$H=\frac{1}{2} \overline{z_{j}} z_{j}$

Show that $(M, H)$ is integrable and describe the invariant tori.

[In this question $j, k=1, \ldots, n$, and the summation convention is understood for these indices.]

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• # Paper 2, Section II, A

(a) Let $\mathcal{L}, \mathcal{A}$ be two families of linear operators, depending on a parameter $t$, which act on a Hilbert space $H$ with inner product $(,$, . Suppose further that for each $t, \mathcal{L}$ is self-adjoint and that $\mathcal{A}$ is anti-self-adjoint. State $L a x$ 's equation for the pair $\mathcal{L}, \mathcal{A}$, and show that if it holds then the eigenvalues of $\mathcal{L}$ are independent of $t$.

(b) For $\psi, \phi: \mathbb{R} \rightarrow \mathbb{C}$, define the inner product:

$(\psi, \phi):=\int_{-\infty}^{\infty} \overline{\psi(x)} \phi(x) d x$

Let $L, A$ be the operators:

$\begin{gathered} L \psi:=i \frac{d^{3} \psi}{d x^{3}}-i\left(q \frac{d \psi}{d x}+\frac{d}{d x}(q \psi)\right)+p \psi \\ A \psi:=3 i \frac{d^{2} \psi}{d x^{2}}-4 i q \psi \end{gathered}$

where $p=p(x, t), q=q(x, t)$ are smooth, real-valued functions. You may assume that the normalised eigenfunctions of $L$ are smooth functions of $x, t$, which decay rapidly as $|x| \rightarrow \infty$ for all $t$.

(i) Show that if $\psi, \phi$ are smooth and rapidly decaying towards infinity then:

$(L \psi, \phi)=(\psi, L \phi), \quad(A \psi, \phi)=-(\psi, A \phi)$

Deduce that the eigenvalues of $L$ are real.

(ii) Show that if Lax's equation holds for $L, A$, then $q$ must satisfy the Boussinesq equation:

$q_{t t}=a q_{x x x x}+b\left(q^{2}\right)_{x x}$

where $a, b$ are constants whose values you should determine. [You may assume without proof that the identity:

$L A \psi=A L \psi-3 i\left(p_{x} \frac{d \psi}{d x}+\frac{d}{d x}\left(p_{x} \psi\right)\right)+\left[q_{x x x}-4\left(q^{2}\right)_{x}\right] \psi$

holds for smooth, rapidly decaying $\psi .]$

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• # Paper 3, Section II, A

Suppose $\psi^{s}:(x, u) \mapsto(\tilde{x}, \tilde{u})$ is a smooth one-parameter group of transformations acting on $\mathbb{R}^{2}$.

(a) Define the generator of the transformation,

$V=\xi(x, u) \frac{\partial}{\partial x}+\eta(x, u) \frac{\partial}{\partial u}$

where you should specify $\xi$ and $\eta$ in terms of $\psi^{s}$.

(b) Define the $n^{\text {th }}$prolongation of $V, \operatorname{Pr}^{(n)} V$ and explicitly compute $\operatorname{Pr}^{(1)} V$ in terms of $\xi, \eta$.

Recall that if $\psi^{s}$ is a Lie point symmetry of the ordinary differential equation:

$\Delta\left(x, u, \frac{d u}{d x}, \ldots, \frac{d^{n} u}{d x^{n}}\right)=0$

then it follows that $\operatorname{Pr}^{(n)} V[\Delta]=0$ whenever $\Delta=0$.

(c) Consider the ordinary differential equation:

$\frac{d u}{d x}=F(x, u),$

for $F$ a smooth function. Show that if $V$ generates a Lie point symmetry of this equation, then:

$0=\eta_{x}+\left(\eta_{u}-\xi_{x}-F \xi_{u}\right) F-\xi F_{x}-\eta F_{u}$

(d) Find all the Lie point symmetries of the equation:

$\frac{d u}{d x}=x G\left(\frac{u}{x^{2}}\right)$

where $G$ is an arbitrary smooth function.

