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 \left\{\begin{aligned} &u_{tt}-c^2u_{xx}=0, \\ &u|_{t=0}=g(x),\quad u_t|_{t=0}=h(x) \end{aligned}\right. with \begin{align} &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; \end{align} \begin{align} &g(x)=0,\ && h(x)=1; \\ &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 $u=u(x,t)$ and describe domain, where it is uniquely defined \begin{align} &u_{tt}-u_{xx}=0, \label{A}\\[2pt] &u|_{t=x^2/2}= x^3, \label{B}\\[2pt] &u_t|_{t=x^2/2}= 2x. \label{C} \end{align}

Problem 4.

1. Prove that if $u$ solves the problem \left\{\begin{aligned} &u_{tt}-c^2u_{xx}=0&&-\infty <x <\infty,\\ &u|_{t=0}=g(x),\\ &u_t|_{t=0}=0, \end{aligned}\right. \label{D} then $v=\int _0^t u(x,t')\,dt'$ solves \left\{\begin{aligned} &v_{tt}-c^2v_{xx}=0&&-\infty <x <\infty,\\ &v|_{t=0}=0,\\ &v_t|_{t=0}=g(x). \end{aligned}\right. \label{E}
2. Also prove that if $v$ solves (\ref{E}) then $u=v_t$ solves (\ref{D}).
3. From formula $$u(x,t)=\frac{1}{2}\bigl( g(x+ct) +g(x-ct)\bigr) \label{F}$$ for the solution of (\ref{D}) derive $$v(x,t)=\frac{1}{2c}\int_{x-ct}^{x+ct}g(x')\,dx' \label{G}$$ for the solution of (\ref{E}).
4. Conversely, from (\ref{G}) for the solution of (\ref{E}) derive (\ref{E}) for the solution of (\ref{D}).

Problem 5. Find solution to equation $$Au_{tt}+2Bu_{tx}+Cu_{xx}=0$$ as $$u=f(x-c_1 t)+ g(x-c_2t)$$ 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 6.

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 $$u_{tt} = c^2 \bigl(u_{rr}+\frac{2}{r}u_r\bigr) \qquad\text{(spherical wave equation).} \label{eq-4a}$$

1. Change variables $v = ru$ to get the equation for $v$: $v_{tt} = c^2 v_{rr}$.
2. Solve for $v$ using $$v = f(r+ct)+g(r-ct)$$ and thereby solve the spherical wave equation.
3. Use $$v(r,t)=\frac{1}{2}\bigl[ \phi (r+ct)+\phi (r-ct)\bigr]+\frac{1}{2c}\int_{r-ct}^{r+ct}\psi (s) \,ds$$ 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 7. 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 long as $g(0)=h(0)$.

Problem 8. Find solution u=u(x,t) and describe domain, where it is uniquely defined \begin{align} &u_{tt}-u_{xx}=0, \\[2pt] &u|_{t=x^2/2}= x^3, &&|x|\le 1,\\[2pt] &u_t|_{t=x^2/2}= 2x &&|x|\le 1. \end{align} Explain, why we imposed restriction $|x|\le 1$?

Problem 9. 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 10. Look for solutions in the form of travelling wave for sine-Gordon equation $$u_{tt}-c^2 u_{xx}+\sin(u)=0.$$ observe that resulting ODE is describing mathematical pendulum which could be explored. Describe all possible velocities $v$.

Problem 11. 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 such solutions (they are called kinks). Describe all possible velocities $v$.
2. Find solitons. Describe all possible velocities $v$.

Problem 12. 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 $$\frac{\partial e}{\partial t} = c\frac{\partial p}{\partial x} \qquad \text{and} \qquad \frac{\partial p}{\partial t} = c\frac{\partial e}{\partial x}. \label{eq-HA2.11}$$
2. Show that both $e(x, t)$ and $p(x,t)$ also satisfy the same wave equation.

Problem 13.

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 14.

Generalize Problem 6:

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 $$u_{tt} = c^2 \bigl(u_{rr}+\frac{n-1}{r}u_r\bigr) \qquad\text{(spherical wave equation)} \label{eq-4b}$$

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 6 write down spherical wave for odd $n$.
3. Describe spherical wave for $n=1$.

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