Completeness of the eigenfunctions for the Caudrey–Beals–Coifman system

Gerdjikov, Vladimir S.; Yanovski, A.B.

doi:10.1063/1.530441

Cited by 86 publications

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“…Beals and Coifman showed that any potential can be approximated to arbitrary accuracy with a potential on a finite support [65]. This result was later generalized to potential with coefficients from any finite-dimensional simple Lie algebra [11]. Reformulating the general results of [11,32], for the particular choice of the algebra D 4 ≃ so(8) we find that 1.…”

Section: Spectral Properties Of L and Fundamental Analytic Solutionsmentioning

confidence: 99%

“…This result was later generalized to potential with coefficients from any finite-dimensional simple Lie algebra [11]. Reformulating the general results of [11,32], for the particular choice of the algebra D 4 ≃ so(8) we find that 1. The continuous spectrum of L fills up the set of rays l ν , ν = 0, .…”

Section: Spectral Properties Of L and Fundamental Analytic Solutionsmentioning

confidence: 99%

“…This approach is based on the Wronskian relations, which are basic tools for the analysis of the mapping between the potential and the scattering data. They allow the construction of the "squared solutions" of the Lax operator, which form a complete set of functions and play the role of the generalized exponentials (see [10,11,12] and the references therein). Another powerful technique we should mention here is that of the so called recursion operators.…”

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Soliton equations related to the affine Kac-Moody algebra D 4 (1)

et al. 2015

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We have derived the hierarchy of soliton equations associated with the untwisted affine Kac-Moody algebra D 4 by calculating the corresponding recursion operators. The Hamiltonian formulation of the equations from the hierarchy is also considered. As an example we have explicitly presented the first non-trivial member of the hierarchy, which is an one-parameter family of mKdV equations. We have also considered the spectral properties of the Lax operator and introduced a minimal set of scattering data. IntroductionAfter the discovery of the inverse scattering method (ISM) in 1967 by the Princeton group of Gardner, Greene, Kruskal and Miura [1] the study of soliton equations and the related integrable nonlinear evolution equations (NLEEs) became one of the most active branches of nonlinear science. In this pioneering paper, the Korteveg-de Vries (KdV) equation was exactly integrated using the ISM. One year later, Peter Lax presented a method, which would later be instrumental in the search of new integrable equations different from the KdV[2]. This started numerous attempts to extend the range of application of the ISM to other equations.Three years had passed, when Gardner started the canonical approach to the NLEEs. He was the first to realize that the KdV equation can be written in a Hamiltonian form [3]. Soon after that, Zakharov and Faddeev proved that KdV equation is a completely 1 integrable Hamiltonian system [4] and Wadati solved the modified Korteweg-de Vries (mKdV) equation [5].This explosion of interest in the soliton science led to the discovery of a completely new world of unknown before integrable equations together with methods and techniques for finding their exact solutions. We should mention some of those methods and techniques, starting with the fundamental analytic solution (FAS), which was introduced by Shabat [6]. Within this approach one can show that the ISM is equivalent to a multiplicative Reimman-Hilbert problem (RHP), which is the basis for the dressing method invented by Zakharov and Shabat [7,8]. Another important result was derived by Ablowitz, Kaup, Newell and Segur [9]. They showed that the ISM can be interpreted as a generalized Fourier transform. This approach is based on the Wronskian relations, which are basic tools for the analysis of the mapping between the potential and the scattering data. They allow the construction of the "squared solutions" of the Lax operator, which form a complete set of functions and play the role of the generalized exponentials (see [10,11,12] and the references therein). Another powerful technique we should mention here is that of the so called recursion operators. This method has in principal a long history but the application to the NLEEs started in [13] (see also [14] and [15]) where with the help of the recursion operator the KdV equation was written in a compact form. Within the study of the Nonlinear Schrödinger (NS) equation, considered in the rapidly decreasing case and imposing the requirement that the squared Jost solutions of th...

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Section: Spectral Properties Of L and Fundamental Analytic Solutionsmentioning

confidence: 99%

Section: Spectral Properties Of L and Fundamental Analytic Solutionsmentioning

confidence: 99%

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confidence: 99%

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Soliton equations related to the affine Kac-Moody algebra D 4 (1)

et al. 2015

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“…Combining the ideas of [7] with the symmetries of the potential (1.6) we can construct a set of 2h fundamental analytic solutions (FAS) m ν (x, t, λ) of (2.1) and prove that:…”

Section: The Spectral Properties Of Lmentioning

confidence: 99%

On Reductions and Real Hamiltonian Forms of Affine Toda Field Theories

Gerdjikov¹,

Grahovski²

2005

JNMP

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A family of real Hamiltonian forms (RHF) for the special class of affine 1 + 1-dimensional Toda field theories is constructed. Thus the method, proposed in [1] for systems with finite number of degrees of freedom is generalized to infinite-dimensional Hamiltonian systems. We show that each of these RHF is related to a special Z 2 -symmetry of the system of roots for the relevant Kac-Moody algebra. A number of explicit nontrivial examples of RHF of ATFT are presented.

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“…Then the construction of the fundamental analytic solutions and the solution of the corresponding inverse scattering problem for (1.2) becomes more difficult [11,12].…”

Section: Introductionmentioning

confidence: 99%

Reductions ofN-wave interactions related to low-rank simple Lie algebras: I. ℤ₂-reductions

Gerdjikov¹,

Grahovski²,

Kostov³

2001

J. Phys. A: Math. Gen.

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Abstract. The analysis and the classification of all reductions for the nonlinear evolution equations solvable by the inverse scattering method is an interesting and still open problem. We show how the second order reductions of the N -wave interactions related to low-rank simple Lie algebras g can be embedded also in the Weyl group of g. This allows us to display along with the well known ones a number of new types of integrable N -wave systems. Some of the reduced systems find applications to nonlinear optics.

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Completeness of the eigenfunctions for the Caudrey–Beals–Coifman system

Cited by 86 publications

References 47 publications

Soliton equations related to the affine Kac-Moody algebra D 4 (1)

Soliton equations related to the affine Kac-Moody algebra D 4 (1)

On Reductions and Real Hamiltonian Forms of Affine Toda Field Theories

Reductions ofN-wave interactions related to low-rank simple Lie algebras: I. ℤ₂-reductions

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Completeness of the eigenfunctions for the Caudrey–Beals–Coifman system

Cited by 86 publications

References 47 publications

Soliton equations related to the affine Kac-Moody algebra D 4 (1)

Soliton equations related to the affine Kac-Moody algebra D 4 (1)

On Reductions and Real Hamiltonian Forms of Affine Toda Field Theories

Reductions ofN-wave interactions related to low-rank simple Lie algebras: I. ℤ2-reductions

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Product

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Reductions ofN-wave interactions related to low-rank simple Lie algebras: I. ℤ₂-reductions