Stability of trapped solutions of a nonlinear Schrödinger equation with a nonlocal nonlinear self-interaction potential

Journal Article · Mon Dec 06 00:00:00 EST 2021 · Journal of Physics. A, Mathematical and Theoretical

DOI:https://doi.org/10.1088/1751-8121/ac265b· OSTI ID:1922759

Charalampidis, Efstathios G. ^[1]; Cooper, Fred ^[2]; Khare, Avinash ^[3]; Dawson, John F. ^[4]; Saxena, Avadh ^[5]

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1. California Polytechnic State University (CalPoly), San Luis Obispo, CA (United States)
2. Santa Fe Institute, NM (United States); Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
3. Savitribai Phule Pune University, Pune (India)
4. University of New Hampshire, Durham, NH (United States)
5. Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

This work focuses on the study of the stability of trapped soliton-like solutions of a (1 + 1)-dimensional nonlinear Schrödinger equation (NLSE) in a nonlocal, nonlinear, self-interaction potential of the form [|Ψ(x,t)|² +|Ψ(-x,t)|²]^κ where κ is an arbitrary nonlinearity parameter. Although the system with κ = 1 (i.e. fully integrable case) was first reported by Yang (2018 Phys. Rev. E 98 042202), here in the present work, we extend this model to the one in which κ is arbitrary. This allows us to compare the stability properties of the now trapped solutions to previously found solutions of the more usual NLSE with κ ≠ 1 which are moving soliton solutions. We show that there is a simple, one-component, nonlocal Lagrangian and corresponding action governing the dynamics of the system. Using a collective coordinate method derived from the action as well as assuming the validity of Derrick's theorem, we find that these trapped solutions are stable for 0 < κ < 2 and unstable when κ > 2. At the critical value of κ, i.e. κ = 2, the solution can either collapse or blowup linearly in time when q₀ = 0, where q₀ is the center of the initial density ρ(x, t = 0) = ψ*ψ of the solution. For q₀ ≠ 0 the displaced solution collapses. When κ > 2 initial small displacements from the origin also lead to collapse of the wave function. This phenomenon is not seen in the usual NLSE.

• Research Organization: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
• Sponsoring Organization: USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA)
• Grant/Contract Number: 89233218CNA000001
• OSTI ID: 1922759
• Report Number(s): LA-UR-21-20590
• Journal Information: Journal of Physics. A, Mathematical and Theoretical, Journal Name: Journal of Physics. A, Mathematical and Theoretical Journal Issue: 1 Vol. 55; ISSN 1751-8113
• Publisher: IOP Publishing
• Country of Publication: United States
• Language: English

Related Subjects

collective coordinates
97 MATHEMATICS AND COMPUTING
Nonlocal nonlinear Schrodinger equation
action principle
existence and spectral stability analysis
variational approximation