A Class of Periodic Solutions of (2+1)-dimensional Boussinesq Equation

Zhang, Jie-Fang; Lai, Xian-Jing

doi:10.1143/jpsj.73.2402

Cited by 10 publications

(5 citation statements)

References 16 publications

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“…In the present work we will extend our study to the two higher dimensional Boussinesq equations, namely, the (2 + 1)-dimensional Boussinesq equation [29,30,36,37]:…”

Section: H Kumarmentioning

confidence: 99%

“…Nevertheless, the Boussinesq equation provides a much superior approximation to such waves. Among the NLEEs, the Boussinesq equation has been resulting in order to describe the long waves transmitting on the surface of shallow water [28][29][30][31]. The Boussinesq-like equations also appear in many physical phenomena, such as electromagnetic waves in nonlinear dielectrics, one-dimensional nonlinear lattice waves, ion sound waves in plasma, and oscillations in a nonlinear string.…”

Section: Introductionmentioning

confidence: 99%

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Dynamics of Shallow Water Waves with Various Boussinesq Equations

Kumar¹,

Malik

Gautam

et al. 2017

Acta Phys. Pol. A

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Attempt has been made to construct the solitary waves and shock wave solutions or domain walls (in higher dimension) for various Boussinesq equations. The method of undetermined coefficients have been used to explore the exact analytical solitary waves and shock wave solutions in terms of bell-shaped sech p function and kinkshaped tanh p function for the considered equations. The Boussinesq equation in the (1 + 1)-dimensional, the (2 + 1)-dimensional and the (3 + 1)-dimensional equations are studied and the parametric constraint conditions and uniqueness in view of both solitary waves and shock wave solutions are determined. Such solutions can be valuable and desirable for explaining some nonlinear physical phenomena in nonlinear science described by the Boussinesq equations. The effect of the varying parameters on the development of solitary waves and shock wave solutions have been demonstrated by direct numerical simulation technique.

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“…In the present work we will extend our study to the two higher dimensional Boussinesq equations, namely, the (2 + 1)-dimensional Boussinesq equation [29,30,36,37]:…”

Section: H Kumarmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Dynamics of Shallow Water Waves with Various Boussinesq Equations

Kumar¹,

Malik

Gautam

et al. 2017

Acta Phys. Pol. A

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“…5 Other examples include time modes of nonlinear systems, 6 Einstein nonlinear electrodynamics (NLE) equations, 7 nonlinear gas governing equations, 8 the Novikov-Veselov equation, 9 Maxwell-Schrödinger equations, 10 the (2 + 1)-dimensional KdV equations, 11 Caudrey-Dodd-Gibbon-Sawada-Kotera (CDGSK) equation and the (2 + 1)-dimensional Nizhnik-Novikov-Veselov (NNV) equation, 12 the (2 + 1)-dimensional Sawada-Kotera (SK) equation, 13 the (3 + 1)-dimensional Jimbo-Miwa (JM) equations, 14 the (2 + 1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation, 15 higher order NLSE, 16 coupled nonlinear Klein-Gordon and Schrödinger equations, 17 the (2 + 1)-dimensional modified Zakharov-Kuznetsov (ZK) equation and the (3 + 1)-dimensional KP equation, 18 the (2 + 1)-dimensional ZK equation and the Davey-Stewartson (DS) equation, 19 generalized KdV equation, the Oliver water wave equation, the k(n; n) equation, 20 the fifth-order KdV equation, 21 the cubic-quintic NLSE, 22 Hirota bilinear equations, 23,24 and the (2 + 1)-dimensional Boussinesq equation. 25 The main property that allows for the application of superposition principle to all of the above-mentioned nonlinear systems is the reduction of the nonlinear cross terms into linear ones, which then combine with other linear terms. Hence, the nonlinear equation reduces to a sum of two uncoupled replicates for each of the superimposed solutions individually.…”

Section: Introductionmentioning

confidence: 99%

“…This was possible due to cyclic identities satisfied by the Jacobi elliptic functions where the nonlinear cross terms reduce to, and combine with, other linear terms [3]. Other examples include time modes of nonlinear systems [4], the Novikov-Veselov equation [5], Maxwell-Schrödinger equations [6], the (2+1)-dimensional KdV Equations [7], Caudrey-Dodd-Gibbon-Sawada-Kotera (CDGSK) equation and the (2+1)-dimensional Nizhnik-Novikov-Veselov (NNV) equation [8], higher order NLSE [9], coupled nonlinear Klein-Gordon and Schrödinger equations [10], the (2+1)dimensional modified Zakharov-Kuznetsov equation and the (3+1)-dimensional Kadomtsev-Petviashvili equation [11], the (2+1)-dimensional Zakharov-Kuznetsov (ZK) equation and the Davey-Stewartson (DS) equation [12], generalized KdV equation, the Oliver water wave equation, the k(n; n) equation [13], fifth-order KdV equation [14], cubic-quintic NLSE [15], and (2+1)-dimensional Boussinesq Equation [16].…”

Section: Introductionmentioning

confidence: 99%

Superposition principle and composite solutions to coupled nonlinear Schrödinger equations

Sakkaf

Khawaja²

2020

Math Methods in App Sciences

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We show that the superposition principle applies to coupled nonlinear Schrödinger equations with cubic nonlinearity where exact solutions may be obtained as a linear combination of other exact solutions. This is possible due to the cancelation of cross terms in the nonlinear coupling. First, we show that a composite solution, which is a linear combination of the two components of a seed solution, is another solution to the same coupled nonlinear Schrödinger equation. Then, we show that a linear combination of two composite solutions is also a solution to the same equation. With emphasis on the case of Manakov system of two-coupled nonlinear Schrödinger equations, the superposition is shown to be equivalent to a rotation operator in a two-dimensional function space with components of the seed solution being its coordinates. Repeated application of the rotation operator, starting with a specific seed solution, generates a series of composite solutions, which may be represented by a generalized solution that defines a family of composite solutions. Applying the rotation operator to almost all known exact seed solutions of the Manakov system, we obtain for each seed solution the corresponding family of composite solutions. Composite solutions turn out, in general, to possess interesting features that do not exist in the seed solution. Using symmetry reductions, we show that the method applies also to systems of N-coupled nonlinear Schrödinger equations. Specific examples for the three-coupled nonlinear Schrödinger equation are given.

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“…It was shown that the approach works for many well-known nonlinear equations in several areas of physics, including both integrable and non-integrable systems. [11] In fact, the unexpected linear superposition approach is a consequence of some remarkable identities involving Jacobian elliptic functions, and depends on these identities to a great extent. However, sometimes there are no relations needed directly, particularly in differential equations with higher order; one then must go on with very complicated and tedious work.…”

Section: Introductionmentioning

confidence: 99%