Multi-Soliton Solutions for the Coupled Fokas–Lenells System via Riemann–Hilbert Approach

Kang, Zhou-Zheng; Xia, Tiecheng; Ma, Xi

doi:10.1088/0256-307x/35/7/070201

Cited by 15 publications

(8 citation statements)

References 20 publications

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“…Constructing Jacobi elliptic doubly periodic solutions of nonlinear PDEs with fractional derivatives is worthy of the study. At the same time, constructing multisoliton solutions via the Riemann-Hilbert approach, for example, see Kang et al's meaningful work [52,53], has been a hot topic. Dealing with initial-boundary problems of nonlinear PDEs by means of the Riemann-Hilbert approach is also worthy of the study.…”

Section: Resultsmentioning

confidence: 99%

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A Novel Approach to a Time‐Dependent‐Coefficient WBK System: Doubly Periodic Waves and Singular Nonlinear Dynamics

Zhang

2018

Complexity

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Under investigation in this paper is a more general time-dependent-coefficient Whitham-Broer-Kaup (tdcWBK) system, which includes some important models as special cases, such as the approximate equations for long water waves, the WBK equations in shallow water, the Boussinesq-Burgers equations, and the variant Boussinesq equations. To construct doubly periodic wave solutions, we extend the generalized F-expansion method for the first time to the tdcWBK system. As a result, many new Jacobi elliptic doubly periodic solutions are obtained; the limit forms of which are the hyperbolic function solutions and trigonometric function solutions. It is shown that the original F-expansion method cannot derive Jacobi elliptic doubly periodic solutions of the tdcWBK system, but the novel approach of this paper is valid. To gain more insight into the doubly periodic waves contained in the tdcWBK system, we simulate the dynamical evolutions of some obtained Jacobi elliptic doubly periodic solutions. The simulations show that the doubly periodic waves possess time-varying amplitudes and velocities as well as singularities in the process of propagations.

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Section: Resultsmentioning

confidence: 99%

“…In Figure 5, the spatial structures of solutions (27) determined by (52) are shown by selecting the parameters as a 0 = 2, A 0 = 0 5, c 0 = −4, c 1 = 3, c 2 = −1, k = 1 5, and m = 0 8, respectively. We shown the spatial structures of solutions (40) in Figure 6.…”

Section: Singular Nonlinear Dynamicsmentioning

confidence: 99%

A Novel Approach to a Time‐Dependent‐Coefficient WBK System: Doubly Periodic Waves and Singular Nonlinear Dynamics

Zhang

2018

Complexity

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“…Constructing exact solutions of nonlinear mathematical physical equations is of theoretical and practical significance. Since the famous Korteweg-de Vries (KdV) equation was solved in 1967 [7], a large number of exact solutions like [2][3][4][5][6][7]10,11,[13][14][15][16][18][19][20][22][23][24][25][26][27][28]31,33,34,36,38,41,43] of nonlinear partial differential equations (PDEs) have been found. The exp-function method [10] proposed by He and Wu has been widely used for constructing exact solutions of nonlinear PDEs.…”

Section: Introductionmentioning

confidence: 99%

Exact Solutions of Nonlinear Equations in Mathematical Physics via Negative Power Expansion Method

Xu¹,

Zhang²

2021

Z. mat. fiz. anal. geom.

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In this paper, a direct method called negative power expansion (NPE) method is presented and extended to construct exact solutions of nonlinear mathematical physical equations. The presented NPE method is also effective for the coupled, variable-coefficient and some other special types of equations. To illustrate the effectiveness, the (2 + 1)-dimensional dispersive long wave (DLW) equations, Maccari's equations, Tzitzeica-Dodd-Bullough (TDB) equation, Sawada-Kotera (SK) equation with variable coefficients and two lattice equations are considered. As a result, some exact solutions are obtained including traveling wave solutions, non-traveling wave solutions and semi-discrete solutions. This paper shows that the NPE method is a simple and effective method for solving nonlinear equations in mathematical physics.

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“…In [8] multi-Hamiltonian structure and infinitely many conservation laws were established for the vector Kaup-Newell hierarchy of the positive and negative orders. Some other methods, such as Darboux transformation method [5,[8][9][10], Hirota bilinear method [11][12][13], Riemann-Hilbert problem [14][15][16], Bäcklund transformation [17] and trial equation method [18] could be found in references.…”

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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Multi-Soliton Solutions for the Coupled Fokas–Lenells System via Riemann–Hilbert Approach

Cited by 15 publications

References 20 publications

A Novel Approach to a Time‐Dependent‐Coefficient WBK System: Doubly Periodic Waves and Singular Nonlinear Dynamics

A Novel Approach to a Time‐Dependent‐Coefficient WBK System: Doubly Periodic Waves and Singular Nonlinear Dynamics

Exact Solutions of Nonlinear Equations in Mathematical Physics via Negative Power Expansion Method

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

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