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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

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    $\begingroup$ It may help to note the RHS can be written as $\frac{1}{3}[(1+u_x)^3]_x$ $\endgroup$
    – DaveNine
    Nov 11, 2017 at 11:55
  • $\begingroup$ @DaveNine May I have more detail, please? $\endgroup$
    – Vui Tinh
    Nov 12, 2017 at 6:28
  • $\begingroup$ @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. $\endgroup$
    – DaveNine
    Nov 12, 2017 at 6:54
  • $\begingroup$ @Harry49 Can you please give more detail to solve this? Thanks $\endgroup$
    – Vui Tinh
    Nov 15, 2017 at 8:28
  • $\begingroup$ 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 $h(x)$. $\endgroup$
    – EditPiAf
    Nov 17, 2017 at 14:02

1 Answer 1

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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.


Note: This example was used in the introduction of the following paper: F. John, "Delayed singularity formation in solutions of nonlinear wave equations in higher dimensions", Comm. Pure Appl. Math. 29, 1976, 649-682. doi:10.1002/cpa.3160290608 In: Moser, J. (eds) Fritz John. Collected Papers Volume 1, Birkhäuser, 1985. doi:10.1007/978-1-4612-5406-5_37

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