Spectral transverse instabilities and soliton dynamics in the higher-order multidimensional nonlinear Schrödinger equation

• Transverse instability of low-dimensional solitons for the biharmonic nonlinear Schrödinger equation is explored.
• Numerical, asymptotic and variational techniques are used to characterize the unstable spectrum.
• Strong higher-order dispersion/diffraction is shown to suppress transverse instability.
• For certain parameter regimes, long wavelength stability is observed.
• Direct numerical simulations of the instability are conducted showing filamentation.

Spectral transverse instabilities of one-dimensional solitary wave solutions to the two-dimensional nonlinear Schrödinger (NLS) equation with fourth-order dispersion/diffraction subject to higher-dimensional perturbations are studied. A linear boundary value problem governing the evolution of the transverse perturbations is derived. The eigenvalues of the perturbations are numerically computed using Fourier and finite difference differentiation matrices. It is found that for both signs of the higher-order dispersion coefficient there exists a finite band of unstable transverse modes. In the long wavelength limit we derive an asymptotic formula for the perturbation growth rate that agrees well with the numerical findings. Using a variational formulation based on Lagrangian model reduction, an approximate expression for the perturbation eigenvalues is obtained and its validity is compared with both the asymptotic and numerical results. The time dynamics of a one-dimensional soliton stripe in the presence of a transverse perturbation is studied using direct numerical simulations. Numerical nonlinear stability analysis is also addressed.