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• # Paper 1, Section II, A

Define a Lie point symmetry of the first order ordinary differential equation $\Delta[t, \mathbf{x}, \dot{\mathbf{x}}]=$ 0. Describe such a Lie point symmetry in terms of the vector field that generates it.

Consider the $2 n$-dimensional Hamiltonian system $(M, H)$ governed by the differential equation

$\frac{\mathrm{d} \mathbf{x}}{\mathrm{d} t}=J \frac{\partial H}{\partial \mathbf{x}}$

Define the Poisson bracket $\{\cdot, \cdot\}$. For smooth functions $f, g: M \rightarrow \mathbf{R}$ show that the associated Hamiltonian vector fields $V_{f}, V_{g}$ satisfy

$\left[V_{f}, V_{g}\right]=-V_{\{f, g\}} .$

If $F: M \rightarrow \mathbf{R}$ is a first integral of $(M, H)$, show that the Hamiltonian vector field $V_{F}$ generates a Lie point symmetry of $(\star)$. Prove the converse is also true if $(\star)$ has a fixed point, i.e. a solution of the form $\mathbf{x}(t)=\mathbf{x}_{0}$.

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• # Paper 2, Section II, A

Let $U$ and $V$ be non-singular $N \times N$ matrices depending on $(x, t, \lambda)$ which are periodic in $x$ with period $2 \pi$. Consider the associated linear problem

$\Psi_{x}=U \Psi, \quad \Psi_{t}=V \Psi$

for the vector $\Psi=\Psi(x, t ; \lambda)$. On the assumption that these equations are compatible, derive the zero curvature equation for $(U, V)$.

Let $W=W(x, t, \lambda)$ denote the $N \times N$ matrix satisfying

$W_{x}=U W, \quad W(0, t, \lambda)=I_{N}$

where $I_{N}$ is the $N \times N$ identity matrix. You should assume $W$ is unique. By considering $\left(W_{t}-V W\right)_{x}$, show that the matrix $w(t, \lambda)=W(2 \pi, t, \lambda)$ satisfies the Lax equation

$w_{t}=[v, w], \quad v(t, \lambda) \equiv V(2 \pi, t, \lambda)$

Deduce that $\left\{\operatorname{tr}\left(w^{k}\right)\right\}_{k \geqslant 1}$ are first integrals.

By considering the matrices

$\frac{1}{2 \mathrm{i} \lambda}\left[\begin{array}{cc} \cos u & -\mathrm{i} \sin u \\ \mathrm{i} \sin u & -\cos u \end{array}\right], \quad \frac{\mathrm{i}}{2}\left[\begin{array}{cc} 2 \lambda & u_{x} \\ u_{x} & -2 \lambda \end{array}\right]$

show that the periodic Sine-Gordon equation $u_{x t}=\sin u$ has infinitely many first integrals. [You need not prove anything about independence.]

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• # Paper 3, Section II, A

Let $u=u(x, t)$ be a smooth solution to the $\mathrm{KdV}$ equation

$u_{t}+u_{x x x}-6 u u_{x}=0$

which decays rapidly as $|x| \rightarrow \infty$ and let $L=-\partial_{x}^{2}+u$ be the associated Schrödinger operator. You may assume $L$ and $A=4 \partial_{x}^{3}-3\left(u \partial_{x}+\partial_{x} u\right)$ constitute a Lax pair for KdV.

Consider a solution to $L \varphi=k^{2} \varphi$ which has the asymptotic form

$\varphi(x, k, t)= \begin{cases}e^{-\mathrm{i} k x}, & \text { as } x \rightarrow-\infty \\ a(k, t) e^{-\mathrm{i} k x}+b(k, t) e^{\mathrm{i} k x}, & \text { as } x \rightarrow+\infty\end{cases}$

Find evolution equations for $a$ and $b$. Deduce that $a(k, t)$ is $t$-independent.

