Some new integrable nonlinear evolution equations in 2 + 1 dimensions

Konopelchenko, B. G.; Dubrovsky, V. G.

doi:10.1016/0375-9601(84)90442-0

Cited by 322 publications

(182 citation statements)

References 3 publications

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“…For instance, the multisoliton solutions, conserved quantities, Bäcklund transformation, and Darboux transformation of the equation have been discussed in [22][23][24][25]. In [29], a (2+1)-dimensional integrable generalization of the Sawada-Kotera (2DSK) equation has the following form:…”

Section: Introductionmentioning

confidence: 99%

Lump Solutions and Resonance Stripe Solitons to the (2+1)-Dimensional Sawada-Kotera Equation

Wang

Chen

et al. 2017

Advances in Mathematical Physics

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Based on the symbolic computation, a class of lump solutions to the (2+1)-dimensional Sawada-Kotera (2DSK) equation is obtained through making use of its Hirota bilinear form and one positive quadratic function. These solutions contain six parameters, four of which satisfy two determinant conditions to guarantee the analyticity and rational localization of the solutions, while the others are free. Then by adding an exponential function into the original positive quadratic function, the interaction solutions between lump solutions and one stripe soliton are derived. Furthermore, by extending this method to a general combination of positive quadratic function and hyperbolic function, the interaction solutions between lump solutions and a pair of resonance stripe solitons are provided. Some figures are given to demonstrate the dynamical properties of the lump solutions, interaction solutions between lump solutions, and stripe solitons by choosing some special parameters.

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Section: Introductionmentioning

confidence: 99%

Lump Solutions and Resonance Stripe Solitons to the (2+1)-Dimensional Sawada-Kotera Equation

Wang

Chen

et al. 2017

Advances in Mathematical Physics

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“…For a = 0, system reduces to mKP equations [20] which can describe the propagation of ion-acoustic waves in a plasma with the non-isothermal electrons. For a = −6γ, c = −1, b = 1.5δ 2 , α = −3 and β = 3δ , equations reduces to KD equations [21,22] which have proposed by the inverse scattering transformation method. So investigation of coupled system (1) and (2) will be of certain interest.…”

Section: Mathematical Analysismentioning

confidence: 99%

Solutions of Konopelchenko-Dubrovsky Equation by Traveling Wave Hypothesis and Lie Symmetry Approach

Kumar¹,

Hama²,

Biswas³

2014

Appl. Math. Inf. Sci.

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“…This equation was presented firstly by Maccari [12]from the Konopelchenko-Dubrovsky (KD) equation [13] by using the reduction method. (6) had been proved to be Lax integrable.…”

Section: The New Integrable Davey-stewartson-type Equationmentioning

confidence: 99%

The Weierstrass elliptic function expansion method and its applications in nonlinear wave equations

Chen

Yan

2006

Chaos, Solitons & Fractals

102

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Abstract. In this paper, based on the close relationship between the Weierstrass elliptic function ℘(ξ; g 2 , g 3 )(g 2 , g 3 , invariants) and nonlinear ordinary differential equation, a Weierstrass elliptic function expansion method is developed in terms of the Weierstrass elliptic function instead of many Jacobi elliptic functions. The mechanism is constructive and can be carried out in computer with the aid of computer algebra (for example, Maple). Many important nonlinear wave equations arising from nonlinear science are chosen to illustrate this technique such as the new integrable Davey-Stewartsontype equation, the (2+1)-dimensional modified KdV equation, the the generalized Hirota equation in 2+1 dimensions, the Generalized KdV equation, the (2+1)-dimensional modified Novikov-Veselov equations, (2+1)-dimensional generalized system of modified KdV equation, the coupled Klein-Gordon equation, and the (2+1)-dimensional generalization of coupled nonlinear Schrodinger equation. As a consequence, some new doubly periodic solutions are obtained in terms of the Weierstrass elliptic function. Moreover solitary wave solutions and singular solitary wave solutions are also given as simple limits of doubly periodic solutions. These solutions may be useful to explain some physical phenomena. The algorithm is also applied to other many nonlinear wave equations. Moreover we also present the general form of the technique.

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Some new integrable nonlinear evolution equations in 2 + 1 dimensions

Cited by 322 publications

References 3 publications

Lump Solutions and Resonance Stripe Solitons to the (2+1)-Dimensional Sawada-Kotera Equation

Lump Solutions and Resonance Stripe Solitons to the (2+1)-Dimensional Sawada-Kotera Equation

Solutions of Konopelchenko-Dubrovsky Equation by Traveling Wave Hypothesis and Lie Symmetry Approach

The Weierstrass elliptic function expansion method and its applications in nonlinear wave equations

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