Stability of exact solutions of the (2 + 1)-dimensional nonlinear Schrödinger equation with arbitrary nonlinearity parameter κ

Journal Article · Tue Dec 13 00:00:00 EST 2022 · Physica Scripta

DOI:https://doi.org/10.1088/1402-4896/aca227· OSTI ID:2441353

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

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

In this work, we consider the nonlinear Schrödinger equation (NLSE) in 2+1 dimensions with arbitrary nonlinearity exponent κ in the presence of an external confining potential. Exact solutions to the system are constructed, and their stability as we increase the 'mass' (i.e., the L² norm) and the nonlinearity parameter κ is explored. Here we observe both theoretically and numerically that the presence of the confining potential leads to wider domains of stability over the parameter space compared to the unconfined case. Our analysis suggests the existence of a stable regime of solutions for all κ as long as their mass is less than a critical value M*(κ). Furthermore, we find that there are two different critical masses, one corresponding to width perturbations and the other one to translational perturbations. The results of Derrick's theorem are also obtained by studying the small amplitude regime of a four-parameter collective coordinate (4CC) approximation. A numerical stability analysis of the NLSE shows that the instability curve M*(κ) versus κ lies below the two curves found by Derrick's theorem and the 4CC approximation. In the absence of the external potential, κ = 1 demarcates the separation between the blowup regime and the stable regime. In this 4CC approximation, for κ < 1, when the mass is above the critical mass for the translational instability, quite complicated motions of the collective coordinates are possible. Energy conservation prevents the blowup of the solution as well as confines the center of the solution to a finite spatial domain. We call this regime the 'frustrated' blowup regime and give some illustrations. In an appendix, we show how to extend these results to arbitrary initial ground state solution data and arbitrary spatial dimension d.

• Research Organization: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
• Sponsoring Organization: National Science Foundation (NSF); USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA)
• Grant/Contract Number: 89233218CNA000001; AC52-06NA25396
• OSTI ID: 2441353
• Report Number(s): LA-UR--22-25694
• Journal Information: Physica Scripta, Journal Name: Physica Scripta Journal Issue: 1 Vol. 98; ISSN 0031-8949
• Publisher: IOP Publishing
• Country of Publication: United States
• Language: English

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