TY - JOUR

T1 - Singular solitons

AU - Sakaguchi, Hidetsugu

AU - Malomed, Boris A.

N1 - Funding Information:
The work of H.S. is supported by the Japan Society for Promotion of Science through KAKENHI Grant No. 18K03462. B.A.M. appreciates support provided by the Israel Science Foundation through Grant No. 1286/17 and the hospitality of the Interdisciplinary Graduate School of Engineering Sciences at Kyushu University.
Publisher Copyright:
© 2020 American Physical Society.

PY - 2020/1/17

Y1 - 2020/1/17

N2 - 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, represented by septimal, quintic, and usual cubic terms in the framework of the one-, two-, and three-dimensional (1D, 2D, and 3D) nonlinear Schrödinger 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 δ-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.

AB - 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, represented by septimal, quintic, and usual cubic terms in the framework of the one-, two-, and three-dimensional (1D, 2D, and 3D) nonlinear Schrödinger 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 δ-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.

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U2 - 10.1103/PhysRevE.101.012211

DO - 10.1103/PhysRevE.101.012211

M3 - Article

C2 - 32069529

AN - SCOPUS:85078841659

VL - 101

JO - Physical Review E

JF - Physical Review E

SN - 2470-0045

IS - 1

M1 - 012211

ER -