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Abstract Dispersive shock waves (DSWs) of the defocusing radial nonlinear Schrödinger (rNLS) equation in two spatial dimensions are studied. This equation arises naturally in Bose‐Einstein condensates, water waves, and nonlinear optics. A unified nonlinear WKB approach, equally applicable to integrable or nonintegrable partial differential equations, is used to find the rNLS Whitham modulation equation system in both physical and hydrodynamic type variables. The description of DSWs obtained via Whitham theory is compared with direct rNLS numerics; the results demonstrate very good quantitative agreement. On the other hand, as expected, comparison with the corresponding DSW solutions of the one‐dimensional NLS equation exhibits significant qualitative and quantitative differences.
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Nonlinear Schrödinger equations and the universal description of dispersive shock wave structure
Abstract The nonlinear Schrödinger (NLS) equation and the Whitham modulation equations both describe slowly varying, locally periodic nonlinear wavetrains, albeit in differing amplitude‐frequency domains. In this paper, we take advantage of the overlapping asymptotic regime that applies to both the NLS and Whitham modulation descriptions in order to develop a universal analytical description of dispersive shock waves (DSWs) generated in Riemann problems for a broad class of integrable and nonintegrable nonlinear dispersive equations. The proposed method extends DSW fitting theory that prescribes the motion of a DSW's edges into the DSW's interior, that is, this work reveals the DSW structure. Our approach also provides a natural framework in which to analyze DSW stability. We consider several representative, physically relevant examples that illustrate the efficacy of the developed general theory. Comparisons with direct numerical simulations show that inclusion of higher order terms in the NLS equation enables a remarkably accurate description of the DSW structure in a broad region that extends from the harmonic, small amplitude edge.
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- PAR ID:
- 10080544
- Publisher / Repository:
- Wiley-Blackwell
- Date Published:
- Journal Name:
- Studies in Applied Mathematics
- Volume:
- 142
- Issue:
- 3
- ISSN:
- 0022-2526
- Page Range / eLocation ID:
- p. 241-268
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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