By writing $\varphi$ in the form

$\varphi(x, k, t)=\exp \left[-\mathrm{i} k x+\int_{-\infty}^{x} S(y, k, t) \mathrm{d} y\right], \quad S(x, k, t)=\sum_{n=1}^{\infty} \frac{S_{n}(x, t)}{(2 \mathrm{i} k)^{n}}$

show that

$a(k, t)=\exp \left[\int_{-\infty}^{\infty} S(x, k, t) \mathrm{d} x\right]$

Deduce that $\left\{\int_{-\infty}^{\infty} S_{n}(x, t) \mathrm{d} x\right\}_{n=1}^{\infty}$ are first integrals of KdV.

By writing a differential equation for $S=X+\mathrm{i} Y$ (with $X, Y$ real), show that these first integrals are trivial when $n$ is even.

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• # Paper 1, Section II, D

What does it mean for an evolution equation $u_{t}=K\left(x, u, u_{x}, \ldots\right)$ to be in Hamiltonian form? Define the associated Poisson bracket.

An evolution equation $u_{t}=K\left(x, u, u_{x}, \ldots\right)$ is said to be bi-Hamiltonian if it can be written in Hamiltonian form in two distinct ways, i.e.

$K=\mathcal{J} \delta H_{0}=\mathcal{E} \delta H_{1}$

for Hamiltonian operators $\mathcal{J}, \mathcal{E}$ and functionals $H_{0}, H_{1}$. By considering the sequence $\left\{H_{m}\right\}_{m \geqslant 0}$ defined by the recurrence relation

$\mathcal{E} \delta H_{m+1}=\mathcal{J} \delta H_{m}$

show that bi-Hamiltonian systems possess infinitely many first integrals in involution. [You may assume that $(*)$ can always be solved for $H_{m+1}$, given $H_{m}$.]

The Harry Dym equation for the function $u=u(x, t)$ is

$u_{t}=\frac{\partial^{3}}{\partial x^{3}}\left(u^{-1 / 2}\right)$

This equation can be written in Hamiltonian form $u_{t}=\mathcal{E} \delta H_{1}$ with

$\mathcal{E}=2 u \frac{\partial}{\partial x}+u_{x}, \quad H_{1}[u]=\frac{1}{8} \int u^{-5 / 2} u_{x}^{2} \mathrm{~d} x$

Show that the Harry Dym equation possesses infinitely many first integrals in involution. [You need not verify the Jacobi identity if your argument involves a Hamiltonian operator.]

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• # Paper 2, Section II, D

What does it mean for $g^{\epsilon}:(x, u) \mapsto(\tilde{x}, \tilde{u})$ to describe a 1-parameter group of transformations? Explain how to compute the vector field

$V=\xi(x, u) \frac{\partial}{\partial x}+\eta(x, u) \frac{\partial}{\partial u}$

that generates such a 1-parameter group of transformations.

Suppose now $u=u(x)$. Define the $n$th prolongation, $\mathrm{pr}^{(n)} g^{\epsilon}$, of $g^{\epsilon}$ and the vector field which generates it. If $V$ is defined by $(*)$ show that

$\mathrm{pr}^{(n)} V=V+\sum_{k=1}^{n} \eta_{k} \frac{\partial}{\partial u^{(k)}}$

where $u^{(k)}=\mathrm{d}^{k} u / \mathrm{d} x^{k}$ and $\eta_{k}$ are functions to be determined.

The curvature of the curve $u=u(x)$ in the $(x, u)$-plane is given by

$\kappa=\frac{u_{x x}}{\left(1+u_{x}^{2}\right)^{3 / 2}}$

Rotations in the $(x, u)$-plane are generated by the vector field

$W=x \frac{\partial}{\partial u}-u \frac{\partial}{\partial x}$

Show that the curvature $\kappa$ at a point along a plane curve is invariant under such rotations. Find two further transformations that leave $\kappa$ invariant.

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• # Paper 3, Section II, D

What is meant by an auto-Bäcklund transformation?

