# Simple wave solutions of $u_{tt} = (1+u_x)^2u_{xx}$

I really need help to solve this problem:

Find all simple wave solutions of the equation: $$u_{tt} = (1+u_x)^2u_{xx}$$ with $$u(x,0)=h(x)$$. [Hint: write above equation as a first order system for the vector $$v =(u_x, u_t)$$ and find the solutions with $$u_x=\theta, u_t=F(\theta)$$].

Here is a answer for this : $$u=\pm \frac{1}{2}\theta ^2 + h(x \mp (1+\theta)t) +ct$$, where $$c$$ is constant and $$\theta$$ is solution of $$\theta = h'(x \pm (1+\theta )t)$$.

I have no idea how to solve this. Any help I really appreciate. Thanks

• It may help to note the RHS can be written as $\frac{1}{3}[(1+u_x)^3]_x$ Nov 11 '17 at 11:55
• @DaveNine May I have more detail, please? Nov 12 '17 at 6:28
• @Harry49 pretty much carried out the rest of the answer. All I did was see that the chain rule was on the RHS. You can see a treatment of conservation laws in Evan's PDE book as well. Nov 12 '17 at 6:54
• @Harry49 Can you please give more detail to solve this? Thanks Nov 15 '17 at 8:28
• If I am not mistaken, setting $t=0$ in the proposed "solution" gives $u(x,0)=±\frac{1}{2}(h′(x))^2+h(x)$, which is different from $h(x)$. Nov 17 '17 at 14:02

## 1 Answer

By analogy with nonlinear elastodynamics (or with the $$p$$-system of fluid dynamics), we introduce the velocity variable $$v = u_t$$ and the strain variable $$\theta = u_x$$. Using the equality of mixed derivatives and the second-order PDE in OP, we rewrite the problem as a nonlinear system of conservation laws of the form $${\bf u}_t + {\bf f}({\bf u})_x = {\bf 0}$$, where $${\bf u} = \begin{pmatrix} \theta\\ v \end{pmatrix}, \qquad {\bf f}({\bf u}) = -\! \begin{pmatrix} v\\ \tfrac{1}{3}(1+\theta)^3 \end{pmatrix} .$$ As suggested in OP, we assume that $$v = F(\theta)$$, where $$F$$ is a smooth function to be determined. Injecting this Ansatz in the PDE system leads to $$\begin{pmatrix} 1 & -F'(\theta)\\ F'(\theta) & -(1+\theta)^2 \end{pmatrix} \begin{pmatrix} \theta_t \\ \theta_x \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} .$$ Nontrivial solutions can be obtained provided that the determinant of the matrix above vanishes, i.e. $$F'(\theta)^2 = (1+\theta)^2$$. Therefore, the (smooth) function $$F$$ satisfies $$F'(\theta) = \pm(1+\theta)$$, which solutions are $$F(\theta) = c \pm (\theta+\frac12 \theta^2)$$ for some constant $$c$$. Note that this constant equals $$v \mp \int_0^\theta (1+\vartheta)\,\text d \vartheta$$ which is a Riemann invariant of the system. With this expression of $$v=F(\theta)$$, the system of conservation laws amounts to the single PDE $$\theta_t \mp (1 + \theta)\, \theta_x = 0 \, .$$ Using the boundary condition $$\theta(x_0,0) = h'(x_0)$$, one deduces that the characteristic curves are the lines $$x = x_0 \mp (1+\theta) t$$ along which $$\theta = h'(x_0)$$ is constant, which can be written in implicit form as $$\theta = h'\big(x \pm (1+\theta) t\big)\, .$$ Since $$\theta$$ is constant along the characteristic lines, the velocity $$v = F(\theta)$$ is constant along those lines too. Therefore, the variable $$u$$ is increasing along those lines with a constant rate. More precisely, we have $$\frac{\text d}{\text d t} u\big(x_0 \mp (1+\theta) t, t\big) = \mp (1+\theta) \theta + F(\theta) = \mp\tfrac12 \theta^2 + c \, ,$$ where $$\theta = h'(x_0)$$ is constant. Finally, one obtains $$u = h\big(x \pm (1+\theta) t\big) \mp\tfrac12 \theta^2 t + c t \, ,$$ where $$\theta$$ satisfies the implicit equation $$\theta = h'(x \pm (1+\theta) t)$$. Note that this expression is different from the one in OP (which obviously doesn't match the boundary condition), and that it is only valid until the breaking time $$t_b = \pm 1/\inf h''$$.

The concept of simple wave solution refers to an Ansatz of the form $${\bf u}(x,t) = {\bf v}(\xi)$$, where $$\xi$$ is a smooth function of $$(x,t)$$ and $${\bf v}(\xi)$$ follows an integral curve. Along the integral curve, the Riemann invariant $$v \mp \int_0^\theta (1+\vartheta)\,\text d \vartheta$$ is constant, and $${\bf v}'(\xi)$$ is the eigenvector of the Jacobian matrix $${\bf f'}({\bf v}(\xi))$$ corresponding to the eigenvalue $$\mp (1+\theta(\xi))$$. The quasi-linear PDE thus yields the scalar conservation law $$\xi_t \mp (1+\theta(\xi))\, \xi_x = {0}$$ for $$\xi$$, which is constant along its characteristic lines deduced from the initial condition $$\theta(\xi(x,0)) = h'(x)$$. Note that $$v$$ and $$\theta$$ are constant along these lines too, which is expressed by the PDE on $$\theta$$ above.