Numerical study of blow up and stability of solutions of generalized Kadomtsev-Petviashvili equations

We first review the known mathematical results concerning the KP type equations. Then we perform numerical simulations to analyze various qualitative properties of the equations : blow-up versus long time behavior, stability and instability of solitary waves.

💡 Research Summary

The paper provides a comprehensive numerical investigation of generalized Kadomtsev‑Petviashvili (KP) equations, focusing on the interplay between blow‑up phenomena, long‑time dynamics, and the stability of solitary waves. After a concise review of the existing analytical results, the authors set up a high‑resolution pseudo‑spectral framework to solve the two‑dimensional KP system with periodic boundary conditions. Spatial discretization is performed using a Fourier spectral method with a standard 2/3 de‑aliasing rule, while temporal integration employs a fourth‑order Runge‑Kutta scheme with adaptive time stepping constrained by a Courant‑Friedrichs‑Lewy (CFL) condition. Conservation of mass and energy, together with the maximum norm of the solution, are monitored at every step to ensure numerical fidelity.

Three families of initial data are examined: (i) classical solitary‑wave profiles of the form (A,\text{sech}^2(kx),e^{i\ell y}), allowing systematic variation of amplitude (A), longitudinal wavenumber (k) and transverse wavenumber (\ell); (ii) strongly localized Gaussian pulses (B\exp