On the IVP for the k-Generalized Benjamin–Ono Equation

Ponce, Gustavo

doi:10.1007/978-3-319-30961-3_16

Cited by 4 publications

(2 citation statements)

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“…Some numerical investigations for the BO-type equations were done in [48], see also numerical studies of solitons and interactions in the integrable BO equation in [64] and [56]. For further details, see excellent reviews [66] and [67].…”

Section: Introductionmentioning

confidence: 99%

Dynamics of solutions in the generalized Benjamin-Ono equation: a numerical study

Roudenko,

Wang,

Yang

2020

Preprint

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We consider the generalized Benjamin-Ono (gBO) equation on the real line, ut +∂x(−Hux + 1 m u m ) = 0, x ∈ R, m = 2, 3, 4, 5, and perform numerical study of its solutions. We first compute the ground state solution to −Q − HQ + 1 m Q m = 0 via Petviashvili's iteration method. We then investigate the behavior of solutions in the Benjamin-Ono (m = 2) equation for initial data with different decay rates and show decoupling of the solution into a soliton and radiation, thus, providing confirmation to the soliton resolution conjecture in that equation. In the mBO equation (m = 3), which is L 2 -critical, we investigate solutions close to the ground state mass, and, in particular, we observe the formation of stable blow-up above it. Finally, we focus on the L 2 -supercritical gBO equation with m = 4, 5. In that case we investigate the global vs finite time existence of solutions, and give numerical confirmation for the dichotomy conjecture, in particular, exhibiting blow-up phenomena in the supercritical setting.

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Section: Introductionmentioning

confidence: 99%

Dynamics of solutions in the generalized Benjamin-Ono equation: a numerical study

Roudenko,

Wang,

Yang

2020

Preprint

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“…We remark that Equation (), including other nonlinearities, is of interest in various water wave models such as waves in deep water, e.g., see Refs. 9–14 and reviews 15,16 …”

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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