• A2.18

(i) Let $u(x, t)$ satisfy the Burgers equation

$\frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}=\nu \frac{\partial^{2} u}{\partial x^{2}}$

where $\nu$ is a positive constant. Consider solutions of the form $u=u(X)$, where $X=x-U t$ and $U$ is a constant, such that

$u \rightarrow u_{2}, \quad \frac{\partial u}{\partial X} \rightarrow 0 \quad \text { as } \quad X \rightarrow-\infty ; \quad u \rightarrow u_{1}, \quad \frac{\partial u}{\partial X} \rightarrow 0 \quad \text { as } \quad X \rightarrow \infty$

with $u_{2}>u_{1}$.

Show that $U$ satisfies the so-called shock condition

$U=\frac{1}{2}\left(u_{2}+u_{1}\right)$

By using the factorisation

$\frac{1}{2} u^{2}-U u+A=\frac{1}{2}\left(u-u_{1}\right)\left(u-u_{2}\right)$

where $A$ is the constant of integration, express $u$ in terms of $X, u_{1}, u_{2}$ and $\nu$.

(ii) According to shallow-water theory, river waves are characterised by the PDEs

$\begin{gathered} \frac{\partial v}{\partial t}+v \frac{\partial v}{\partial x}+g \cos \alpha \frac{\partial h}{\partial x}=g \sin \alpha-C_{f} \frac{v^{2}}{h} \\ \frac{\partial h}{\partial t}+v \frac{\partial h}{\partial x}+h \frac{\partial v}{\partial x}=0 \end{gathered}$

where $h(x, t)$ denotes the depth of the river, $v(x, t)$ denotes the mean velocity, $\alpha$ is the constant angle of inclination, and $C_{f}$ is the constant friction coefficient.

Find the characteristic velocities and the characteristic form of the equations. Find the Riemann variables and show that if $C_{f}=0$ then the Riemann variables vary linearly with $t$ on the characteristics.

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• A3.18

(i) Let $\Phi^{+}(t)$ and $\Phi^{-}(t)$ denote the boundary values of functions which are analytic inside and outside the unit disc centred on the origin, respectively. Let $C$ denote the boundary of this disc. Suppose that $\Phi^{+}(t)$ and $\Phi^{-}(t)$ satisfy the jump condition

$\Phi^{+}(t)=t^{-2} \Phi^{-}(t)+t^{-1}+\alpha\left(t^{-1}+t-t^{-3}\right), \quad t \in C,$

where $\alpha$ is a constant.

Find the canonical solution of the associated homogeneous Riemann-Hilbert problem. Write down the orthogonality conditions.

(ii) Consider the linear singular integral equation

$\left(t+t^{-1}\right) \psi(t)+\frac{t-t^{-1}}{\pi i} \oint_{C} \frac{\psi(\tau)}{\tau-t} d \tau=2+2 \alpha\left(1+t^{2}-t^{-2}\right)$

where $\oint$ denotes the principal value integral.

Show that the associated Riemann-Hilbert problem has the jump condition defined in Part (i) above. Using this fact, find the value of the constant $\alpha$ that allows equation $(*)$ to have a solution. For this particular value of $\alpha$ find the unique solution $\psi(t)$.

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• A4.23

Let $\psi(k ; x, t)$ satisfy the linear integral equation

$\psi(k ; x, t)+i e^{i\left(k x+k^{3} t\right)} \int_{L} \frac{\psi(l ; x, t)}{l+k} d \lambda(l)=e^{i\left(k x+k^{3} t\right)}$

where the measure $d \lambda(k)$ and the contour $L$ are such that $\psi(k ; x, t)$ exists and is unique.

Let $q(x, t)$ be defined in terms of $\psi(k ; x, t)$ by

$q(x, t)=-\frac{\partial}{\partial x} \int_{L} \psi(k ; x, t) d \lambda(k)$

(a) Show that

$(M \psi)+i e^{i\left(k x+k^{3} t\right)} \int_{L} \frac{(M \psi)(l ; x, t)}{l+k} d \lambda(l)=0$

where

$M \psi \equiv \frac{\partial^{2} \psi}{\partial x^{2}}-i k \frac{\partial \psi}{\partial x}+q \psi$

(b) Show that

$(N \psi)+i e^{i\left(k x+k^{3} t\right)} \int_{L} \frac{(N \psi)(l ; x, t)}{l+k} d \lambda(l)=3 k e^{i\left(k x+k^{3} t\right)} \int_{L} \frac{(M \psi)(l ; x, t)}{l+k} d \lambda(l)$,

where

$N \psi \equiv \frac{\partial \psi}{\partial t}+\frac{\partial^{3} \psi}{\partial x^{3}}+3 q \frac{\partial \psi}{\partial x}$

(c) By recalling that the $\mathrm{KdV}$ equation

$\frac{\partial q}{\partial t}+\frac{\partial^{3} q}{\partial x^{3}}+6 q \frac{\partial q}{\partial x}=0$

$M \psi=0, \quad N \psi=0,$

write down an expression for $d \lambda(l)$ which gives rise to the one-soliton solution of the $\mathrm{KdV}$ equation. Write down an expression for $\psi(k ; x, t)$ and for $q(x, t)$.

