How to get the periodic solution of the nonlinear PDE?

Question

How to get the periodic solution of the nonlinear PDE? i.e. the equation

$iq_{t} +q_{xx} = i(|q|^{2}q)_{x}$ has the priodic solution $q = ke^{ia[x-(a-k^{2})t]}$, where $a$ and $k$ are real numbers.

How to find this solution?

Answer 1

This is not an easy question in general, but let me give you what I think would be a kind of satisfactory answer. You are actually looking for periodic traveling wave solutions, that is, periodic solutions of the form $u(t,x)=\varphi(x-ct)$, for some speed $c>0$. Hence, if $u(t,x)$ solves your "derivative Schrödinger" equation, then $\varphi$ must to satisfy the following nonlinear ODE: $$ \qquad \qquad \varphi''-ic\varphi'-i(\vert \varphi\vert^2\varphi)'=0. \qquad \qquad (*) $$ Assuming some decay on the profile $\varphi$ you can integrate the previous equation to conclude that $\varphi$ satisfies the following first order ODE: $$ \qquad \qquad \varphi'-ic\varphi-i\vert \varphi\vert^2\varphi=0. \qquad \qquad (**) $$ So, now the problem is reduced to find solutions of the previous ODE, which is far easier than your initial problem, still difficult though. Finding solutions of the latter ODE is a different story and in general, is not an easy task, but I think now you understand where does this come from. I hope that this have helped you.

Edit: Notice that in the periodic case, once you integrate equation $(*)$ you might get an integration constant in the right-hand side of $(**)$, so the correct equation is $\varphi'-ic\varphi-i\vert \varphi\vert^2\varphi=A_c$. However, in the non-periodic case, the right-hand is actually zero.

Note: I kept thinking about the solutions of the ODE $(**)$ and I think you can solve this particular case by using the classical separation variable's method. In the general case, that is, given some PDE, find the traveling wave solutions (if they exist) is not an easy task and requires some knowledge on elliptic PDEs (in the $\mathbb{R}^n$ case, if $n=1$ this is an ODE).