Singular solitons

Abstract

We demonstrate that the commonly known concept, which treats solitons as nonsingular solutions produced by the interplay of nonlinear self-attraction and linear dispersion, may be extended to include modes with a relatively weak singularity at the central point, which keeps their integral norm convergent. Such states are generated by self-repulsion, which should be strong enough, namely, represented by septimal, quintic, and usual cubic terms in the framework of the one-, two-, and three-dimensional (1D, 2D, and 3D) nonlinear Schroedinger equations (NLSEs), respectively. Although such solutions seem counterintuitive, we demonstrate that they admit a straightforward interpretation as a result of screening of an additionally introduced attractive delta-functional potential by the defocusing nonlinearity. The strength ("bare charge") of the attractive potential is infinite in 1D, finite in 2D, and vanishingly small in 3D. Analytical asymptotics of the singular solitons at small and large distances are found, entire shapes of the solitons being produced in a numerical form. Complete stability of the singular modes is accurately predicted by the anti-Vakhitov-Kolokolov criterion (under the assumption that it applies to the model), as verified by means of numerical methods. In 2D, the NLSE with a quintic self-focusing term admits singular soliton solutions with intrinsic vorticity too, but they are fully unstable. We also mention that dissipative singular solitons can be produced by the model with a complex coefficient in front of the nonlinear term.

to be published in Physical Review E

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• Existence, structure, and robustness of ground states of a NLSE in 3D with a point defect
• Singular mean-field states: A brief review of recent results
• Breather stripes and radial breathers of the two-dimensional sine-Gordon equation
• Singular Soliton Molecules of the Nonlinear Schrodinger Equation
• A general review on the NLS equation with point-concentrated nonlinearity
• Singular and regular vortices on top of a background pulled to the center
• Well-posedness of the three-dimensional NLS equation with sphere-concentrated nonlinearity
• Minimizers of mass-constrained functionals involving a nonattractive point interaction
• Nonlinear scalar field equation with point interaction