Problems to Section 2.3

$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$

### Problems

Problem 1. Find the general solutions of \begin{gather} u_{tt}-u_{xx}=0; \\ u_{tt}-4u_{xx}=0; \\ u_{tt}-9u_{xx}=0; \\ 4u_{tt}-u_{xx}=0; \\ 4u_{tt}-9u_{xx}=0. \end{gather}

Problem 2. Solve IVP \begin{align} &u_{tt}-c^2u_{xx}=0, \\ &u|_{t=0}=g(x),\quad u_t|_{t=0}=h(x) \end{align} with \begin{align*} &g(x)=0,\ && h(x)=1; \\ &g(x)=\left\{\begin{aligned} &0 &&x < 0,\\ &1 &&x \ge 0. \end{aligned}\right. &&h(x)=0; \\ &g(x)=\left\{\begin{aligned} &1 &&|x| < 1,\\ &0 &&|x| \ge 1. \end{aligned}\right. &&h(x)=0; \\ &g(x)=\left\{\begin{aligned} &\cos (x) &&|x| < \pi/2,\\ &0 &&|x| \ge \pi/2. \end{aligned}\right. &&h(x)=0; \\ &g(x)=0, &&h(x)=\left\{\begin{aligned} &0 &&x < 0,\\ &1 &&x \ge 0. \end{aligned}\right. \\ &g(x)=0, &&h(x)=\left\{\begin{aligned} &1 &&|x| < 1,\\ &0 &&|x| \ge 1. \end{aligned}\right. \\ &g(x)=0, &&h(x)=\left\{\begin{aligned} &\cos (x) &&|x| < \pi/2,\\ &0 &&|x| \ge \pi/2. \end{aligned}\right. \end{align*}

Problem 3. Find solution to equation \begin{equation} Au_{tt}+2Bu_{tx}+Cu_{xx}=0 \end{equation} as \begin{equation} u=f(x-c_1 t)+ g(x-c_2t) \end{equation} with arbitrary $f,g$ and real $c_1< c_2$.

1. What equation should satisfy $c_1$ and $c_2$?
2. When this equation has such roots?

Problem 4.

A spherical wave is a solution of the three-dimensional wave equation of the form $u(r, t)$, where r is the distance to the origin (the spherical coordinate). The wave equation takes the form \begin{equation} u_{tt} = c^2 \bigl(u_{rr}+\frac{2}{r}u_r\bigr) \qquad\text{("spherical wave equation").} \label{eq-4a} \end{equation}

1. Change variables $v = ru$ to get the equation for $v$: $v_{tt} = c^2 v_{rr}$.
2. Solve for $v$ using \begin{equation} v = f(r+ct)+g(r-ct) \end{equation} and thereby solve the spherical wave equation.
3. Use \begin{equation} v(r,t)=\frac{1}{2}\bigl[ \phi (r+ct)+\phi (r-ct)\bigr]+\frac{1}{2c}\int_{r-ct}^{r+ct}\psi (s) \,ds \end{equation} with $\phi(r)=v(r,0)$, $\psi(r)=v_t(r,0)$ to solve it with initial conditions $u(r, 0) = \Phi (r)$, $u_t(r, 0) = \Psi(r)$.
4. Find the general form of solution $u$ to (\ref{eq-4a}) which is continuous as $r=0$.

Problem 5. Find formula for solution of the Goursat problem \begin{align} &u_{tt} - c^2 u_{xx}=0, && x > c|t|, \\ &u|_{x=-ct}=g(t), && t<0, \\ &u|_{x=ct}=h(t), &&t > 0 \end{align} as $g(0)=h(0)$.

Problem 6. Often solution in the form of travelling wave $u=\phi (x-vt)$ is sought for more general equations. Here we are interested in the bounded solutions, especially in those with $\phi(x)$ either tending to $0$ as $|x|\to \infty$ (solitons) or periodic (kinks). Plugging such solution to equation we get ODE for function $\phi$, which could be either solved or at least explored. Sure we are not interested in the trivial solution which is identically equal to $0$.

1. Find such solutions for each of the following equations \begin{align} &u_{tt}-c^2 u_{xx}+m^2 u=0;\\ &u_{tt}-c^2 u_{xx}-m^2 u=0; \end{align} the former is Klein-Gordon equation. Describe all possible velocities $v$.
2. Find such solutions for each of the following equations \begin{align} &u_{t}-K u_{xxx}=0;\\ &u_{t} - iKu_{xx} =0;\\ &u_{tt}+Ku_{xxxx}=0; \end{align}

Problem 7. Look for solutions in the form of travelling wave for sine-Gordon equation \begin{equation} u_{tt}-c^2 u_{xx}+\sin(u)=0. \end{equation} observe that resulting ODE is describing mathematical pendulum which could be explored. Describe all possible velocities $v$.

Problem 8. Look for solutions in the form of travelling wave for each of the following equations \begin{align} u_{tt}-u_{xx}+u -2u^3=0; \\ u_{tt}-u_{xx}-u +2u^3=0; \end{align}

1. Describe kinks. Describe all possible velocities $v$.
2. Find solitons. Describe all possible velocities $v$.

Problem 9. For a solution $u(x, t)$ of the wave equation $u_{tt}=c^2u_{xx}$, the energy density is defined as $e=\frac{1}{2}\bigl(u_t^2+c^2 u_x^2\bigr)$ and the momentum density as $p =c u_t u_x$.

1. Show that \begin{equation} \frac{\partial e}{\partial t} = c\frac{\partial p}{\partial x} \qquad \text{and} \qquad \frac{\partial p}{\partial t} = c\frac{\partial e}{\partial x}. \label{eq-HA2.11} \end{equation}
2. Show that both $e(x, t)$ and $p(x,t)$ also satisfy the same wave equation.

Problem 10.

1. Consider wave equation $u_{tt}-u_{xx}=0$ in the rectangle $0< x< a$, $0< t < b$ and prove that if $a$ and $b$ are not commensurable (i.e. $a:b$ is not rational) then Dirichlet problem $u|_{t=0}=u_{t=b}=u|_{x=0}=u|_{x=a}=0$ has only trivial solution.
2. On the other hand, prove that if $a$ and $b$ are commensurable then there exists a nontrivial solution $u=\sin (px/a)\sin (qt/b)$.

Problem 11.

Generalize Problem 4:

A spherical wave is a solution of the $n$-dimensional wave equation of the form $u(r, t)$, where r is the distance to the origin (the spherical coordinate). The wave equation takes the form \begin{equation} u_{tt} = c^2 \bigl(u_{rr}+\frac{n-1}{r}u_r\bigr) \qquad\text{("spherical wave equation").} \label{eq-4b} \end{equation}

1. Show that if $u$ satisfies (\ref{eq-4b}) then $r^{-1}\partial_r u(r,t)$ also satisfies (\ref{eq-4b}) but with $n$ replaced by $n+2$.
2. Using this and Problem 4 write down spherical wave for odd $n$.
3. Describe spherical wave for $n=1$.

Remark 1. For even $n$ spherical waves do not exist