The multicomponent Eckhaus equation

Calogero, F.; Degasperis, A.; Lillo, S. De

doi:10.1088/0305-4470/30/16/021

Cited by 15 publications

(24 citation statements)

References 10 publications

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“…The transformations can be calculated recursively and they lead to an infinite number of conserved quantities. We have also shown that all equations (25) admit travelling wave solutions which are given implicitly in terms of the hypergeometric function, except for α = ±1, ±2 when the solutions are given in explicit form. However, it is an open problem to determine if equation (25) is integrable for all α ∈ Z \ {0}.…”

Section: Resultsmentioning

confidence: 88%

“…Any smooth solution of equation (27) generates a residual gauge transformation of (25). We noted earlier that equation (25) is integrable for α = 1, −2.…”

Section: General Frameworkmentioning

confidence: 94%

“…It is also noted that for α = 1 the matrices A can be extended to matrix polynomials in the loop algebra sl(2, C) ⊗ C[λ, λ −1 ] appearing in the Riemann-Hilbert factorization for the KdV equation (see [12]). It would be a very interesting problem to determine if this construction carries over for all α and if (25) can be integrated by solving a Riemann-Hilbert factorization problem.…”

Section: General Frameworkmentioning

confidence: 99%

“…If we choose w so that the (2, 1) element of A yields the Calogero KdV (CKdV) equation [25] (83) w t = w xxx − 1 2 3w…”

Section: Evaluation Of the Condition (70) Withmentioning

confidence: 99%

“…is obtained from (25) by setting α = −2 and s = 1. It was discovered by Harry Dym when trying to transfer some results about the isospectral theory to the string equation [29].…”

Section: The Harry Dym Equationmentioning

confidence: 99%

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Gauge transformations and symmetries of integrable systems

Fukuyama

Kamimura

Krešić–Jurić

et al. 2007

J. Phys. A: Math. Theor.

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Abstract. We analyze several integrable systems in zero-curvature form within the framework of SL(2, R) invariant gauge theory. In the Drinfeld-Sokolov gauge we derive a two-parameter family of nonlinear evolution equations which as special cases include the Kortweg-de Vries (KdV) and Harry Dym equations. We find residual gauge transformations which lead to infinitesimal symmetries of this family of equations. For KdV and Harry Dym equations we find an infinite hierarchy of such symmetry transformations, and we investigate their relation with local conservation laws, constants of the motion and the bi-Hamiltonian structure of the equations. Applying successive gauge transformations of Miura type we obtain a sequence of gauge equivalent integrable systems, among them the modified KdV and Calogero KdV equations.

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Section: Resultsmentioning

confidence: 88%

“…Any smooth solution of equation (27) generates a residual gauge transformation of (25). We noted earlier that equation (25) is integrable for α = 1, −2.…”

Section: General Frameworkmentioning

confidence: 94%

Section: General Frameworkmentioning

confidence: 99%

“…If we choose w so that the (2, 1) element of A yields the Calogero KdV (CKdV) equation [25] (83) w t = w xxx − 1 2 3w…”

Section: Evaluation Of the Condition (70) Withmentioning

confidence: 99%

“…is obtained from (25) by setting α = −2 and s = 1. It was discovered by Harry Dym when trying to transfer some results about the isospectral theory to the string equation [29].…”

Section: The Harry Dym Equationmentioning

confidence: 99%

See 3 more Smart Citations

Gauge transformations and symmetries of integrable systems

Fukuyama

Kamimura

Krešić–Jurić

et al. 2007

J. Phys. A: Math. Theor.

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Gauge Equivalence among Quantum Nonlinear Many Body Systems

Scarfone

2008

Acta Appl Math

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Transformations performing on the dependent and/or the independent variables are an useful method used to classify PDE in class of equivalence. In this paper we consider a large class of U(1)-invariant nonlinear Schrödinger equations containing complex nonlinearities. The U(1) symmetry implies the existence of a continuity equation for the particle density ρ ≡ |ψ| 2 where the current j ψ has, in general, a nonlinear structure. We introduce a nonlinear gauge transformation on the dependent variables ρ and j ψ which changes the evolution equation in another one containing only a real nonlinearity and transforms the particle current j ψ in the standard bilinear form. We extend the method to U(1)-invariant coupled nonlinear Schrödinger equations where the most general nonlinearity is taken into account through the sum of an Hermitian matrix and an anti-Hermitian matrix. By means of the nonlinear gauge transformation we change the nonlinear system in another one containing only a purely Hermitian nonlinearity. Finally, we consider nonlinear Schrödinger equations minimally coupled with an Abelian gauge field whose dynamics is governed, in the most general fashion, through the Maxwell-Chern-Simons equation. It is shown that the nonlinear transformation we are introducing can be applied, in this case, separately to the gauge field or to the matter field with the same final result. In conclusion, some relevant examples are presented to show the applicability of the method.

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