Mixed dispersion nonlinear Schr\"odinger equation in higher dimensions: theoretical analysis and numerical computations

TL;DR

This paper analyzes the stability of ground states in a higher-dimensional nonlinear Schrödinger equation with mixed dispersion, revealing thresholds for stability and instability depending on nonlinearity power and dispersion parameters.

Contribution

It provides a theoretical and numerical characterization of ground state stability in a generalized mixed dispersion nonlinear Schrödinger model across different dimensions and parameters.

Findings

Instability occurs beyond a certain Laplacian coefficient threshold.

All solutions are stable for powers below the cubic.

Above the critical nonlinearity exponent, all solutions are unstable.

Abstract

In the present work we provide a characterization of the ground states of a higher-dimensional quadratic-quartic model of the nonlinear Schr{\"o}dinger class with a combination of a focusing biharmonic operator with either an isotropic or an anisotropic defocusing Laplacian operator (at the linear level) and power-law nonlinearity. Examining principally the prototypical example of dimension $d=2$, we find that instability arises beyond a certain threshold coefficient of the Laplacian between the cubic and quintic cases, while all solutions are stable for powers below the cubic. Above the quintic, and up to a critical nonlinearity exponent $p$, there exists a progressively narrowing range of stable frequencies. Finally, above the critical $p$ all solutions are unstable. The picture is rather similar in the anisotropic case, with the difference that even before the cubic case, the numerical computations suggest an interval of unstable frequencies. Our analysis generalizes the relevant observations for arbitrary combinations of Laplacian prefactor $b$ and nonlinearity power $p$.