Two-Field Integrable Evolutionary Systems of the Third Order and Their Differential Substitutions

Meshkov, A. G.

doi:10.3842/sigma.2008.018

Cited by 7 publications

(11 citation statements)

References 35 publications

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“…The coupled KdV Eqs. (1)- (3) and (6) presented in [6,7,10,11,15,16] are not new, and under some transformations, they are nothing but the coupled KdV Eqs. (4) and (5) given early by Drinfeld and Sokolov [1] in 1981.…”

Section: Discussionmentioning

confidence: 99%

“…Recently, Meshkov and Balakhnev proposed a coupled KdV equation

({, \begin{matrix} u_{t} - u_{xxx} - (u + v) u_{x} - \frac{1}{2} ((3 - c) u + (5 c - 11) v) v_{x} = 0, \\ v_{t} - \frac{1}{2} (3 c - 7) v_{xxx} - ((c + 2) u - v) u_{x} - \frac{1}{2} (c - 3) (u + v) v_{x} = 0, c^{2} = 5 \end{matrix})

and showed that it could be reduced to Eqs. and by certain transformations.…”

Section: Equivalence Relationships Among the Six Coupled Kdv Equationsmentioning

confidence: 99%

“…which is conjectured that it is new and should be integrable. Foursov [10] pointed out that this equation possesses generalized symmetries of orders 7,9,11,13, and 17, as well as conserved densities of weights 2, 4, 8, 10, 12, and 14. Foursov [10] did an extensive search in the literature but was unable to find any explicit equation equivalent to Eq.…”

Section: Equivalence Relationships Among the Six Coupled Kdv Equationsmentioning

confidence: 99%

“…The scalar Korteweg–de Vries (KdV) equation admits various multifield extensions that come up in many physical issues. Since about 1981, many integrable coupled KdV‐type equations have been found in the literature . In 1981, Drinfel'd and Sokolov presented some Lax integrable coupled KdV equations along with their Lax pairs.…”

Section: Introductionmentioning

confidence: 99%

“…which is derived from two-layer fluids. In this paper, we point out that the six integrable coupled KdV equations presented in Drinfel'd and Sokolov [1], Sakovich [6,7], Foursov [10], Meshkov and Balakhnev [11], Wang [15], and Lou [16] are equivalent to each other under invertible transformations. We also show that all the six integrable coupled KdV equations admit fourth-order Lax pairs instead of the fifth-order ones proposed in Drinfel'd and Sokolov [1], Sakovich [7], and Popowicz [12].…”

Section: Introductionmentioning

confidence: 99%

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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.

show abstract

Section: Discussionmentioning

confidence: 99%

“…Recently, Meshkov and Balakhnev proposed a coupled KdV equation

({, \begin{matrix} u_{t} - u_{xxx} - (u + v) u_{x} - \frac{1}{2} ((3 - c) u + (5 c - 11) v) v_{x} = 0, \\ v_{t} - \frac{1}{2} (3 c - 7) v_{xxx} - ((c + 2) u - v) u_{x} - \frac{1}{2} (c - 3) (u + v) v_{x} = 0, c^{2} = 5 \end{matrix})

and showed that it could be reduced to Eqs. and by certain transformations.…”

Section: Equivalence Relationships Among the Six Coupled Kdv Equationsmentioning

confidence: 99%

Section: Equivalence Relationships Among the Six Coupled Kdv Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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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Miura-type transformations for lattice equations and Lie group actions associated with Darboux–Lax representations

Berkeley¹,

Igonin²

2016

J. Phys. A: Math. Theor.

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Abstract. Miura-type transformations (MTs) are an essential tool in the theory of integrable nonlinear partial differential and difference equations. We present a geometric method to construct MTs for differentialdifference (lattice) equations from Darboux-Lax representations (DLRs) of such equations.The method is applicable to parameter-dependent DLRs satisfying certain conditions. We construct MTs and modified lattice equations from invariants of some Lie group actions on manifolds associated with such DLRs.Using this construction, from a given suitable DLR one can obtain many MTs of different orders. The main idea behind this method is closely related to the results of Drinfeld and Sokolov on MTs for the partial differential KdV equation.Considered examples include the Volterra, Narita-Itoh-Bogoyavlensky, Toda, and Adler-Postnikov lattices. Some of the constructed MTs and modified lattice equations seem to be new.

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First-order differential substitutions for equations integrable on % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGcbaWefv3ySLgznfgDOjdarCqr1ngBPrginfgDObcv39ga % iyaacqWFsc-udaahaaWcbeqaaGqaciab+5gaUbaaaaa!4862! $$ \mathbb{S}^n $$

Balakhnev

2011

Math Notes

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Two-Field Integrable Evolutionary Systems of the Third Order and Their Differential Substitutions

Cited by 7 publications

References 35 publications

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

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

Miura-type transformations for lattice equations and Lie group actions associated with Darboux–Lax representations

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