Recently, we have derived a new equation from Geophysical Fluid mechanics with the following presentation in the attached file.
We wonder about two facts:
(1) When is it integrable?
{2} Could we find the solutions for the equation? such as the solitons, N-solitons etc..?
Have you ever encountered the equation, and I am looking forward to your help. Thank you for your suggestive references!
It looks like an extended KP equation (2-D) instead of revealing the KdV family (1-D) features. It can be further analyzed in a reduced form as per the basic directional suggestions given in the following.
[1] A. C. Newell, “Perturbative theory of solitons,” Solitons in Mathematics and Physics (Society for Industrial and Applied Mathematics, Philadelphia, PA, 1985).
[2] M. Lakshamanan and S. Rajasekhar, Nonlinear Dynamics: Integrability, Chaos, and Patterns (Springer, New York, 2002).
I agree that it has some generalization of KdV, but it is also different from KP; see https://en.wikipedia.org/wiki/Kadomtsev%E2%80%93Petviashvili_equation .
I have never seen this equation before. If you have derived it on your own, what were the initial equations and what assumptions have you used? For example, here I derive generalization of KdV (2nd order terms and uneven bottom); see Conference Paper Evolution of long nonlinear waves on shelves
It is in continuation with our earlier discussion on the topic. Since, derived in a realistic geo-fluidic configuration presumed to be closed by usual basic equations, such a derived PDE is indeed believed to exist in Nature without loss of any usual conservation law. The Painleve analysis may be executed to detect and confirm global integrability. An asymptotic analytic (approximate, traveling wave) solution may also be tried to check further reliability and validation of the derived PDE.
The following book may be useful for the purpose:
"Nonlinear PDE's in Condensed Matter and Reactive Flows",
edited by
Henri Berestycki et al.
Dear P. K. Karmakar,Yuri Yegorov
Thank you so much for your helpful suggestions. I have read the paper or book you have suggested, and we will do the integrability and solutions for the derived equation.
Thank you too.
Hope all success.
Please let me know if anything else needed.
Sincerely,-
PK Karmakar
I never encountered exactly this type of PDE even if it looks similar to other equations, it seems having dispersion terms. I assume is written in non-dimensional form or the coefficients ai are dimensionally such that all terms are congruent.
In order to understand the type of boundary conditions we cannot use the standard classification for second order PDE (elliptic, parabolic, hyperbolic). Maybe you can try to reduce to a system of first order PDE and do an eigenvalue analysis.
Dear Filippo Maria Denaro
Thanks a lot for your valuable suggestions on the derived equation.
Best wishes!
Ruigang Zhang
Just because it looks similar to KdV or KP does not mean it has the same properties,, like interoperability. You should also show the derivation of the equation just to make sure it is correct. There have so many equations derived in Fluids that finding a new one is hard,, most likely it has been derived under one name or another.
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