The IVP for a higher dimensional version of the Benjamin-Ono equation in weighted Sobolev spaces

Riaño, Oscar

doi:10.1016/j.jfa.2020.108707

Cited by 10 publications

(6 citation statements)

References 22 publications

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“…We have also included an extra weight in our arguments to consider solutions with lower regularity in the hypothesis of theorems 1.3 and 1.4. We apply a similar approach in [39].…”

Section: Remark 12mentioning

confidence: 99%

On persistence properties in weighted spaces for solutions of the fractional Korteweg–de Vries equation

Riaño

2021

Nonlinearity

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Persistence problems in weighted spaces have been studied for different dispersive models involving non-local operators. Generally, these models do not propagate polynomial weights of arbitrary magnitude, and the maximum decay rate is associated with the dispersive part of the equation. Altogether, this analysis is complemented by unique continuation principles that determine optimal spatial decay. This work is intended to establish the above questions for a weakly dispersive perturbation of the inviscid Burgers equation. More precisely, we consider the fractional Korteweg–de Vries equation, which comprises the Burgers–Hilbert equation and dispersive effects weaker than those of the Benjamin–Ono equation.

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“…We have also included an extra weight in our arguments to consider solutions with lower regularity in the hypothesis of theorems 1.3 and 1.4. We apply a similar approach in [39].…”

Section: Remark 12mentioning

confidence: 99%

On persistence properties in weighted spaces for solutions of the fractional Korteweg–de Vries equation

Riaño

2021

Nonlinearity

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“…In general, the fKdV equation in dimention d = 1 has been used in a variety of wave phenomena, and the lower dimensional case 0 < α < 1 has been investigated as a model to measure the influence of dispersive effects on the dynamics of Burger's equation. For other studies on the 1d fKdV equation, for example, see [7,45,64,74,65,77], for the higher dimensional fKdV, see [76,38,79,75], and reference therein.…”

Section: Introductionmentioning

confidence: 99%

“…Later, for α = 1 and d = 2 , the local well-posedness was improved by Schippa [79], who also extended the local well-posedness to 1 ≤ α < 2 in H s (R d ) with s > d+3 2 − α. We also mention that for d ≥ 1 and 0 < α < 2, the well-posedness of the fKdV equation has been studied in weighted spaces H s (R d ) ∩ L 2 (ω(x) dx) with certain weights ω(x), e.g., (1 + |x|) a , see [41,31,30,76,74]. Now, that we reviewed the existence of solutions, we point a very specific solution, a solitary wave, traveling in time preserving its shape.…”

Section: Introductionmentioning

confidence: 99%

Stability and instability of solitary waves in fractional generalized KdV equation in all dimensions

Riaño¹,

Roudenko²

2022

Preprint

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We study stability properties of solitary wave solutions for the fractional generalized Korteweg-de Vries equationα + 1, we prove the orbital stability of solitary waves using the concentration-compactness argument, the commutator estimates and expansions of nonlocal operator D α in several variables. In the L 2 -supercritical case, 2d α + 1 < m < m * , we show that solitary waves are unstable. The instability is obtained by making use of an explicit sequence of initial conditions in conjunction with modulation and truncation arguments. This work maybe be of interest for studying solitary waves and other coherent structures in dispersive equations that involve nonlocal operators. The arguments developed here are independent of the spatial dimension and rely on a careful analysis of spatial decay of ground states and their regularity. In particular, in the one-dimensional setting, we show the instability of solitary waves of the supercritical generalized Benjamin-Ono equation and the dispersion-generalized Benjamin-Ono equation, furthermore, new results on the instability are obtained in the weaker dispersion regime 1 2 < α < 1.Contents 2 -instability Appendix References

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“…Furthermore, the well‐posedness results in weighted Sobolev spaces as well as some unique continuation principles for Equation () were studied in Ref. 24. We remark that the above lwp results were obtained via compactness methods as one cannot solve the initial value problem associated to () by a Picard iterative method implemented on its integral formulation for any initial data in the Sobolev space

H^r(\mathbb {R}^2)

r\in \mathbb {R}

(see Ref.…”

Section: Introductionmentioning

confidence: 99%

Higher dimensional generalization of the Benjamin‐Ono equation: 2D case

2021

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We consider a higher‐dimensional version of the Benjamin‐Ono (HBO) equation in the 2D setting: ut−R1normalΔu+12false(u2false)x=0,(x,y)∈R2$u_t- \mathcal {R}_1 \Delta u + \frac{1}{2}(u^2)_x=0, (x,y) \in \mathbb {R}^2$, which is L2$L^2$‐critical, and investigate properties of solutions both analytically and numerically. For a generalized equation (fractional 2D gKdV) after deriving the Pohozaev identities, we obtain nonexistence conditions for solitary wave solutions, then prove uniform bounds in the energy space or conditional global existence, and investigate the radiation region, a specific wedge in the negative x$x$‐direction. We then introduce our numerical approach in a general context, and apply it to obtain the ground state solution in the 2D critical HBO equation, then show that its mass is a threshold for global versus finite time existing solutions, which is typical in the focusing (mass‐)critical dispersive equations. We also observe that globally existing solutions tend to disperse completely into the radiation in this nonlocal equation. The blow‐up solutions travel in the positive x$x$‐direction with the rescaled ground state profile while also radiating dispersive oscillations into the radiative wedge. We conclude with examples of different interactions of two solitary wave solutions, including weak and strong interactions.

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The IVP for a higher dimensional version of the Benjamin-Ono equation in weighted Sobolev spaces

Cited by 10 publications

References 22 publications

On persistence properties in weighted spaces for solutions of the fractional Korteweg–de Vries equation

On persistence properties in weighted spaces for solutions of the fractional Korteweg–de Vries equation

Stability and instability of solitary waves in fractional generalized KdV equation in all dimensions

Higher dimensional generalization of the Benjamin‐Ono equation: 2D case

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