Long-Time Asymptotics for the Nonlocal MKdV Equation*

He, Feng-Jing; Fan, Engui; Jian, Xu

doi:10.1088/0253-6102/71/5/475

Cited by 38 publications

(25 citation statements)

References 33 publications

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“…In the case of the NNLS equation, the reflection coefficients r 1 (k) and r 2 (k) (see Section 2.2) are not connected (in contrast with the classical NLS equation); this "lack of symmetry" implies, in particular, that F ∞ (k 1 ) and G ∞ (k 0 , α) in (1.3) and (1.4) can be complexvalued and thus the modulus of the asymptotics depends on the initial data. A similar lack of symmetry holds for other integrable nonlocal equations (see, e.g., [3,5,31,35,55]), which allows us to conjecture that the modulation instability in other nonlocal models should also exhibit a kind of non-universal behavior.…”

Section: Theorem 1 (Plane Wave Region)mentioning

confidence: 56%

Asymptotic stage of modulation instability for the nonlocal nonlinear Schrödinger equation

Rybalko¹,

Shepelsky²

2021

Preprint

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We study the initial value problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equation iq t (x, t) + q xx (x, t) + 2q 2 (x, t)q(−x, t) = 0 with symmetric boundary conditions: q(x, t) → Ae 2iA 2 t as x → ±∞, where A > 0 is an arbitrary constant. We describe the asymptotic stage of modulation instability for the NNLS equation by computing the large-time asymptotics of the solution q(x, t) of this initial value problem. We shown that it exhibits a non-universal, in a sense, behavior: the asymptotics of |q(x, t)| depends on details of the initial data q(x, 0). This is in a sharp contrast with the local classical NLS equation, where the long-time asymptotics of the solution depends on the initial value through the phase parameters only. The main tool used in this work is the inverse scattering transform method applied in the form of the matrix Riemann-Hilbert problem. The Riemann-Hilbert problem associated with the original initial value problem is analyzed asymptotically by the nonlinear steepest decent method.

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Section: Theorem 1 (Plane Wave Region)mentioning

confidence: 56%

Asymptotic stage of modulation instability for the nonlocal nonlinear Schrödinger equation

Rybalko¹,

Shepelsky²

2021

Preprint

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“…27 It has become clear that the Riemann-Hilbert approach is applicable to the construction of exact solutions and asymptotic analysis of solutions for a wide class of integrable systems. [28][29][30][31][32][33][34][35] In this paper, we apply the Riemann-Hilbert approach to study discrete sine-Gordon equation (1) and obtain solutions in simple-pole and double-pole cases. In 1980s, Levi et al studied inverse spectral transform of (1) by classical inverse scattering method.…”

Section: Introductionmentioning

confidence: 99%

Riemann–Hilbert approach for discrete sine‐Gordon equation with simple and double poles

Chen

Fan

2021

Stud Appl Math

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In this paper, we present the inverse scattering transformation (IST) for discrete sine-Gordon equation via the Riemann-Hilbert approach. In the direct scattering part, we establish analyticity, symmetries, and asymptotic properties of Jost solutions and reflection coefficients. In the inverse part, we solve the discrete sine-Gordon equation by establishing a Riemann-Hilbert problem with simple and double poles, respectively. Nsoliton solutions are obtained via the reconstruction formula correspondent to the Riemann-Hilbert problem.As examples of N-solitons, we present some explicit solutions such as breathers, degenerate solitons for discrete sine-Gordon equation, and the dynamical properties of these solutions are further analyzed.

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“…Yang et al proposed the localized wave solutions of the reverse space nonlocal Lakshmanan-Porsezian-Daniel equation by the Darboux transformations [25]. He, Fan and Xu studied the Cauchy problem with decaying initial data for the reverse space-time nonlocal modified KdV equation by Riemann-Hilbert method [26]. Feng et al [27] considered a nonlocal nonlinear Schrödinger equation with PT-symmetry for both zero and nonzero boundary conditions via the combination of Hirota's bilinear method and the Kadomtsev-Petviashvili hierarchy reduction method.…”

Section: Introductionmentioning

confidence: 99%

Integrability and Multisoliton Solutions of the Reverse Space or/and Time Nonlocal Fokas-Lenells (FL) Equation

Zhang

Liu

2021

Preprint

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This paper studies reverse space or/and time nonlocal Fokas-Lenells (FL) equation, which describes the propagation of nonlinear light pulses in monomode optical fibers when certain higher-order nonlinear effects are considered, by Hirota bilinear method. Firstly, variable transformations from reverse space nonlocal FL equation to reverse time and reverse space-time nonlocal FL equations are constructed. Secondly, the one-, two- and three-soliton solutions of the reverse space nonlocal FL equation are derived through Hirota bilinear method, and the soliton solutions of reverse time and reverse space-time nonlocal FL equations are given through variable transformations. Dynamical behaviors of the multisoliton solutions are discussed in detail by analyzing their wave structures. Thirdly, asymptotic analysis of two- and three-soliton solutions of reverse space nonlocal FL equation is used to investigated the elastic interaction and inelastic interaction. At last, the Lax integrability and conservation laws of three types of nonlocal FL equations is studied. The results obtained in this paper possess new properties that different from the ones for FL equation, which are useful in exploring novel physical phenomena of nonlocal systems in nonlinear media.

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Long-Time Asymptotics for the Nonlocal MKdV Equation*

Cited by 38 publications

References 33 publications

Asymptotic stage of modulation instability for the nonlocal nonlinear Schrödinger equation

Asymptotic stage of modulation instability for the nonlocal nonlinear Schrödinger equation

Riemann–Hilbert approach for discrete sine‐Gordon equation with simple and double poles

Integrability and Multisoliton Solutions of the Reverse Space or/and Time Nonlocal Fokas-Lenells (FL) Equation

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