Systematic method of generating new integrable systems via inverse Miura maps

Tsuchida, Takayuki

doi:10.1063/1.3563585

Cited by 15 publications

(17 citation statements)

References 93 publications

(136 reference statements)

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“…In our previous paper [17], we proposed a systematic method of generating new integrable systems from known systems through inverse Miura maps. 3 As a result, two derivative NLS systems, namely, the Gerdjikov-Ivanov (also known as Ablowitz-Ramani-Segur) system [10,11] and the Chen-Lee-Liu system [9], were constructed from the Lax representation 4 for the NLS system; the same prescription applies to the space-discrete case.…”

Section: Introductionmentioning

confidence: 99%

“…In section 2, we start with the Lax representation for the Ablowitz-Ladik lattice; using its linear eigenfunctions, we derive two derivative NLS lattices. First, we apply the method proposed in [17] and derive the space-discrete Gerdjikov-Ivanov system. Second, we propose a new systematic method and obtain the spacediscrete Kaup-Newell system.…”

Section: Introductionmentioning

confidence: 99%

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A refined and unified version of the inverse scattering method for the Ablowitz-Ladik lattice and derivative NLS lattices

Tsuchida¹

2012

Preprint

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We refine and develop the inverse scattering theory on a lattice in such a way that the Ablowitz-Ladik lattice and derivative NLS lattices as well as their matrix analogs can be solved in a unified way. The inverse scattering method for the (matrix analog of the) Ablowitz-Ladik lattice is simplified to the same level as that for the continuous NLS system. Using the linear eigenfunctions of the Lax pair for the Ablowitz-Ladik lattice, we can construct solutions of the derivative NLS lattices such as the discrete Gerdjikov-Ivanov (also known as Ablowitz-Ramani-Segur) system and the discrete Kaup-Newell system. Thus, explicit solutions such as the multisoliton solutions for these systems can be obtained by solving linear summation equations of the Gel'fand-Levitan-Marchenko type. The derivation of the discrete Kaup-Newell system from the Ablowitz-Ladik lattice is based on a new method that allows us to generate new integrable systems from known systems in a systematic manner. In an appendix, we describe the reduction of the matrix Ablowitz-Ladik lattice to a vector analog of the modified Volterra lattice from the point of view of the inverse scattering method.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A refined and unified version of the inverse scattering method for the Ablowitz-Ladik lattice and derivative NLS lattices

Tsuchida¹

2012

Preprint

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“…One new development that we have introduced here is an extension the theory to obtain integrable systems of NLS-type by exploiting a U(1) subgroup given by the center of the unitary equivalence group of the U(n)-parallel frame in the symmetric spaces M = SU(n + 1)/U(n) and M = SO(2n)/U(n). In the case of M = SU(n + 1)/U(n), this leads to a scalar-vector version of the Yajima-Oikawa system [31,30], whereas in the case of M = SO(2n)/U(n), we obtain a novel nonlocal NLS system.…”

Section: Discussionmentioning

confidence: 99%

Unitarily-invariant integrable systems and geometric curve flows inSU(n + 1)/U(n) andSO(2n)/U(n)

Ahmed¹,

Anco²,

Asadi³

2018

J. Phys. A: Math. Theor.

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1 department of mathematics university of south florida tampa, fl 33620, usa 2 department of mathematics and statistics brock university st. catharines, on l2s3a1, canada 3 department of mathematics institute for advance studies in basic science (iasbs) 45137-66731, zanjan, iran Abstract. Bi-Hamiltonian hierarchies of soliton equations are derived from geometric nonstretching (inelastic) curve flows in the Hermitian symmetric spaces SU (n + 1)/U (n) and SO(2n)/U (n). The derivation uses Hasimoto variables defined by a moving parallel frame along the curves. As main results, new integrable multi-component versions of the Sine-Gordon (SG) equation and the modified Korteveg-de Vries (mKdV) equation, as well as a novel nonlocal multi-component version of the nonlinear Schrödinger (NLS) equation are obtained, along with their bi-Hamiltonian structures and recursion operators. These integrable systems are unitarily invariant and correspond to geometric curve flows given by a non-stretching wave map and a mKdV analog of a non-stretching Schrödinger map in the case of the SG and mKdV systems, and a generalization of the vortex filament bi-normal equation in the case of the NLS systems. N x = −κT + τB,B x = −τ N x , while the tangential coefficient a is determined by a x = κb due to the non-stretching property of the curve. When b = a = 0 and c = κ, the curve γ undergoes a bi-normal flow γ t = κB. This flow equation physically describes the motion of a vortex filament in incompressible fluids [5]. The induced flow on (κ, τ ) turns out to be equivalent to the NLS equation for the Hasimoto variable u = κ exp(i τ dx) [5]. Moreover, the Lax pair and bi-Hamiltonian operators for the NLS equation turn out to be encoded in a simple way in the structure equations of a moving frame formulation of the curve flow [3, 4], where the Hasimoto transformation from (κ, τ ) to u corresponds geometrically to a gauge transformation from a Frenet frame to a parallel frame given by rotating the vectors (N,B) in the normal plane by an angle θ(x) = − τ dx along the curve [6]. Unlike a Frenet frame, a parallel frame has a rigid gauge freedom consisting of a constant rotation φ applied to the vectors (N,B) in the normal plane. Under this rigid gauge transformation, u transforms to e iφ u by a constant phase rotation, and so u is not an invariant of the curve like (κ, τ ) but instead has the geometrical meaning of a U(1)-covariant of the curve [4].Similarly, all of the symmetries of the NLS equation themselves correspond to geometrical curve flows in Euclidean space, and in particular the first higher symmetry in the NLS hierarchy is given by γ t = κτB + κ xN + 1 2T with the Hasimoto variable u satisfying the complex modified Korteveg-de Vries (mKdV) equation u t = u xxx + 3 2 |u| 2 u x . This flow equation physically describes axial motion of a vortex filament [7]. It is also an integrable system, sharing the same integrability properties as the NLS equation.A broad generalization of parallel frames and Hasimoto variables has been obtained in work...

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“…The (1+2)-dimensional systems (5.18) and (5.22) have been mentioned in the literature, see [10,15,[18][19][20]24]. However, these papers are devoted mostly to the interrelations between various integrable models and the algebraic structures behind them.…”

Section: Discussionmentioning

confidence: 99%

Functional representation of the negative DNLS hierarchy

Vekslerchik¹

2021

JNMP

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This paper is devoted to the negative flows of the derivative nonlinear Schrödinger hierarchy (DNLSH). The main result of this work is the functional representation of the extended DNLSH, composed of both positive (classical) and negative flows. We derive a finite set of functional equations, constructed by means of the Miwa's shifts, which contains all equations of the hierarchy. Using the obtained functional representation we convert the nonlocal equations of the negative subhierarchy into local ones of higher order, derive the generating function of the conservation laws and the N-soliton solutions for the extended DNLSH under non-vanishing boundary conditions.

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Systematic method of generating new integrable systems via inverse Miura maps

Cited by 15 publications

References 93 publications

A refined and unified version of the inverse scattering method for the Ablowitz-Ladik lattice and derivative NLS lattices

A refined and unified version of the inverse scattering method for the Ablowitz-Ladik lattice and derivative NLS lattices

Unitarily-invariant integrable systems and geometric curve flows inSU(n + 1)/U(n) andSO(2n)/U(n)

Functional representation of the negative DNLS hierarchy

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