On a class of solutions to the generalized KdV type equation

Linares, Felipe; Miyazaki, Hayato; Ponce, Gustavo

doi:10.1142/s0219199718500566

Cited by 17 publications

(18 citation statements)

References 8 publications

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“…These results were useful, via the pseudo-conformal transformation, to study the scattering problem for NLS with α ≥ 2/N close to the critical power α = 2/N . The highly-regular solutions were also used to prove local existence for the generalized derivative Schrödinger equation [26,27] and to the generalized Korteweg-de Vries equation [25]. We expect that the results in the present paper will be useful to derive similar results for the nonlinear Klein-Gordon equation (1.1).…”

mentioning

confidence: 63%

Local smooth solutions of the nonlinear Klein-gordon equation

Cazenave¹,

Naumkin²

2021

DCDS-S

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Given any µ 1 , µ 2 ∈ C and α > 0, we prove the local existence of arbitrarily smooth solutions of the nonlinear Klein-Gordon equation ∂ttu − ∆u + µ 1 u = µ 2 |u| α u on R N , N ≥ 1, that do not vanish, i.e. |u(t, x)| > 0 for all x ∈ R N and all sufficiently small t. We write the equation in the form of a first-order system associated with a pseudo-differential operator, then use a method adapted from [Commun. Contemp. Math. 19 (2017), no. 2, 1650038]. We also apply a similar (but simpler than in the case of the Klein-Gordon equation) argument to prove an analogous result for a class of nonlinear Dirac equations.

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confidence: 63%

Local smooth solutions of the nonlinear Klein-gordon equation

Cazenave¹,

Naumkin²

2021

DCDS-S

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“…Recently, Linares, Miyazaki and Ponce in [33] considered the following IVP associated to the generalized KdV equation with low degree of non-linearity [33]). We would like to extend this result for the high dimensional models with non-linearity of fractional order such as generalized BO-ZK equation…”

Section: A C Nascimentomentioning

confidence: 99%

On special regularity properties of solutions of the benjamin-ono-zakharov-kuznetsov (bo-zk) equation

Nascimento¹

2020

Communications on Pure &Amp; Applied Analysis

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In this paper we study special properties of solutions of the initial value problem (IVP) associated to the Benjamin-Ono-Zakharov-Kuznetsov (BO-ZK) equation. We prove that if initial data has some prescribed regularity on the right hand side of the real line, then this regularity is propagated with infinite speed by the flow solution. In other words, the extra regularity on the data propagates in the solutions in the direction of the dispersion. The method of proof to obtain our result uses weighted energy estimates arguments combined with the smoothing properties of the solutions. Hence we need to have local well-posedness for the associated IVP via compactness method. In particular, we establish a local well-posedness in the usual L 2 (R 2)-based Sobolev spaces H s (R 2) for s > 5 4 which coincides with the best available result in the literature proved employing more complicated tools.

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“…The strategy of constructing solutions of (1) that do not vanish was adapted to the derivative Schrödinger equations [23], and to generalized KdV equations [22].…”

Section: Exp N O I-spatial Behavior For Nls and Applications To Scatmentioning

confidence: 99%