Nonlinear Waves And Integrable Systems

2003

• A2.18

  Nonlinear Waves and Integrable Systems | Part II, 2003

  (i) Write down the shock condition associated with the equation

  $$\rho_{t}+q_{x}=0$$

  where $q=q(\rho)$. Discuss briefly two possible heuristic approaches to justifying this shock condition.

  (ii) According to shallow water theory, waves on a uniformly sloping beach are described by the equations

  $$\begin{aligned} &\frac{\partial h}{\partial t}+\frac{\partial}{\partial x}(h u)=0, \\ &\frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}+g \frac{\partial \eta}{\partial x}=0, \quad h=\alpha x+\eta, \end{aligned}$$

  where $\alpha$ is the constant slope of the beach, $g$ is the gravitational acceleration, $u(x, t)$ is the fluid velocity, and $\eta(x, t)$ is the elevation of the fluid surface above the undisturbed level.

  Find the characteristic velocities and the characteristic form of the equations.

  What are the Riemann variables and how do they vary with $t$ on the characteristics?

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• A3.18

  Nonlinear Waves and Integrable Systems | Part II, 2003

  (i) Write down a Lax pair for the equation

  $$i q_{t}+q_{x x}=0 .$$

  Discuss briefly, without giving mathematical details, how this pair can be used to solve the Cauchy problem on the infinite line for this equation. Discuss how this approach can be used to solve the analogous problem for the nonlinear Schrödinger equation.

  (ii) Let $q(\zeta, \eta), \tilde{q}(\zeta, \eta)$ satisfy the equations

  $$\begin{aligned} &\tilde{q}_{\zeta}=q_{\zeta}+2 \lambda \sin \frac{\tilde{q}+q}{2} \\ &\tilde{q}_{\eta}=-q_{\eta}+\frac{2}{\lambda} \sin \frac{\tilde{q}-q}{2} \end{aligned}$$

  where $\lambda$ is a constant.

  (a) Show that the above equations are compatible provided that $q, \tilde{q}$ both satisfy the Sine-Gordon equation

  $$q_{\zeta \eta}=\sin q$$

  (b) Use the above result together with the fact that

  $$\int \frac{d x}{\sin x}=\ln \left(\tan \frac{x}{2}\right)+\text { constant }$$

  to show that the one-soliton solution of the Sine-Gordon equation is given by

  $$\tan \frac{q}{4}=c \exp \left(\lambda \zeta+\frac{\eta}{\lambda}\right)$$

  where $c$ is a constant.

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• A4.22

  Nonlinear Waves and Integrable Systems | Part II, 2003

  Let $\Phi^{+}(t), \Phi^{-}(t)$ denote the boundary values of functions which are analytic inside and outside a disc of radius $\frac{1}{2}$ centred at the origin. Let $C$ denote the boundary of this disc.

  Suppose that $\Phi^{+}, \Phi^{-}$satisfy the jump condition

  $$\Phi^{+}(t)=\frac{t}{t^{2}-1} \Phi^{-}(t)+\frac{t^{3}-t^{2}+1}{t^{2}-t}, \quad t \in C .$$

  (a) Show that the associated index is 1 .

  (b) Find the canonical solution of the homogeneous problem, i.e. the solution satisfying

  $$X(z) \sim z^{-1}, \quad z \rightarrow \infty$$

  (c) Find the general solution of the Riemann-Hilbert problem satisfying the above jump condition as well as

  $$\Phi(z)=O\left(z^{-1}\right), \quad z \rightarrow \infty$$

  (d) Use the above result to solve the linear singular integral problem

  $$\left(t^{2}+t-1\right) \phi(t)+\frac{t^{2}-t-1}{\pi i} \oint_{C} \frac{\phi(\tau)}{\tau-t} d \tau=\frac{2\left(t^{3}-t^{2}+1\right)(t+1)}{t}, \quad t \in C .$$

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