Two-component generalizations of the Novikov equation

Li, Hongmin

doi:10.1080/14029251.2019.1613048

Cited by 16 publications

(7 citation statements)

References 34 publications

(39 reference statements)

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“…(1.1) The two-component Novikov equation (1.1) was found by Li [46]. It was shown in [46] that the system (1.1) appears in the bi-Hamiltonian form…”

Section: Introductionmentioning

confidence: 97%

On the Cauchy problem for the new integrable two-component Novikov equation

2019

Annali di Matematica

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This paper is devoted to a new integrable two-component Novikov equation, lax pairs and bi-Hamiltonian structures. Firstly, the local well-posedness in nonhomogeneous Besov spaces is established by using the Littlewood-Paley theory and transport equations theory. Then, we verify the blow-up that occurs for this system only in the form of breaking waves. Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. Finally, we prove that the strong solutions of the system maintain corresponding properties at infinity within its lifespan provided the initial data decay exponentially and algebraically, respectively.

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“…(1.1) The two-component Novikov equation (1.1) was found by Li [46]. It was shown in [46] that the system (1.1) appears in the bi-Hamiltonian form…”

Section: Introductionmentioning

confidence: 97%

On the Cauchy problem for the new integrable two-component Novikov equation

2019

Annali di Matematica

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“…Recently, another kind of two-component integrable generalization (1.1) of the Novikov equation (1.2) was introduced in [33] and we find that this system admits the two-component peakon structure, which are given by…”

Section: Introductionmentioning

confidence: 97%

“…In this paper, we are devoted to the orbital stability of the two-component peakon solutions and the corresponding configuration of train-profiles. The system we are concerned with is the following integrable two-component Novikov system [33] m t + uvm x + (2vu x + uv x )m = 0, m = u − u xx , n t + uvn x + (2uv x + vu x )n = 0, n = v − v xx .…”

Section: Introductionmentioning

confidence: 99%

Orbital stability of two-component peakons

He¹,

Liu²,

Qu³

2021

Preprint

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We prove that the two-component peakon solutions are orbitally stable in the energy space. The system concerned here is a two-component Novikov system, which is an integrable multicomponent extension of the integrable Novikov equation. We improve the method for the scalar peakons to the two-component case with genuine nonlinear interactions by establishing optimal inequalities for the conserved quantities involving the coupled structures. Moreover, we also establish the orbital stability for the train-profiles of these two-component peakons by using the refined analysis based on monotonicity of the local energy and an induction method.

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“…This equation is a completely integrable system with a Lax pair and bi-Hamiltonian structure [12,22]. It is mentioned that the homogeneous and local properties of the Hamiltonian functionals were discussed [20]. Also, the Geng-Xue equation is related to a negative flow in a modified Boussinesq hierarchy by a reciprocal transformation [23] and the behaviour of the bi-Hamiltonian structures under the transformation was studied [21].…”

Section: Introductionmentioning

confidence: 99%

Smooth multisoliton solutions of the Geng-Xue equation

Li¹,

Liu²

2022

Preprint

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We present a reciprocal transformation which links the Geng-Xue equation to a particular reduction of the first negative flow of the Boussinesq hierarchy. We discuss two reductions of the reciprocal transformation for the Degasperis-Procesi and Novikov equations, respectively. With the aid of the Darboux transformation and the reciprocal transformation, we obtain a compact parametric representation for the smooth soliton solutions such as multi-kink solutions of the Geng-Xue equation.

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Two-component generalizations of the Novikov equation

Abstract: Some two-component generalizations of the Novikov equation, except the Geng-Xue equation, are presented, as well as their Lax pairs and bi-Hamiltonian structures. Furthermore, we study the Hamiltonians of the Geng-Xue equation and discuss the homogeneous and local properties of them.

Cited by 16 publications

References 34 publications

On the Cauchy problem for the new integrable two-component Novikov equation

On the Cauchy problem for the new integrable two-component Novikov equation

Orbital stability of two-component peakons

Smooth multisoliton solutions of the Geng-Xue equation

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