The Modified Korteweg-de Vries Hierarchy: Lax Pair Representation and Bi-Hamiltonian Structure

Choudhuri, Amitava; Talukdar, B.; Das, U.

doi:10.1515/zna-2009-3-403

Cited by 6 publications

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“…Like the equations in the KdV hierarchy, all equations in the modified KdV hierarchy satisfy the property of the complete integrability. Recently, Choudhuri, Talukdar and Das [6] derived the Lax representation and constructed the bi-Hamiltonian structure of the equations in the modified KdV hierarchy. The followings are a few equations and their associated Hamiltonians with respect to bi-Hamiltonian structures:…”

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confidence: 99%

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Low regularity Cauchy problem for the fifth-order modified KdV equations on 𝕋

Kwak

2018

J. Hyper. Differential Equations

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In this paper, we consider the fifth-order modified Korteweg-de Vries (modified KdV) equation under the periodic boundary condition. We prove the local well-posedness in H s (T), s > 2, via the energy method. The main tool is the short-time Fourier restriction norm method, which was first introduced in its current form by Ionescu, Kenig and Tataru [Global well-posedness of the KP-I initial-value problem in the energy space, Invent. Math. 173 (2) (2008) 265-304]. Besides, we use the frequency localized modified energy to control the high-low interaction component in the energy estimate. We remark that under the periodic setting, the integrable structure is very useful (but not necessary) to remove harmful terms in the nonlinearity and this work is the first low regularity well-posedness result for the fifth-order modified KdV equation.2010 Mathematics Subject Classification. 35Q53, 37K10.

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confidence: 99%

“…Remark that it, in fact, is not necessary to use the integrable structure for this problem thanks to the suitable nonlinear transformation, but it should be necessary for higher-order equations, due to the higher degree nonlinearity with derivatives, see Remark 7. 6.…”

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confidence: 99%

Low regularity Cauchy problem for the fifth-order modified KdV equations on 𝕋

Kwak

2018

J. Hyper. Differential Equations

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“…To realize the bi-Hamiltonian structure of the ccmKdV equation we consider equations obtained from the complex form of the linear equation in (11) in the mKdV hierarchy with n = 0. For these equations, the direct and indirect Hamiltonian densities…”

Section: Bi-hamiltonian Structurementioning

confidence: 99%

“…The Miura transformation (12) [11]. The mKdV equation was studied extensively because of its simplicity and physical significance [12].…”

Section: Introductionmentioning

confidence: 99%

“…More specifically, we shall envisage a variational formulation for the coupled system implied by the ccmKdV equation. The approach to the present inverse problem is somewhat different from that used for higher mKdV equations [11]. Here the system of equations is of lowest order such that we can proceed simply by applying the Helmholtz theorem as realized in the calculus of differential forms [18].…”

Section: Introductionmentioning

confidence: 99%

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On complexly coupled modified KdV equations

Choudhuri

2010

Pramana - J Phys

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We introduced complexly coupled modified KdV (ccmKdV) equations, which could be derived from a two-layer fluid model [Yang and Mao, Chin. Phys. Lett. 25, 1527 (2008); Hu, J. Phys. A: Math. Theor. 43, 185207 (2009)], and used the Miura transformation to construct expressions for their alternative Lax pair representations. We derived a Lagrangian-based approach to study the Hamiltonian structures of the ccmKdV equations and observed that the complexly coupled mKdV equations have an additional analytic structure. The coupled equations were characterized by two alternative Lagrangians not connected by a gauge term. We examined how the alternative Lagrangian descriptions of the system affect the bi-Hamiltonian structures.

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CRE Solvability, Nonlocal Symmetry and Exact Interaction Solutions of the Fifth-Order Modified Korteweg-de Vries Equation

Cheng

Qiu

2017

Commun. Theor. Phys.

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This paper is concerned with the fifth-order modified Korteweg-de Vries (fmKdV) equation. It is proved that the fmKdV equation is consistent Riccati expansion (CRE) solvable. Three special form of soliton-cnoidal wave interaction solutions are discussed analytically and shown graphically. Furthermore, based on the consistent tanh expansion (CTE) method, the nonlocal symmetry related to the consistent tanh expansion (CTE) is investigated, we also give the relationship between this kind of nonlocal symmetry and the residual symmetry which can be obtained with the truncated Painlevé method. We further study the spectral function symmetry and derive the Lax pair of the fmKdV equation. The residual symmetry can be localized to the Lie point symmetry of an enlarged system and the corresponding finite transformation group is computed.

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The Modified Korteweg-de Vries Hierarchy: Lax Pair Representation and Bi-Hamiltonian Structure

Cited by 6 publications

References 0 publications

Low regularity Cauchy problem for the fifth-order modified KdV equations on 𝕋

Low regularity Cauchy problem for the fifth-order modified KdV equations on 𝕋

On complexly coupled modified KdV equations

CRE Solvability, Nonlocal Symmetry and Exact Interaction Solutions of the Fifth-Order Modified Korteweg-de Vries Equation

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