Doubly periodic patterns of modulated hydrodynamic waves: Exact solutions of the Davey-Stewartson system

Li, J. H.; Sen-Yue, Lou; Chow, K. W.

doi:10.1007/s10409-011-0468-2

Cited by 4 publications

(2 citation statements)

References 27 publications

(41 reference statements)

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“…The Hirota bilinear method is a powerful tool for constructing various exact solutions for NLEEs, which include soliton, negaton, rogue waves, rational solutions, and quasiperiodic solutions [25][26][27][28][29][30][31][32][33][34][35]. Recently, by means of Hirota bilinear method and theta function identities [36][37][38], Fan et al obtained a class of doubly periodic standing wave solutions of (1) [39], which was expressed as rational functions of elliptic/theta functions of different moduli. A significant portion of these solutions represents travelling wave, that is, those which will remain steady in an appropriate frame of reference.…”

Section: Introductionmentioning

confidence: 99%

New Types of Doubly Periodic Standing Wave Solutions for the Coupled Higgs Field Equation

2014

Abstract and Applied Analysis

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Based on the Hirota bilinear method and theta function identities, we obtain a new type of doubly periodic standing wave solutions for a coupled Higgs field equation. The Jacobi elliptic function expression and long wave limits of the periodic solutions are also presented. By selecting appropriate parameter values, we analyze the interaction properties of periodic-periodic waves and periodic-solitary waves by some figures.

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Section: Introductionmentioning

confidence: 99%

New Types of Doubly Periodic Standing Wave Solutions for the Coupled Higgs Field Equation

2014

Abstract and Applied Analysis

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“…Recently, doubly periodic wave patterns were studied by Chow and Lou and Li et al . . Tajiri and Arai studied the existence of long‐range interaction between two quasi‐line solitons through a periodic soliton.…”

Section: Introductionmentioning

confidence: 99%

Qualitative analysis and explicit traveling wave solutions for the Davey–Stewartson equation

Song

Liu

2013

Math Methods in App Sciences

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In this paper, we use the bifurcation method of dynamical systems to study the traveling wave solutions for the Davey-Stewartson equation. A number of traveling wave solutions are obtained. Those solutions contain explicit periodic wave solutions, periodic blow-up wave solutions, unbounded wave solutions, kink profile solitary wave solutions, and solitary wave solutions. Relations of the traveling wave solutions are given. Some previous results are extended.where u is the (complex valued) envelope of the wave packet associated with the fast oscillations, and v is the induced mean flow, x and y are the slow, horizontal scales parallel and perpendicular to the fast oscillations, respectively, whereas t is the slow time in the group velocity frame. As usual, irrotational flow of an inviscid fluid is studied, and u is connected with the velocity potential. Of particular significance is the parameter a, and a D 1 and a D 1 is called the DS I and DS II equations, respectively. The constant b measures the cubic nonlinearity. The DS equation is the two-dimensional generalization of the nonlinear Schrödinger (NLS) equation.The solutions to the DS equation have been studied previously in various aspects. The N-soliton solution of the DS equation was studied by Anker and Freeman [3] in the inverse scattering transform method and later by Satsuma and Ablowitz [4] in the Hirota method. Satsuma and Ablowitz also revealed that the solutions describing multiple collisions of rational solitons are constructed from the N-soliton solutions for the equation. The interaction of two dark line solitons, which are skewed with respect to each other, has been studied by Anker and Freeman [5]. They found the existence of soliton resonance. The dromion solutions to the DS equation, which decay exponentially in all directions, were obtained in various aspects [6][7][8][9]. The DS equation is known to have solutions called solitoffs, and solitoff-solitoff and solitoff-dromion resonant behaviors have been studied [10]. Watanabe and Tajiri [11] and Tajiri et al. [12] investigated the interactions between y-periodic soliton and line soliton and between y-periodic soliton and algebraic soliton. Recently, doubly periodic wave patterns were studied by Chow and Lou [13] and Li et al. [14]. Tajiri and Arai [15] studied the existence of long-range interaction between two quasi-line solitons through a periodic soliton.The aim of this paper is to study the traveling wave solutions and their relations for Equation (1.1) by using the bifurcation method and qualitative theory of dynamical systems [16][17][18][19][20]. Through some special phase orbits, we obtain many smooth periodic wave solutions and periodic blow-up solutions. Their limits contain kink profile solitary wave solutions, unbounded wave solutions, periodic blow-up solutions, and solitary wave solutions.

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Optical soliton wave solutions to the resonant Davey–Stewartson system

Aghdaei

Manafian

2016

Opt Quant Electron

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Doubly periodic patterns of modulated hydrodynamic waves: Exact solutions of the Davey-Stewartson system

Cited by 4 publications

References 27 publications

New Types of Doubly Periodic Standing Wave Solutions for the Coupled Higgs Field Equation

New Types of Doubly Periodic Standing Wave Solutions for the Coupled Higgs Field Equation

Qualitative analysis and explicit traveling wave solutions for the Davey–Stewartson equation

Optical soliton wave solutions to the resonant Davey–Stewartson system

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