The Initial-boundary Value Problem for the Ostrovsky-Vakhnenko Equation on the Half-line

Jian, Xu; Fan, Engui

doi:10.1007/s11040-016-9223-z

Cited by 11 publications

(11 citation statements)

References 22 publications

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“…In 2012, Lenells first develops a methodology for analyzing IBV problems on the half-line for integrable evolution equations with Lax pairs involving 3 × 3 matrices [18]. Although the transition from 2 × 2 to 3 × 3 matrix Lax pairs involves a number of novelties, the two main steps of the method of [3,6] remain the same: After Lenells' work, IBV problems on the half-line for other integrable evolution equations such as the Degasperis-Procesi [19], Sasa-Satsuma [20], three wave [21], the two-component nonlinear Schrödinger [22,23,24], the Ostrovsky-Vakhnenko [25] equations, are analyzed. However, within the knowledge of the authors, the IBV problems for integrable equations with 3 × 3 matrices Lax pair on the finite interval have not been studied yet.…”

Section: Introductionmentioning

confidence: 99%

Initial‐Boundary Value Problem for Integrable Nonlinear Evolution Equation with 3 × 3 Lax Pairs on the Interval

Jian

Fan

2016

Stud Appl Math

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We present an approach for analyzing initial-boundary value problems which are formulated on the finite interval (0 ≤ x ≤ L, where L is a positive constant) for integrable equation whose Lax pairs involve 3 × 3 matrices. Boundary value problems for integrable nonlinear evolution partial differential equations (PDEs) can be analyzed by the unified method introduced by Fokas and developed by him and his collaborators. In this paper, we show that the solution can be expressed in terms of the solution of a 3 × 3 Riemann-Hilbert problem (RHP). The relevant jump matrices are explicitly given in terms of the three matrix-value spectral functions s(k), S(k), and S L (k), which in turn are defined in terms of the initial values, boundary values at x = 0, and boundary values at x = L, respectively. However, these spectral functions are not independent; they satisfy a global relation. Here, we show that the characterization of the unknown boundary values in terms of the given initial and boundary data is explicitly described for a nonlinear evolution PDE defined on the interval. Also, we show that in the limit when the length of the interval tends to infinity, the relevant formulas reduce to the analogous formulas obtained for the case of boundary value problems formulated on the half-line.

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Section: Introductionmentioning

confidence: 99%

Initial‐Boundary Value Problem for Integrable Nonlinear Evolution Equation with 3 × 3 Lax Pairs on the Interval

Jian

Fan

2016

Stud Appl Math

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“…Recently, Lenells further developed the Fokas method to analyze the IBV problems for integrable NLEEs with 3 × 3 Lax pairs on the half-line [32]. After that, the modified approach was applied in the IBV problems of other integrable NLEEs with 3 × 3 Lax pairs on the half-line or the finite interval, such as the Degasperis-Procesi equation [33], the Sasa-Satsuma equation [34], the coupled NLS equations [35][36][37], and the Ostrovsky-Vakhnenko equation [38].…”

Section: Introductionmentioning

confidence: 99%

Initial-boundary value problem for the spin-1 Gross-Pitaevskii system with a 4 × 4 Lax pair on a finite interval

Yan

2019

Journal of Mathematical Physics

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In this paper, we explore the initial-boundary value (IBV) problem for an integrable spin-1 Gross-Pitaevskii system with a 4 × 4 Lax pair on the finite interval x ∈ [0, L] by extending the Fokas unified transform approach. The solution of this system can be expressed in terms of the solution of a 4 × 4 matrix Riemann-Hilbert (RH) problem formulated in the complex k-plane. Furthermore, the relevant jump matrices with explicit (x, t)-dependence of the matrix RH problem can be explicitly found via three spectral functions {s(k), S(k), S L (k)} arising from the initial data and the Dirichlet-Neumann boundary conditions at x = 0 and x = L, respectively. The global relation is also found to deduce two distinct but equivalent types of representations (i.e., one via the large k of asymptotics of the eigenfunctions and another one in terms of the Gel'fand-Levitan-Marchenko (GLM) approach) for the Dirichlet and Neumann boundary value problems. In particular, the formulae for IBV problems on the finite interval can reduce to ones on a half-line as the length L of the interval approaches to infinity. Moreover, we also present the linearizable boundary conditions for the GLM representations.(c) S L (k) is determined using the boundary data at x = L, q j (x = L, t) = v 0j (t), q jx (x = L, t) = v 1j (t), j = 1, 0, −1, 0 < t < T ;• Show that these above-mentioned spectral functions satisfy a global relation, which implies that the initial-boundary value conditions can not be chosen arbitrary.Step 2. Use the spectral functions {s(k), S(k), S L (k)} to determine a regular Reimann-Hilbert problem, whose solution can generate a solution of the spin-1 GP system (1).Step 3. The Gel'fand-Levitan-Marchenko (GLM) representations can also given for the IBV problem of system (1).The rest of this paper is organized as follows. In Sec. 2, we introduce the 4 × 4 Lax pair of Eq. (1) and explore its spectral analysis such as the eigenfunctions, the jump matrices, and the global relation. Sec. 3 exhibits the corresponding 4 × 4 matrix RH problem in terms of the jump

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“…The former has a complicated calculation procedure, while the latter can provide an equivalent and simpler method for solving integrable equations, especially soliton solutions [10][11][12][13][14]. R-H approach [15][16][17][18][19] is extensively applied in lots of nonlinear PDEs, for example, the coupled mKdV equation [20], the generalized Sasa-Satsuma equation [21], the general coupled nonlinear Schrödinger equation [22], which plays a vital role in dealing with initial boundary value problems [23][24][25], discussing long-time asymptotic behavior [26] and investigating the lump solutions [27].…”

Section: Introductionmentioning

confidence: 99%

Prolongation Structures and N‐Soliton Solutions for a New Nonlinear Schrödinger‐Type Equation via Riemann‐Hilbert Approach

Lin

Fang

Dong

2019

Mathematical Problems in Engineering

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In this paper, a new integrable nonlinear Schrödinger-type (NLST) equation is investigated by prolongation structures theory and Riemann-Hilbert (R-H) approach. Via prolongation structures theory, the Lax pair of the NLST equation, a 2×2 matrix spectral problem, is derived. Depending on the analysis of red the spectral problem, a R-H problem of the NLST equation is formulated. Furthermore, through a specific R-H problem with the vanishing scattering coefficient, N-soliton solutions of the NLST equation are expressed explicitly. Moreover, a few key differences are presented, which exist in the implementation of the inverse scattering transform for NLST equation and cubic nonlinear Schrödinger (NLS) equation. Finally, the dynamic behaviors of soliton solutions are shown by selecting appropriate spectral parameter λ, respectively.

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The Initial-boundary Value Problem for the Ostrovsky-Vakhnenko Equation on the Half-line

Cited by 11 publications

References 22 publications

Initial‐Boundary Value Problem for Integrable Nonlinear Evolution Equation with 3 × 3 Lax Pairs on the Interval

Initial‐Boundary Value Problem for Integrable Nonlinear Evolution Equation with 3 × 3 Lax Pairs on the Interval

Initial-boundary value problem for the spin-1 Gross-Pitaevskii system with a 4 × 4 Lax pair on a finite interval

Prolongation Structures and N‐Soliton Solutions for a New Nonlinear Schrödinger‐Type Equation via Riemann‐Hilbert Approach

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