Integrable Hierarchy of Higher Nonlinear Schrödinger Type Equations

Kundu, Anjan

doi:10.3842/sigma.2006.078

Cited by 14 publications

(10 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The KE equation is an integrable equation and introduced independently during 1984 by Kundu [29] and later by Calogero and Eckhaus [30] from different perspective. The KE equation has been studied extensively on the integrability associated with explicit form of the Lax pair and Painlevé property [31], Hamiltonian structure [32], higher order extension [33], Miura transformation [34], infinitely many conservation laws [35], soliton given by the bilinear method [36] and other methods [37][38][39][40], rogue waves solutions given by the DT method [41][42][43][44][45].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Riemann-Hilbert approach to the initial-boundary value problem for Kundu-Eckhaus equation on the half line

Xia

2019

Complex Variables and Elliptic Equations

View full text Add to dashboard Cite

In this paper, we consider the initial-boundary value problem of the Kundu-Eckhaus equation on the half-line by using of the Fokas unified transform method. Assuming that the solution u(x, t) exists, we show that it can be expressed in terms of the unique solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter λ. Moreover, we also get that some spectral functions are not independent and satisfy the so-called global relation.Assume that the solution u(x; t) of the KE equation exists, and the initial-boundary values data are defined as follows( 1.2) We will show that u(x, t) can be expressed in terms of the unique solution of a matrix RHP formulated in the plane of the complex spectral parameter λ. The jump matrix has explicit (x, t) dependence and is given in terms of the spectral functions a(λ), b(λ) and A(λ), B(λ), which obtained from the initial data u 0 (x) = u(x, 0) and the boundary data g 0 (t) = u(0, t), g 1 (t) = u x (0, t), respectively. The problem has the jump across {Imλ 2 = 0}. The spectral functions are not independent, but related by a compatibility condition, the so-called global relation, which is an algebraic equation coupling a(λ), b(λ) and A(λ), B(λ). Where a different RHP was formulated and a different representation of the solution of the KE equation was given, which are more convenient for studying the long-time asymptotic behavior for the solutions of the KE equation with the decay initial and boundary values lie in the Schwartz class.Organization of this paper is as follows. In section 2, some summary results and the basic RHP of the KE equation are given. In section 3, the spectral functions a(λ), b(λ) and A(λ), B(λ) are investigated and the RHP is presented. The last section is devoted to conclusions and discussions.

show abstract

Section: Introductionmentioning

confidence: 99%

“…Hamiltonian structure [32], higher order extension [33], Miura transformation [34], infinitely many conservation laws [35], soliton given by the bilinear method [36] and other methods [37][38][39][40], rogue waves solutions given by the DT method [41][42][43][44][45].…”

Section: Introductionmentioning

confidence: 99%

A Riemann-Hilbert approach to the initial-boundary value problem for Kundu-Eckhaus equation on the half line

Xia

2019

Complex Variables and Elliptic Equations

View full text Add to dashboard Cite

show abstract

“…The soliton solutions, rogue wave solutions and the flow wave of the KE equation can be obtained by the Darboux transformations [4,16] and the bilinear method [17]. Via a direct method [18], higher order extension [19] and the tan(φ(ζ))-expansion method [20], the abundant soliton solutions of the KE equation, etc, were obtained. Recently, the bright soliton solutions and the long-time asymptotics with zero boundary conditions of the KE equation were obtained in [1,25].…”

Section: Introductionmentioning

confidence: 99%

Inverse scattering transform for the Kundu-Eckhaus Equation with nonzero boundary condition

Guo¹,

Jian²

2019

Preprint

View full text Add to dashboard Cite

In this paper, we consider the initial value problem for both of the defocusing and focusing Kundu-Eckhaus (KE) equation with non-zero boundary conditions (NZBCs) at infinity by inverse scattering transform method. The solutions of the KE equation with NZBCs can be reconstructed in terms of the solution of an associated 2×2 matrix Riemann-Hilbert problem (RHP). In our formulation, both the direct and the inverse problems are posed in terms of a suitable uniformization variable which allows us to develop the IST on the standard complex plane instead of a two-sheeted Riemann surface or the cut plane with discontinuities along the cuts. Furthermore, on the one hand, we obtain the N-soliton solutions with simple pole of the defocusing and focusing KE equation with the NZBCs, especially, the explicit one-soliton solutions are given in details. And we prove that the scattering data a(ζ) of the defocusing KE equation can only have simple zeros. On the other hand, we also obtain the soliton solutions with double pole of the focusing KE equation with NZBCs. And we show that the double pole solutions can be viewed as some proper limit of the two simple pole soliton solutions. Some dynamical behaviors and typical collisions of the soliton solutions of both of the defocusing and focusing KE equation are shown graphically.

show abstract

“…Considering the significance of the higher order nonlinearities in physical system, the DNLS equation yields an integrable higher nonlinear equation, i.e. Kundu-DNLS equation [3,4] …”

Section: Introductionmentioning

confidence: 99%