A Riemann-Hilbert approach to the initial-boundary problem for derivative nonlinear Schrödinger equation

Jian, Xu; Fan, Engui

doi:10.1016/s0252-9602(14)60063-1

Cited by 33 publications

(18 citation statements)

References 30 publications

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“…It should be pointed out that the two-component mYOLS Equation (10) is different from the long-wave-short-wave resonance Equation (1), and the two-component mYOLS Equation (10) is equivalent to the long-wave-short-wave model (2) under some transformation [21]. Then, with the help of Riccati equations for the Lax pair associated with the vmYOLS Equation (8) [22][23][24], Bäcklund transformation [25,26], Darboux transformation [27][28][29][30][31][32][33][34][35][36][37][38][39], and others [40][41][42][43][44][45][46][47][48][49][50][51]. Some interesting explicit solutions have been found, the most important among which are pure-soliton solutions, quasi-periodic solutions, and rogue waves solutions.…”

Section: Introductionmentioning

confidence: 99%

On a Vector Modified Yajima–Oikawa Long-Wave–Short-Wave Equation

Geng

2019

Mathematics

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A vector modified Yajima–Oikawa long-wave–short-wave equation is proposed using the zero-curvature presentation. On the basis of the Riccati equations associated with the Lax pair, a method is developed to construct multi-fold classical and generalized Darboux transformations for the vector modified Yajima–Oikawa long-wave–short-wave equation. As applications of the multi-fold classical Darboux transformations and generalized Darboux transformations, various exact solutions for the vector modified long-wave–short-wave equation are obtained, including soliton, breather, and rogue wave solutions.

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Section: Introductionmentioning

confidence: 99%

On a Vector Modified Yajima–Oikawa Long-Wave–Short-Wave Equation

Geng

2019

Mathematics

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“…Reviewing what we did before, when (21) is a solution of (19), we have Ψ o 1 = i 2 QDσ 3 . Suppose (32).…”

Section: The Riemann-hilbert Problem and Some Relationsmentioning

confidence: 99%

“…The Riemann-Hilbert approach was introduced by Fokas to analyze the initial-boundary values problem for linear and nonlinear partial differential equations [23,24]. In the past 20 years, many researchers have discussed a lot of nonlinear integrable equations for the initial-boundary values problem [25][26][27][28][29][30][31][32][32][33][34][35][36][37][38][39][40][41]. They have all made a great contribution to the development of this method.…”

Section: Introductionmentioning

confidence: 99%

The Prolongation Structure of the Modified Nonlinear Schrödinger Equation and Its Initial-Boundary Value Problem on the Half Line via the Riemann-Hilbert Approach

Liu

Dong

2019

Mathematics

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In this paper, the Lax pair of the modified nonlinear Schrödinger equation (mNLS) is derived by means of the prolongation structure theory. Based on the obtained Lax pair, the mNLS equation on the half line is analyzed with the assistance of Fokas method. A Riemann-Hilbert problem is formulated in the complex plane with respect to the spectral parameter. According to the initial-boundary values, the spectral function can be defined. Furthermore, the jump matrices and the global relations can be obtained. Finally, the potential q ( x , t ) can be represented by the solution of this Riemann-Hilbert problem.

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“…Remark 3. 13 We can use similar method with [11] to prove that initial values u(x, 0) = u 0 (x) and boundary values u(0, t) = g 0 (t), u x (0, t) = g 1 (t), we omit this proof in here because of the length of this article.…”

Section: Remark 39mentioning

confidence: 99%

“…In 1997, Fokas announced a new unified approach based on the Riemann-Hilbert factorization problem to analysis the IBV problems for linear and nonlinear integrable PDEs [1,2,3], we call that Fokas unified transform method. This method provides an important generalization of the IST formalism from initial value to IBV problems, and over the last 20 years, this method has been used to analyse boundary value problems for several of the most important integrable equations possessing 2 × 2 Lax pairs, such as the KdV, the nonlinear Schrödinger(NLS), the sine-Gordon and the stationary axisymmetric Einstein equations and so on [4][5][6][7][8][9][10][11][12][13][14][15]. In 2012, Lenells first extended the Fokas unified transform method to the IBV problem for the 3 × 3 matrix Lax pair [16,17].…”

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

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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.

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A Riemann-Hilbert approach to the initial-boundary problem for derivative nonlinear Schrödinger equation

Cited by 33 publications

References 30 publications

On a Vector Modified Yajima–Oikawa Long-Wave–Short-Wave Equation

On a Vector Modified Yajima–Oikawa Long-Wave–Short-Wave Equation

The Prolongation Structure of the Modified Nonlinear Schrödinger Equation and Its Initial-Boundary Value Problem on the Half Line via the Riemann-Hilbert Approach

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

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