New findings for the old problem: Exact solutions for domain walls in coupled real Ginzburg-Landau equations

Malomed, Boris A.

doi:10.1016/j.physleta.2021.127802

Cited by 33 publications

(9 citation statements)

References 47 publications

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“…Thus, asymmetric DWs only exist in the focusing regime, while symmetric DWs only exist in the defocusing regime. Note that prior work on DWs between ZN and NZ in binary BECs define asymmetric and symmetric DWs depending on the symmetry of their spatial profiles [69,73]. In our language, both are symmetric DWs that are typically homotopically equivalent.…”

Section: Dwsmentioning

confidence: 99%

Domain walls and vector solitons in the coupled nonlinear Schrödinger equation

Snee,

2024

J. Phys. A: Math. Theor.

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We outline a program to classify domain walls (DWs) and vector solitons in the 1D two-component coupled nonlinear Schrödinger (CNLS) equation without restricting the signs or magnitudes of any coeﬃcients. The CNLS equation is reduced ﬁrst to a complex ordinary diﬀerential equation (ODE), and then to a real ODE after imposing a restriction. In the real ODE, we identify four possible equilibria including ZZ, ZN, NZ, and NN, with Z (N) denoting a zero (nonzero) value in a component, and analyze their spatial stability. We identify two types of DWs including asymmetric DWs between ZZ and NN and symmetric DWs between ZN and NZ. We identify three codimension-1 mechanisms for generating vector solitons in the real ODE including heteroclinic cycles, local bifurcations, and exact solutions. Heteroclinic cycles are formed by assembling two DWs back-to-back and generate extended bright-bright (BB), dark-dark (DD), and dark-bright (DB) solitons. Local bifurcations include the Turing (Hamiltonian-Hopf) bifurcation that generates Turing solitons with oscillatory tails and the pitchfork bifurcation that generates DB, bright-antidark, DD, and dark-antidark solitons with monotonic tails. Exact solutions include scalar bright and dark solitons with vector amplitudes. Any codimension-1 real vector soliton can be numerically continued into a codimension-0 family. Complex vector solitons have two more parameters: a dark or antidark component can be numerically continued in the wavenumber, while a bright component can be multiplied by a constant phase factor. We introduce a numerical continuation method to ﬁnd real and complex vector solitons and show that DWs and DB solitons in the immiscible regime can be related by varying bifurcation parameters. We show that collisions between two DB solitons with a nonzero phase diﬀerence in their bright components typically feature a mass exchange that changes the frequencies and phases of the two bright components and the two soliton velocities.

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Section: Dwsmentioning

confidence: 99%

Domain walls and vector solitons in the coupled nonlinear Schrödinger equation

Snee,

2024

J. Phys. A: Math. Theor.

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“…In mathematical physics, it is very important to find exact solutions to nonlinear differential equations [1,2,3,4,5]. Exact solutions play a crucial role in understanding various qualitative and quantitative features of nonlinear phenomena [6]. Among them, the rogue wave solutions are a kind of rational solutions localised in time and space.…”

Section: Introductionmentioning

confidence: 99%

A simplified symbolic computation method based on Hirota bilinear equations for constructing multiple-order rogue waves

Tang,

Zeng

2024

Preprint

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The computation of higher-order rogue waves can be a time-consuming process.Hirota bilinear equations with controllable centers for all orders of rogue wave solutions can be solved by a simplified symbolic computation method.Only one-third of the computational effort needed by the original method is required by the simplified one, for very high orders. For instance, simpler first- to fourth-order rogue waves, as well as fifth-order rogue waves that were previously unpublished, are yielded by the simplified method for a generalized (2+1)-dimensional Kadomtsev-Petviashvili equation. To be honest, the obtained results in this paper substantially improve or complement the corresponding conditions in the references [17,18]. As a consequence,~this paper gives a new idea to derive high-order rogue waves for the generalized (2+1)-dimensional Kadomtsev-Petviashvili equation.

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“…Inductive components containing the ferromagnetic cores, like the transformers, filters, chokes, resonant circuits, etc., have been seen in almost all modern electronic devices [6][7][8]. Nonlinear evolution equations have been proposed to describe the nonlinear phenomena in optics, fluid mechanics, plasma physics and other fields [9][10][11][12][13]. It has been known that some nonlinear phenomena in the ferromagnetic materials can be modelled via those equations [14][15][16].…”

Section: Introductionmentioning

confidence: 99%

Complex Kraenkel-Manna-Merle system in a ferrite: N-fold Darboux transformation, generalized Darboux transformations and solitons

Shen

Tian

Zhou

et al. 2023

Preprint

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Ferromagnetic materials such as the ferrites are used in the electronic and energy industries. Here, we concentrate on a complex Kraenkel-Manna-Merle system for the ultra-short waves in a ferrite. An N-fold Darboux transformation of that system is presented via an existing Lax pair, where N is a positive integer. A generalized (1, N − 1)- fold Darboux transformation, which admits one spectral parameter, and a generalized (m, N − m)-fold Darboux transformation, which admits m spectral parameters, are proposed based on that N-fold Darboux transformation, where m is a positive integer. One-, two- and three-soliton solutions of that system are determined via that N-fold Darboux transformation. Double-pole and triple-pole soliton solutions of that system are derived via that (1, N − 1)-fold Darboux transformation. Those solitons are graph- ically represented for the magnetization and external magnetic field related to a ferrite.

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New findings for the old problem: Exact solutions for domain walls in coupled real Ginzburg-Landau equations

Cited by 33 publications

References 47 publications

Domain walls and vector solitons in the coupled nonlinear Schrödinger equation

Domain walls and vector solitons in the coupled nonlinear Schrödinger equation

A simplified symbolic computation method based on Hirota bilinear equations for constructing multiple-order rogue waves

Complex Kraenkel-Manna-Merle system in a ferrite: N-fold Darboux transformation, generalized Darboux transformations and solitons

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