A generalized Hirota–Satsuma coupled Korteweg–de Vries equation and Miura transformations

Wu, Yongtang; Geng, Xianguo; Hu, Xing−Biao; Zhu, Siming

doi:10.1016/s0375-9601(99)00163-2

Cited by 165 publications

(87 citation statements)

References 9 publications

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“…A Lax representation for the Hirota-Satsuma system was constructed in [59] 17 . Recently, it was generalized to the three-component case [(3.6) with N = 2] by Wu et al [61]. Let us demonstrate that (3.6) admits a Lax representation in the general case of N-component vector U.…”

Section: System (36)mentioning

confidence: 94%

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Classification of polynomial integrable systems of mixed scalar and vector evolution equations: I

Tsuchida¹,

Wolf²

2005

J. Phys. A: Math. Gen.

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We perform a classification of integrable systems of mixed scalar and vector evolution equations with respect to higher symmetries. We consider polynomial systems that are homogeneous under a suitable weighting of variables. This paper deals with the KdV weighting, the Burgers (or potential KdV or modified KdV) weighting, the Ibragimov-Shabat weighting and two unfamiliar weightings. The case of other weightings will be studied in a subsequent paper. Making an ansatz for undetermined coefficients and using a computer package for solving bilinear algebraic systems, we give the complete lists of 2 nd order systems with a 3 rd order or a 4 th order symmetry and 3 rd order systems with a 5 th order symmetry. For all but a few systems in the lists, we show that the system (or, at least a subsystem of it) admits either a Lax representation or a linearizing transformation. A thorough comparison with recent work of Foursov and Olver is made.

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Section: System (36)mentioning

confidence: 94%

“…The Miura map (4.74) is a multi-component generalization of that for the case of scalar U in [43,44] and that for the case of two-component vector U in [61]. …”

Section: System (430)mentioning

confidence: 99%

Classification of polynomial integrable systems of mixed scalar and vector evolution equations: I

Tsuchida¹,

Wolf²

2005

J. Phys. A: Math. Gen.

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“…In view of the solution procedure from equation (21) to equation (41) and using the initial conditions (60-62), we get the following approximations…”

Section: Second Kind Initial Conditionsmentioning

confidence: 99%

“…In this article, we investigate a general form of Hirota-Satsuma coupled KdV equation which was initiated by Wu et al [41]. Among these equations a generalized form of Hirota-Satsuma coupled KdV equation is written in the following form: u t = 1 2 uxxx − 3 u ux + 3(vw)x , v t = −vxxx + 3 u vx , w t = −wxxx + 3 u wx .…”

Section: Introductionmentioning

confidence: 99%

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An efficient computational approach for generalized Hirota-Satsuma coupled KdV equations arising in shallow water waves

Gupta

Kumar

Singh

2017

Waves, Wavelets and Fractals

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Abstract:The aim of this letter is to present a numerical algorithm for generalized Hirota-Satsuma coupled KdV equations arising in unidirectional propagation of shallow water waves by using the homotopy analysis transform technique. The computational approach is the merged form of the homotopy analysis technique and Laplace transform scheme. The technique provides a series solution, which converges very fast, components are very easily calculated, and it does not require linearization or small perturbation. The numerical and graphical results derived by making use of proposed approach indicate that the scheme is very user friendly and easy to implement.

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Integrability and equivalence relationships of six integrable coupled Korteweg–de Vries equations

Wang

Liu

Zhang

2016

Math Methods in App Sciences

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In this paper, we investigate the integrability and equivalence relationships of six coupled Korteweg–de Vries equations. It is shown that the six coupled Korteweg–de Vries equations are identical under certain invertible transformations. We reconsider the matrix representations of the prolongation algebra for the Painlevé integrable coupled Korteweg–de Vries equation in [Appl. Math. Lett. 23 (2010) 665‐669] and propose a new Lax pair of this equation that can be used to construct exact solutions with vanishing boundary conditions. It is also pointed out that all the six coupled Korteweg–de Vries equations have fourth‐order Lax pairs instead of the fifth‐order ones. Moreover, the Painlevé integrability of the six coupled Korteweg–de Vries equations are examined. It is proved that the six coupled Korteweg–de Vries equations are all Painlevé integrable and have the same resonant points, which further determines the equivalence among them. Finally, the auto‐Bäcklund transformation and exact solutions of one of the six coupled Korteweg–de Vries equations are proposed explicitly. Copyright © 2016 John Wiley & Sons, Ltd.

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A generalized Hirota–Satsuma coupled Korteweg–de Vries equation and Miura transformations

Cited by 165 publications

References 9 publications

Classification of polynomial integrable systems of mixed scalar and vector evolution equations: I

Classification of polynomial integrable systems of mixed scalar and vector evolution equations: I

An efficient computational approach for generalized Hirota-Satsuma coupled KdV equations arising in shallow water waves

Integrability and equivalence relationships of six integrable coupled Korteweg–de Vries equations

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