The Coupled Modified Korteweg-de Vries Equations

Tsuchida, Takayuki; Wadati, Miki

doi:10.1143/jpsj.67.1175

Cited by 167 publications

(145 citation statements)

References 23 publications

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“…admits a Lax representation [52,75,78,79]. Then, system (4.27) is also integrable, because it is obtained from (4.63) through the following reduction:…”

Section: System (427)mentioning

confidence: 99%

Classification of polynomial integrable systems of mixed scalar and vector evolution equations: I

Tsuchida¹,

Wolf²

2005

J. Phys. A: Math. Gen.

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We perform a classification of integrable systems of mixed scalar and vector evolution equations with respect to higher symmetries. We consider polynomial systems that are homogeneous under a suitable weighting of variables. This paper deals with the KdV weighting, the Burgers (or potential KdV or modified KdV) weighting, the Ibragimov-Shabat weighting and two unfamiliar weightings. The case of other weightings will be studied in a subsequent paper. Making an ansatz for undetermined coefficients and using a computer package for solving bilinear algebraic systems, we give the complete lists of 2 nd order systems with a 3 rd order or a 4 th order symmetry and 3 rd order systems with a 5 th order symmetry. For all but a few systems in the lists, we show that the system (or, at least a subsystem of it) admits either a Lax representation or a linearizing transformation. A thorough comparison with recent work of Foursov and Olver is made.

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“…admits a Lax representation [52,75,78,79]. Then, system (4.27) is also integrable, because it is obtained from (4.63) through the following reduction:…”

Section: System (427)mentioning

confidence: 99%

Classification of polynomial integrable systems of mixed scalar and vector evolution equations: I

Tsuchida¹,

Wolf²

2005

J. Phys. A: Math. Gen.

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“…In particular, we address the general problem of spin soliton collisions. For one specific ratio of the scattering lengths, Wadati and coworkers in [14,15] found a complete classification of the one soliton solution with respect to the spin states and even presented an explicit formula of the two-soliton solution. One soliton solutions come in two classes: polar, and ferromagnetic solitons [14,15].…”

Section: Introductionmentioning

confidence: 99%

Class of compact entities in three-component Bose-Einstein condensates

et al. 2011

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We introduce a new class of soliton-like entities in spinor three component BECs. These entities generalize well known solitons. For special values of coupling constants, the system considered is Completely Integrable and supports N soliton solutions. The one-soliton solutions can be generalized to systems with different values of coupling constants. However, they no longer interact elastically. When two so generalized solitons collide, a spin component oscillation is observed in both emerging entities. We propose to call these newly found entities oscillatons. They propagate without dispersion and retain their character after collisions. We derived an exact mathematical model for oscillatons and showed that the well known one soliton solutions are a particular case.

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“…Notice that the soliton solution of the matrix NLS equation was previously obtained in Ref. [27] by means of the Gelfand-Levitan integral equations, while our derivation is purely algebraic. The soliton (5.11) was also derived by Gerdjikov and coworkers via the dressing procedure [41].…”

Section: Soliton Solutions a Rank-one Solitonmentioning

confidence: 81%

Perturbation theory for bright spinor Bose-Einstein condensate solitons

2008

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We develop a perturbation theory for bright solitons of the F = 1 integrable spinor Bose-Einstein condensate (BEC) model. The formalism is based on using the Riemann-Hilbert problem and provides the means to analytically calculate evolution of the soliton parameters. Both rank-one and rank-two soliton solutions of the model are obtained. We prove equivalence of the rank-one soliton and the ferromagnetic rank-two soliton. Taking into account a splitting of a perturbed polar rank-two soliton into two ferromagnetic solitons, it is sufficient to elaborate a perturbation theory for the rank-one solitons only. Treating a small deviation from the integrability condition as a perturbation, we describe the spinor BEC soliton dynamics in the adiabatic approximation. It is shown that the soliton is quite robust against such a perturbation and preserves its velocity, amplitude, and population of different spin components, only the soliton frequency acquires a small shift. Results of numerical simulations agree well with the analytical predictions, demonstrating only slight soliton profile deformation.

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The Coupled Modified Korteweg-de Vries Equations

Cited by 167 publications

References 23 publications

Classification of polynomial integrable systems of mixed scalar and vector evolution equations: I

Classification of polynomial integrable systems of mixed scalar and vector evolution equations: I

Class of compact entities in three-component Bose-Einstein condensates

Perturbation theory for bright spinor Bose-Einstein condensate solitons

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