On the Cauchy problem for the Zakharov-Rubenchik/ Benney-Roskes system

Luong, Hung Truyen; Mauser, Norbert J.; Saut, Jean‐Claude

doi:10.3934/cpaa.2018075

Cited by 9 publications

(5 citation statements)

References 29 publications

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“…Remark 1.5. In [40,22], the authors study the asymptotic behavior of the Zakharov-Rubenchik system. They prove that solutions blow up if the energy is negative and give inestability results for the solitary wave in the case d = 3.…”

Section: Main Results For Zakharov Systemmentioning

confidence: 99%

On the decay problem for the Zakharov and Klein-Gordon Zakharov systems in one dimension

Martínez¹

2020

Preprint

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We are interested in the long time asymptotic behavoir of solutions to the scalar Zakharov system iut + ∆u = nu, ntt − ∆n = ∆|u| 2 and the Klein-Gordon Zakharov systemin one dimension of space. For these two systems, we give two results proving decay of solutions for initial data in the energy space. The first result deals with decay over compact intervals asuming smallness and parity conditions (u odd). The second result proves decay in far field regions along curves for solutions whose growth can be dominated by an increasing C 1 function. No smallness condition is needed to prove this last result for the Zakharov system. We argue relying on the use of suitable virial identities appropiate for the equations and follow the technics of [20, 23] and [32].

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Section: Main Results For Zakharov Systemmentioning

confidence: 99%

On the decay problem for the Zakharov and Klein-Gordon Zakharov systems in one dimension

Martínez¹

2020

Preprint

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“…, where the space-dimension d = 2, 3. Lastly, we mention that Luong et al have recently proved the well-posedness (under some extra conditions) of system (1.2) in the background of a line-soliton [7].…”

Section: The Modelmentioning

confidence: 91%

“…On the other hand, recently in [7] Luong et al studied the existence of the so-called bright and dark solitons for system (1.1). They proved their existence under some conditions on the coefficients of the equations (similar to the one in (1.4)).…”

Section: The Modelmentioning

confidence: 99%

“…Then, they used these solitons to construct line-solitons for the higher-dimensional case. However, none of these solitons belongs to the energy space since they do not decay at ±∞ (see [7] for further details). Finally, concerning the well-posedness for system (1.1), Oliveira [15] proved local and global well-posedness for the one-dimensional case in H 2 (R) × H 1 (R) × H 1 (R).…”

Section: The Modelmentioning

confidence: 99%

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On long-time behavior of solutions of the Zakharov–Rubenchik/Benney–Roskes system

Martínez

Palacios

2021

Nonlinearity

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We study decay properties for solutions to the initial value problem associated with the one-dimensional Zakharov–Rubenchik/Benney–Roskes (ZR/BR) system. We prove time-integrability in growing compact intervals of size t r , r < 2/3, centered on some characteristic curves coming from the underlying transport equations associated with the ZR/BR system. Additionally, we prove decay to zero of the local energy-norm in so-called far-field regions. Our results are independent of the size of the initial data and do not require any parity condition.

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“…Regarding the system (1.2) there are a few works but this one started to get attention in the last years, see for instance [23,24,19]. As far as we know the best local and global well-posedness for (1.2) was established in [19] and in dimensions d = 2, 3 we refer to the works [28,12,21,13] for recent advances concerning existence of solutions, asymptotic behavior, instability of standing waves and blow-up solutions.…”

Section: Introductionmentioning

confidence: 99%

Uniform Adiabatic Limit of Benney type Systems

Corcho¹,

Cordero²

2020

Preprint

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In this paper we show that solutions of the cubic nonlinear Schrödinger equation are asymptotic limit of solutions to the Benney system. Due to the special characteristic of the one-dimensional transport equation same result is obtained for solutions of the onedimensional Zakharov and 1d-Zakharov-Rubenchik systems. Convergence is reached in the topology L 2 (R) × L 2 (R) and with an approximation in the energy space H 1 (R) × L 2 (R). In the case of the Zakharov system this is achieved without the condition ∂tn(x, 0) ∈ Ḣ−1 (R) for the wave component, improving previous results.

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On the Cauchy problem for the Zakharov-Rubenchik/ Benney-Roskes system

Cited by 9 publications

References 29 publications

On the decay problem for the Zakharov and Klein-Gordon Zakharov systems in one dimension

On the decay problem for the Zakharov and Klein-Gordon Zakharov systems in one dimension

On long-time behavior of solutions of the Zakharov–Rubenchik/Benney–Roskes system

Uniform Adiabatic Limit of Benney type Systems

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