The sine-Gordon equation in light-cone coordinates is

$\frac{\partial^{2} \varphi}{\partial \xi \partial \tau}=\sin \varphi$

where $\xi=\frac{1}{2}(x-t), \tau=\frac{1}{2}(x+t)$ and $\varphi$ is to be understood modulo $2 \pi$. Show that the pair of equations

$\partial_{\xi}\left(\varphi_{1}-\varphi_{0}\right)=2 \epsilon \sin \left(\frac{\varphi_{1}+\varphi_{0}}{2}\right), \quad \partial_{\tau}\left(\varphi_{1}+\varphi_{0}\right)=\frac{2}{\epsilon} \sin \left(\frac{\varphi_{1}-\varphi_{0}}{2}\right)$

constitute an auto-Bäcklund transformation for (1).

By noting that $\varphi=0$ is a solution to (1), use the transformation (2) to derive the soliton (or 'kink') solution to the sine-Gordon equation. Show that this solution can be expressed as

$\varphi(x, t)=4 \arctan \left[\exp \left(\pm \frac{x-c t}{\sqrt{1-c^{2}}}+x_{0}\right)\right]$

for appropriate constants $c$ and $x_{0}$.

[Hint: You may use the fact that $\int \operatorname{cosec} x \mathrm{~d} x=\log \tan (x / 2)+$ const.]

The following function is a solution to the sine-Gordon equation:

$\varphi(x, t)=4 \arctan \left[c \frac{\sinh \left(x / \sqrt{1-c^{2}}\right)}{\cosh \left(c t / \sqrt{1-c^{2}}\right)}\right] \quad(c>0) .$

Verify that this represents two solitons travelling towards each other at the same speed by considering $x \pm c t=$ constant and taking an appropriate limit.

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• # Paper 1, Section II, D

Let $u_{t}=K\left(x, u, u_{x}, \ldots\right)$ be an evolution equation for the function $u=u(x, t)$. Assume $u$ and all its derivatives decay rapidly as $|x| \rightarrow \infty$. What does it mean to say that the evolution equation for $u$ can be written in Hamiltonian form?

The modified KdV (mKdV) equation for $u$ is

$u_{t}+u_{x x x}-6 u^{2} u_{x}=0 .$

Show that small amplitude solutions to this equation are dispersive.

Demonstrate that the mKdV equation can be written in Hamiltonian form and define the associated Poisson bracket $\{,$,} on the space of functionals of u. Verify that the Poisson bracket is linear in each argument and anti-symmetric.

Show that a functional $I=I[u]$ is a first integral of the mKdV equation if and only if $\{I, H\}=0$, where $H=H[u]$ is the Hamiltonian.

Show that if $u$ satisfies the mKdV equation then

$\frac{\partial}{\partial t}\left(u^{2}\right)+\frac{\partial}{\partial x}\left(2 u u_{x x}-u_{x}^{2}-3 u^{4}\right)=0$

Using this equation, show that the functional

$I[u]=\int u^{2} d x$

Poisson-commutes with the Hamiltonian.

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• # Paper 2, Section II, D

(a) Explain how a vector field

$V=\xi(x, u) \frac{\partial}{\partial x}+\eta(x, u) \frac{\partial}{\partial u}$

generates a 1-parameter group of transformations $g^{\epsilon}:(x, u) \mapsto(\tilde{x}, \tilde{u})$ in terms of the solution to an appropriate differential equation. [You may assume the solution to the relevant equation exists and is unique.]

(b) Suppose now that $u=u(x)$. Define what is meant by a Lie-point symmetry of the ordinary differential equation

$\Delta\left[x, u, u^{(1)}, \ldots, u^{(n)}\right]=0, \quad \text { where } \quad u^{(k)} \equiv \frac{d^{k} u}{d x^{k}}, \quad k=1, \ldots, n$

(c) Prove that every homogeneous, linear ordinary differential equation for $u=u(x)$ admits a Lie-point symmetry generated by the vector field

$V=u \frac{\partial}{\partial u}$

By introducing new coordinates

$s=s(x, u), \quad t=t(x, u)$

which satisfy $V(s)=1$ and $V(t)=0$, show that every differential equation of the form

$\frac{d^{2} u}{d x^{2}}+p(x) \frac{d u}{d x}+q(x) u=0$

can be reduced to a first-order differential equation for an appropriate function.

