The effects of the singular lines on the traveling wave solutions of modified dispersive water wave equations

Han, Maoan; Zhang, Lijun; Wang, Yue; Khalique, Chaudry Masood

doi:10.1016/j.nonrwa.2018.10.012

Cited by 41 publications

(15 citation statements)

References 16 publications

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“…Bounded smooth traveling wave solutions of (1). We know that the orbits of systems (8), (9) and (12), with ϕ = c, are all determined by the first integral (13) which can be rewritten as…”

Section: Case (2) C < Bmentioning

confidence: 99%

“…For the proof one can refer to [12,27]. Note that there are four parameters A, B, c and h involving in equation (17).…”

Section: Case (2) C < Bmentioning

confidence: 99%

“…Integrating along this closed orbit, we derived a periodic traveling wave solution in last section. Remind that this class of orbits intersecting with the singular line ϕ = c which may effect the solutions of equation and arise some singular traveling wave solutions [12,16,27]. According to the analysis in the previous two sections, we see that for arbitrary c there exists a family of compacted orbits intersecting with the singular line ϕ = c (see Fig 2(1)).…”

mentioning

confidence: 93%

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Smooth and singular traveling wave solutions for the Serre-Green-Naghdi equations

Zhang¹,

Shi²,

Han³

2020

Discrete &Amp; Continuous Dynamical Systems - S

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In this paper, we consider the traveling wave solutions of the onedimensional Serre-Green-Naghdi (SGN) equations which are proposed to model dispersive nonlinear long water waves in a one-layer flow over flat bottom. We decouple the traveling wave system of SGN equations into two ordinary differential equations. By studying the bifurcations and phase portraits of each bifurcation set of one equation, we obtain the exact traveling wave solutions of SGN equations for the variable u(x, t) which represents average horizontal velocity of water wave. For the compacted orbits intersecting with the singular line in phase plane, we obtained two families of solutions: a family of smooth traveling wave solutions including periodic wave solutions and solitary wave solutions, and a family of compacted singular solutions which have continuous first-order derivative but discontinuous second-order derivative.

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“…Bounded smooth traveling wave solutions of (1). We know that the orbits of systems (8), (9) and (12), with ϕ = c, are all determined by the first integral (13) which can be rewritten as…”

Section: Case (2) C < Bmentioning

confidence: 99%

“…For the proof one can refer to [12,27]. Note that there are four parameters A, B, c and h involving in equation (17).…”

Section: Case (2) C < Bmentioning

confidence: 99%

mentioning

confidence: 93%

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Smooth and singular traveling wave solutions for the Serre-Green-Naghdi equations

Zhang¹,

Shi²,

Han³

2020

Discrete &Amp; Continuous Dynamical Systems - S

Self Cite

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show abstract

“…For a long time, people have been committed to the study of integral calculus. Therefore, researchers conduct scientific research based on integer order models [13][14][15][16][17][18]. However, with the continuous development of calculus theory, the appearance and development of fractional order have become a major trend of integral science [19][20][21].…”

Section: Introductionmentioning

confidence: 99%

An Application of (3+1)-Dimensional Time-Space Fractional ZK Model to Analyze the Complex Dust Acoustic Waves

Yang

2019

Complexity

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Dust plasma is a new field of physics which has developed rapidly in recent decades. The study of dust plasma has received much attention due to its importance in the environment of space and the Earth. Dust acoustic waves are generated because of the inertia of dust mass while the restoring force is provided by the thermal pressure of electrons and ions. Since dust acoustic waves were first reported theoretically in unmagnetized dust plasma by Rao et al., they have become a research hot spot. In this paper, the excitation of dust acoustic waves by a gravity field in a dust plasma is analyzed. According to the control equations of dust plasma motion and employing multiscale analysis and perturbation method, we have obtained a (3+1)-dimensional ZK model. Because of the space property of dust plasma, (3+1)-dimensional ZK equation is more suitable than KdV equation and (2+1)-dimensional ZK equation to describe the real dust acoustic waves. Then, the (3+1)-dimensional time-space fractional ZK (TSF-ZK) equation describing the fractal process of nonlinear dust acoustic waves is given for the first time. To further explore how dust acoustic waves change energy as they travel, we discuss the conservation laws of the new model. Moreover, we study the exact solution of (3+1)-dimensional TSF-ZK equation by using extended Kudryashov method. Finally, based on the exact solution, we further investigate the effect of the parameter k, the charge properties of dust particle Zd0, the fractional order values α, β, γ, and θ, the temperature Td, the gravity g, and the collision frequency β0 and β1 on the properties of dust acoustic waves by a gravity field in dust plasma.

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“…There are two typical techniques for gaining inverse scattering: one is using Gel 耠 fand-Levitan-Marchenko equations and the other is formatting a Riemann-Hilbert(R-H) problem. 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 andN-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 effects of the singular lines on the traveling wave solutions of modified dispersive water wave equations

Cited by 41 publications

References 16 publications

Smooth and singular traveling wave solutions for the Serre-Green-Naghdi equations

Smooth and singular traveling wave solutions for the Serre-Green-Naghdi equations

An Application of (3+1)-Dimensional Time-Space Fractional ZK Model to Analyze the Complex Dust Acoustic Waves

Prolongation Structures andN-Soliton Solutions for a New Nonlinear Schrödinger-Type Equation via Riemann-Hilbert Approach

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