Transverse instability of plane wave soliton solutions of the Novikov-Veselov equation

Abstract

Abstract. The Novikov-Veselov (NV) equation is a dispersive (2+1)-dimensional nonlinear evolution equation that generalizes the (1+1)-dimensional Korteweg-deVries (KdV) equation. This paper considers the stability of plane wave soliton solutions of the NV equation to transverse perturbations. To investigate the behavior of the perturbations, a hybrid semi-implicit/spectral numerical scheme was developed, applicable to other nonlinear PDE systems. Numerical simulations of the evolution of transversely perturbed plane wave solutions are presented. In particular, it is established that plane wave soliton solutions are not stable for transverse perturbations.

The Novikov-Veselov Equation:Theory and Computation

“…An example of this are the solutions generated by Hirota's method, see section 4.2.1. This solution is unstable with respect to perturbations periodic in y direction with period 2 π k √ c for 0.363 < k < 1, see [16]. This is confirmed by numerical evolution of the NV equation using the above spectral method.…”

“…There is a close connection between plane wave solutions to NV and solutions to KdV (see [16]): for some direction vector n = (n 1 , n 2 ) = (cos(α), sin(α)). Then the bounded solutions to the ∂ equation (4.2) are given by…”

“…As in [16] we use a FFT-based method to solve the equations on the square −L ≤ x, y ≤ +L with periodic boundary conditions. To examine the linear contributions we introduce a function q(t, x, y) = exp(i (ξ x + η y)).…”

“…We use a Crank-Nicolson scheme for the linear part of NV and an explicit method for the nonlinear contribution div(q u). For details see [16,17].…”

The Cauchy Problem for the Generalized Hyperbolic Novikov–Veselov Equation via the Moutard Symmetries

“…Many articles were dedicated to unusual and fascinating properties of the multi-dimensional solutions, including those for seemingly ordinary flat waves. In particular, in [24] it has been shown that plane wave soliton solutions of NV equation are not stable for transverse perturbations; the paper [18] demonstrates that NV equation permits such interesting solutions as multi-solitons, ring solitons, and the breathers; while the authors of [25] construct a Mach-type soliton of the NV equation. One of the most effective mathematical tools for studying the NV equation is the inverse scattering method.…”