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• # Paper 3, Section II, D

Let $L=L(t)$ and $A=A(t)$ be real $N \times N$ matrices, with $L$ symmetric and $A$ antisymmetric. Suppose that

$\frac{d L}{d t}=L A-A L$

Show that all eigenvalues of the matrix $L(t)$ are $t$-independent. Deduce that the coefficients of the polynomial

$P(x)=\operatorname{det}(x I-L(t))$

are first integrals of the system.

What does it mean for a $2 n$-dimensional Hamiltonian system to be integrable? Consider the Toda system with coordinates $\left(q_{1}, q_{2}, q_{3}\right)$ obeying

$\frac{d^{2} q_{i}}{d t^{2}}=\mathrm{e}^{q_{i-1}-q_{i}}-\mathrm{e}^{q_{i}-q_{i+1}}, \quad i=1,2,3$

where here and throughout the subscripts are to be determined modulo 3 so that $q_{4} \equiv q_{1}$ and $q_{0} \equiv q_{3}$. Show that

$H\left(q_{i}, p_{i}\right)=\frac{1}{2} \sum_{i=1}^{3} p_{i}^{2}+\sum_{i=1}^{3} \mathrm{e}^{q_{i}-q_{i+1}}$

is a Hamiltonian for the Toda system.

Set $a_{i}=\frac{1}{2} \exp \left(\frac{q_{i}-q_{i+1}}{2}\right)$ and $b_{i}=-\frac{1}{2} p_{i}$. Show that

$\frac{d a_{i}}{d t}=\left(b_{i+1}-b_{i}\right) a_{i}, \quad \frac{d b_{i}}{d t}=2\left(a_{i}^{2}-a_{i-1}^{2}\right), \quad i=1,2,3$

Is this coordinate transformation canonical?

By considering the matrices

$L=\left(\begin{array}{lll} b_{1} & a_{1} & a_{3} \\ a_{1} & b_{2} & a_{2} \\ a_{3} & a_{2} & b_{3} \end{array}\right), \quad A=\left(\begin{array}{ccc} 0 & -a_{1} & a_{3} \\ a_{1} & 0 & -a_{2} \\ -a_{3} & a_{2} & 0 \end{array}\right)$

or otherwise, compute three independent first integrals of the Toda system. [Proof of independence is not required.]

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• # Paper 1, Section II, D

Consider the coordinate transformation

$g^{\epsilon}:(x, u) \mapsto(\tilde{x}, \tilde{u})=(x \cos \epsilon-u \sin \epsilon, x \sin \epsilon+u \cos \epsilon)$

Show that $g^{\epsilon}$ defines a one-parameter group of transformations. Define what is meant by the generator $V$ of a one-parameter group of transformations and compute it for the above case.

Now suppose $u=u(x)$. Explain what is meant by the first prolongation $\mathrm{pr}^{(1)} g^{\epsilon}$ of $g^{\epsilon}$. Compute $\mathrm{pr}^{(1)} g^{\epsilon}$ in this case and deduce that

$\mathrm{pr}^{(1)} V=V+\left(1+u_{x}^{2}\right) \frac{\partial}{\partial u_{x}}$

Similarly find $\mathrm{pr}^{(2)} V$.

Define what is meant by a Lie point symmetry of the first-order differential equation $\Delta\left[x, u, u_{x}\right]=0$. Describe this condition in terms of the vector field that generates the Lie point symmetry. Consider the case

$\Delta\left[x, u, u_{x}\right] \equiv u_{x}-\frac{u+x f\left(x^{2}+u^{2}\right)}{x-u f\left(x^{2}+u^{2}\right)}$

where $f$ is an arbitrary smooth function of one variable. Using $(\star)$, show that $g^{\epsilon}$ generates a Lie point symmetry of the corresponding differential equation.

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• # Paper 2, Section II, D

Let $u=u(x)$ be a smooth function that decays rapidly as $|x| \rightarrow \infty$ and let $L=-\partial_{x}^{2}+u(x)$ denote the associated Schrödinger operator. Explain very briefly each of the terms appearing in the scattering data

$S=\left\{\left\{\chi_{n}, c_{n}\right\}_{n=1}^{N}, R(k)\right\},$