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

Atanas Stefanov, G. A. Tsolias+2 more|arXiv (Cornell University)|Mar 12, 2022

Nonlinear Photonic Systems22 references4 citations

TL;DR

This paper investigates the existence and spectral stability of ground state solutions for a higher-dimensional mixed dispersion nonlinear Schrödinger equation with a focusing biharmonic operator and a defocusing Laplacian (isotropic or anisotropic), focusing on dimension d=2. It establishes that solutions are stable for nonlinearity powers p below cubic, become unstable beyond a critical threshold for p between cubic and quintic, and remain unstable above a critical p, with stability windows narrowing as p increases.

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 $p$.

Motivation & Objective

• To characterize the ground states of a higher-dimensional nonlinear Schrödinger equation with competing biharmonic and Laplacian dispersion.
• To analyze the spectral stability of these ground states as a function of nonlinearity power p and Laplacian coefficient b.
• To compare isotropic (uniform Laplacian) and anisotropic (directional Laplacian) dispersion cases in two spatial dimensions.
• To extend theoretical and numerical analysis to arbitrary combinations of b and p, particularly in d=2.
• To explore the implications of these findings for higher-dimensional systems and experimental realizations of higher-order dispersion.

Proposed method

• Formulates the isotropic and anisotropic models: iu_t + ∆^2 u + b∆u - |u|^{p-1}u = 0 and iu_t + ∆^2 u + b∂_{x1}^2 u - |u|^{p-1}u = 0 in R^d.
• Reduces the problem to stationary elliptic equations via standing wave ansatz u = e^{-iωt} Φ, yielding ∆^2Φ + b∆Φ + ωΦ - |Φ|^{p-1}Φ = 0 and similar for the anisotropic case.
• Applies spectral stability analysis using the eigenvalue problem J L v = μ v, where J and L are defined operators based on the linearized system.
• Employs rigorous theoretical analysis for existence and stability of ground states, particularly for p < 3 and p > 5.
• Conducts extensive numerical computations in d=2 to map stability regions across p and b, using semi-logarithmic plots for frequency vs. p.
• Compares isotropic and anisotropic cases, noting differences in instability intervals even below cubic nonlinearity in the anisotropic setting.

Experimental results

Research questions

• RQ1 What is the stability behavior of ground states in the higher-dimensional mixed dispersion NLS equation with isotropic Laplacian dispersion?
• RQ2 How does the stability of solutions depend on the nonlinearity power p and the Laplacian coefficient b in the isotropic case?
• RQ3 How does anisotropic dispersion (only along x1) alter the stability landscape compared to the isotropic case?
• RQ4 What is the critical nonlinearity exponent p above which all solutions become unstable, and how does this threshold vary with b?
• RQ5 Are there stable frequency windows for p > 5, and how do they narrow as p increases?

Key findings

• All solutions are spectrally stable for nonlinearity powers p < 3, regardless of the Laplacian coefficient b.
• For p between cubic and quintic (3 < p < 5), instability arises beyond a critical threshold of the Laplacian coefficient b.
• For p > 5, a progressively narrowing range of stable frequencies exists, with stability windows shrinking as p increases.
• Above a critical nonlinearity exponent p_c, all solutions become unstable, regardless of b.
• In the anisotropic case, numerical results suggest an interval of unstable frequencies even for p < 3, a feature absent in the isotropic case.
• The anisotropic model exhibits a separable solution structure near the linear limit (ω → 0.25), with uniform nodal lines along the y-direction.