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• A2.18

(i) Find a travelling wave solution of unchanging shape for the modified Burgers equation (with $\alpha>0$ )

$\frac{\partial u}{\partial t}+u^{2} \frac{\partial u}{\partial x}=\alpha \frac{\partial^{2} u}{\partial x^{2}}$

with $u=0$ far ahead of the wave and $u=1$ far behind. What is the velocity of the wave? Sketch the shape of the wave.

(ii) Explain why the method of characteristics, when applied to an equation of the type

$\frac{\partial u}{\partial t}+c(u) \frac{\partial u}{\partial x}=0$

with initial data $u(x, 0)=f(x)$, sometimes gives a multi-valued solution. State the shockfitting algorithm that gives a single-valued solution, and explain how it is justified.

Consider the equation above, with $c(u)=u^{2}$. Suppose that

$u(x, 0)=\left\{\begin{array}{ll} 0 & x \geq 0 \\ 1 & x<0 \end{array} .\right.$

Sketch the characteristics in the $(x, t)$ plane. Show that a shock forms immediately, and calculate the velocity at which it moves.

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• A3.18

(i) Show that the equation

$\frac{\partial \phi}{\partial t}-\frac{\partial^{2} \phi}{\partial x^{2}}+1-\phi^{2}=0$

has two solutions which are independent of both $x$ and $t$. Show that one of these is linearly stable. Show that the other solution is linearly unstable, and find the range of wavenumbers that exhibit the instability.

Sketch the nonlinear evolution of the unstable solution after it receives a small, smooth, localized perturbation in the direction towards the stable solution.

(ii) Show that the equations

$\begin{array}{r} \frac{\partial u}{\partial x}+\frac{\partial v}{\partial x}=e^{-u+v} \\ -\frac{\partial u}{\partial y}+\frac{\partial v}{\partial y}=e^{-u-v} \end{array}$

are a Bäcklund pair for the equations

$\frac{\partial^{2} u}{\partial x \partial y}=e^{-2 u}, \quad \frac{\partial^{2} v}{\partial x \partial y}=0$

By choosing $v$ to be a suitable constant, and using the Bäcklund pair, find a solution of the equation

$\frac{\partial^{2} u}{\partial x \partial y}=e^{-2 u}$

which is non-singular in the region $y<4 x$ of the $(x, y)$ plane and has the value $u=0$ at $x=\frac{1}{2}, y=0$.

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• A2.18

(i) Establish two conservation laws for the $\mathrm{MKdV}$ equation

$\frac{\partial u}{\partial t}+u^{2} \frac{\partial u}{\partial x}+\frac{\partial^{3} u}{\partial x^{3}}=0$

State sufficient boundary conditions that $u$ should satisfy for the conservation laws to be valid.

(ii) The equation

$\frac{\partial \rho}{\partial t}+\frac{\partial}{\partial x}(\rho V)=0$

models traffic flow on a single-lane road, where $\rho(x, t)$ represents the density of cars, and $V$ is a given function of $\rho$. By considering the rate of change of the integral

$\int_{a}^{b} \rho d x,$

show that $V$ represents the velocity of the cars.

Suppose now that $V=1-\rho$ (in suitable units), and that $0 \leqslant \rho \leqslant 1$ everywhere. Assume that a queue is building up at a traffic light at $x=1$, so that, when the light turns green at $t=0$,

$\rho(x, 0)=\left\{\begin{array}{l} 0 \text { for } x<0 \text { and } x>1 \\ x \text { for } 0 \leqslant x<1 . \end{array}\right.$

For this problem, find and sketch the characteristics in the $(x, t)$ plane, for $t>0$, paying particular attention to those emerging from the point $(1,0)$. Show that a shock forms at $t=\frac{1}{2}$. Find the density of cars $\rho(x, t)$ for $0, and all $x$.

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• A3.18

(i) The so-called breather solution of the sine-Gordon equation is

$\phi(x, t)=4 \tan ^{-1}\left(\frac{\left(1-\lambda^{2}\right)^{\frac{1}{2}}}{\lambda} \frac{\sin \lambda t}{\cosh \left(1-\lambda^{2}\right)^{\frac{1}{2}} x}\right), \quad 0<\lambda<1$

Describe qualitatively the behaviour of $\phi(x, t)$, for $\lambda \ll 1$, when $|x| \gg \ln (2 / \lambda)$, when $|x| \ll 1$, and when $\cosh x \approx \frac{1}{\lambda}|\sin \lambda t|$. Explain how this solution can be interpreted in terms of motion of a kink and an antikink. Estimate the greatest separation of the kink and antikink.

(ii) The field $\psi(x, t)$ obeys the nonlinear wave equation

$\frac{\partial^{2} \psi}{\partial t^{2}}-\frac{\partial^{2} \psi}{\partial x^{2}}+\frac{d U}{d \psi}=0$

where the potential $U$ has the form

$U(\psi)=\frac{1}{2}\left(\psi-\psi^{3}\right)^{2} .$

Show that $\psi=0$ and $\psi=1$ are stable constant solutions.

Find a steady wave solution $\psi=f(x-v t)$ satisfying the boundary conditions $\psi \rightarrow 0$ as $x \rightarrow-\infty, \psi \rightarrow 1$ as $x \rightarrow \infty$. What constraint is there on the velocity $v ?$

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