3
$\begingroup$

It is well known that some dispersive non--linear equations admit traveling wave solutions $$ u(t,x)=u_0(x-ct)\in L^2_x\,, \qquad (t,x)\in \mathbb{R}\times \mathbb{R}\,\text{ or }\, \mathbb{R}\times\mathbb{T}\,, $$ where $u_0$ is the profile and $c$ is a real constant. Sometimes these traveling waves can be obtained as ground states (minimizers) of the energy, mass... functionals of the equation, leading to say that there exists an $L^2$--threshold below (or above) which the existence of traveling waves cannot occur.

My question : Do there exist nonlinear dispersive PDEs, which have traveling waves with large and small $L^2$ norms ?

$\endgroup$
2
  • $\begingroup$ your interest is in solitonic waves, right, not general traveling waves? $\endgroup$ Commented Feb 12, 2023 at 14:22
  • $\begingroup$ @CarloBeenakker I was first thinking about traveling waves. But, since you mentioned solitonic waves, I also have the same question for solitons... $\endgroup$
    – Niser
    Commented Feb 13, 2023 at 9:43

2 Answers 2

2
$\begingroup$

This addresses the case of a traveling wave (not a solitonic wave).

If I consider the one-dimensional nonlinear Schrödinger equation $$i\partial_t\Psi+\partial_x^2\Psi+\Psi(f(|\Psi|^2)=0,$$ a travelling wave $\Psi(t,x)=U(x-ct)$ which tends to $|U(x)|\rightarrow r_0>0$ for $x\rightarrow\pm\infty$ exists if the velocity $c$ is not too large, $$c<\sqrt{2r_0^2|f'(r_0^2)|}\equiv c^\ast(r_0).$$ So the question whether $U$ can have arbitrarily large norm in a finite system (with periodic boundary conditions) becomes the question whether $c^\ast(r_0)$ can become arbitrarily large with increasing $r_0$, which is possible, for example, if $f(x)=1-x$ (Gross-Pitaevskii equation).

See Traveling waves for the Nonlinear Schrödinger Equation with general nonlinearity in dimension one.

$\endgroup$
5
  • $\begingroup$ My following question might be silly: Suppose that $c^*(r_0)$ is bounded. I don't understand why if $c<c^*(r_0)$ then this implies the $L^2$-norm of $U$ cannot be arbitrarily large. $\endgroup$
    – Niser
    Commented Feb 13, 2023 at 11:26
  • $\begingroup$ perhaps I misunderstood your question, but first of all, for a travelling wave we need a finite system for a finite norm (since by definition, $u_0(x)$ does not go to zero for large $x$). We would typically place the system on a torus, and impose periodic boundary conditions on a length $L$; then we ask whether, for a fixed $L$, the norm of $u_0$ can become arbitrarily large; this norm will be of order $r_0 L$, so for a large norm we need a large $r_0$; this will fail if $c^\ast$ vanishes for large $r_0$, but for the case of the Gross-Pitaevskii equation it will not. $\endgroup$ Commented Feb 13, 2023 at 11:52
  • $\begingroup$ Thank you. I still have a question. In the case of Gross-Pitaevskii, If I am not mistaken, the $L^2$ norm of the profile $U$ cannot $<\!<1$, so there exists a lower bound>0 where the $L^2$ norm of the traveling waves of Gross-Pitaevskii cannot go below this bound. $\endgroup$
    – Niser
    Commented Feb 13, 2023 at 13:02
  • $\begingroup$ yes, there is a lower bound but no upper bound. $\endgroup$ Commented Feb 13, 2023 at 13:07
  • $\begingroup$ Sorry, I will not mark the answer as accepted, because my initial question (which probably I didn't formulate clearly) was to know if there exists a PDE admitting traveling waves without lower and upper bound in their $L^2$ norm... $\endgroup$
    – Niser
    Commented Feb 13, 2023 at 13:23
1
$\begingroup$

I recently came across a paper on arXiv that addresses the question. I am sharing it here if it may be of interest to someone else.

The author seems to consider a nonlocal nonlinear schrödinger equation, referred to as the Calogero-Sutherland DNLS equation. And she finds periodic (in the space variable) traveling waves with small and large $L^2$-norms.

See Remark 1.3 (1) of page 5.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.