Numerical solution of a highly nonlinear PDE and the absence of the soliton solution
Posted Jun 27, 2020, 7:56 AM EDT General, Acoustics & Vibrations, Equation-Based Modeling Version 5.3 3 Replies
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Hello. I try to solve the fourth order stationary PDE equation:
for right trangle geometry.
Following the procedure here and introducing and I rewrite the equation for the simulation
This equation can be represented as a General form PDE for the variables u, P, and Q: Gamma1=(ux+Px+Qx+1/3(ux)^3+ux(uy)^2), uy+Qy+1/3(uy)^3+uy(ux)^2) and F1=0;
Gamma2=(ux,0) and F2=P;
Gamma3=(0,uy) and F3=Q,
where all coefficients are equal to 1.
The equation is supplemented by Dirichlet conditions on the boundaries u=0 and the second derivatives uxx=0, uyy=0 also. Of course also I can introduce Neumann conditions.
It is well-known that for some approximations the equation gives rise soliton solution similar to a solution of the nonlinear Schrodinger equation. However and this is my problem after the Comsol simulation I obtain an obvious trivial solution with u=0. How to avoid this numerical solution? Is it possible? Thank you in advance.