On the propagation of regularity for solutions of the fractional Korteweg-de Vries equation

Mendez, Argenis J.

doi:10.1016/j.jde.2020.06.027

Cited by 24 publications

(16 citation statements)

References 19 publications

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“…The property described in Theorem 1.1 is intrinsic to suitable solutions of some nonlinear dispersive models (see for instance [35]). This result was also obtained for the Benjamin-Ono equation [18] and more recently, for the dispersion generalized Benjamin-Ono equation [37] and the fractional Korteweg-de Vries equation [38]. In the context of 2D models, analogous results for the Kadomtsev-Petviashvili II equation [19], Zakharov-Kuznetsov [36], fifth order Kadomtsev-Petviashvili II [41] and Kadomtsev-Petviashvili-Benjamin-Ono [42] equations were proved.…”

Section: A C Nascimentosupporting

confidence: 67%

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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Section: A C Nascimentosupporting

confidence: 67%

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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“…The proof of Theorem 1.2 follows in spirit the techniques and arguments presented in [16,27,28,29] regarding propagation of local derivatives, and the conclusions in [20] for the fractional setting. However, in the case of (1.1), we face several additional difficulties expected from the interaction between the dispersion ∂ x D α+1…”

Section: Introductionmentioning

confidence: 93%

“…Our analysis follows the technique introduced in [16] (see also [20,27,28,29]), so that our starting point will be basically to obtain weighted energy estimate by localizing the regions in R 2 where the information concerning the regularity is available.…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

On some regularity properties for the dispersive generalized Benjamin-Ono-Zakharov-Kuznetsov Equation

C.¹,

Mendez²,

Riaño³

2020

Preprint

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This work aims to study some smoothness properties concerning the initial value problem associated to the dispersive generalized Benjamin-Ono-Zakharov-Kuznetsov equation. More precisely, we prove that the solutions to this model satisfy the so-called propagation of regularity. Roughly speaking, this principle states that if the initial data enjoys some extra smoothness prescribed on a family of half-spaces, then the regularity is propagated with infinite speed. In this sense, we prove that regardless of the scale measuring the extra regularity in such hyperplane collection, then all this regularity is also propagated by solutions of this model. Our analysis is mainly based on the deduction of propagation formulas relating homogeneous and non-homogeneous derivatives in certain regions of the plane.

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“…The KdV equation has been applied in the description of dynamical effects such as longitudinal astigmatic, ion sound, and magnetic fluid waves [4][5][6][7][8][9]. The convergence properties, existence and the regularity of solutions of KdV-type equation have been discussed in [10][11][12].…”

Section: Introductionmentioning

confidence: 99%

Solitary Wave Solutions of the Generalized Rosenau-KdV-RLW Equation

2020

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This paper investigates the solitary wave solutions of the generalized Rosenau–Korteweg-de Vries-regularized-long wave equation. This model is obtained by coupling the Rosenau–Korteweg-de Vries and Rosenau-regularized-long wave equations. The solution of the equation is approximated by a local meshless technique called radial basis function (RBF) and the finite-difference (FD) method. The association of the two techniques leads to a meshless algorithm that does not requires the linearization of the nonlinear terms. First, the partial differential equation is transformed into a system of ordinary differential equations (ODEs) using radial kernels. Then, the ODE system is solved by means of an ODE solver of higher-order. It is shown that the proposed method is stable. In order to illustrate the validity and the efficiency of the technique, five problems are tested and the results compared with those provided by other schemes.

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On the propagation of regularity for solutions of the fractional Korteweg-de Vries equation

Cited by 24 publications

References 19 publications

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

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

On some regularity properties for the dispersive generalized Benjamin-Ono-Zakharov-Kuznetsov Equation

Solitary Wave Solutions of the Generalized Rosenau-KdV-RLW Equation

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