Discrete Fourier restriction associated with KdV equations

We consider the initial value problem for the KdV equations with low regularity forcing terms. The case that the forcing term f(x) equals pδ′(x) appears in the study of the excitation of long nonlinear water waves by a moving pressure distribution, where δ′(x) is the first derivative of the Dirac delta function and p is a constant. We have the time global well-posedness with f(x) ∈ L2 by the L2a priori estimate. However, we cannot apply it to the case f(x) ∈ Hσ, σ < 0. To overcome this difficulty, we divide f into the high frequency part and the low frequency part and use the scaling argument. Our results include the time local well-posedness with f(x) ∈ Hσ, σ ≥ -3 and the time global well-posedness with f = pδ′(x) or f(x) ∈ Hσ, σ ≥ -3/2. Our main tools are the Fourier restriction norm method and the I-method.

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Fourier Restriction, Decoupling, and Applications

10.1017/9781108584401 ◽

2019 ◽

Cited By ~ 1

Author(s):

Ciprian Demeter

Keyword(s):

Fourier Restriction

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N-soliton solutions and the Hirota conditions in (1 + 1)-dimensions

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2020-0214 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Wen-Xiu Ma

Keyword(s):

Common Factors ◽

Soliton Solutions ◽

Higher Order ◽

Kdv Equations ◽

The Common ◽

Bilinear Formulation ◽

Generalized Kdv Equations ◽

Hirota Bilinear

Abstract We analyze N-soliton solutions and explore the Hirota N-soliton conditions for scalar (1 + 1)-dimensional equations, within the Hirota bilinear formulation. An algorithm to verify the Hirota conditions is proposed by factoring out common factors out of the Hirota function in N wave vectors and comparing degrees of the involved polynomials containing the common factors. Applications to a class of generalized KdV equations and a class of generalized higher-order KdV equations are made, together with all proofs of the existence of N-soliton solutions to all equations in two classes.

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Soliton turbulence in electronegative plasma due to head-on collision of multi solitons

Zeitschrift für Naturforschung A ◽

10.1515/zna-2020-0186 ◽

2020 ◽

Vol 75 (12) ◽

pp. 999-1007

Author(s):

Rustam Ali ◽

Anjali Sharma ◽

Prasanta Chatterjee

Keyword(s):

Wave Field ◽

Soliton Solutions ◽

Negative Ions ◽

First Integral ◽

Statistical Properties ◽

Charged Dust ◽

Bilinear Method ◽

Electronegative Plasma ◽

Kdv Equations ◽

First Four Moments

AbstractHead-on interaction of four dust ion acoustic (DIA) solitons and the statistical properties of the wave field due to head-on interaction of solitons moving in opposite direction is studied in the framework of two Korteweg de Vries (KdV) equations. The extended Poincaré–Lighthill–Kuo (PLK) method is applied to obtain two opposite moving KdV equations from an unmagnetized four component plasma model consisting of Maxwellian negative ions, cold mobile positive ions, κ-distributed electrons and positively charged dust grains. Hirota’s bilinear method is adopted to obtain two-soliton solutions of both the KdV equations and accordingly act of soliton turbulence is presented due to head-on collision of four solitons. The amplitude and shape of the resultant wave profile at the point of strongest interaction are obtained. To see the effect of head-on collision on the statistical properties of wave field the first four moments are computed. It is observed that the head-on collision has no effect on the first integral moment while the second, third and fourth moments increase in the dominant interaction region of four solitons, which is a clean indication of soliton turbulence.

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Lie symmetry reductions and conservation laws for fractional order coupled KdV system

Advances in Difference Equations ◽

10.1186/s13662-020-03149-z ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Hossein Jafari ◽

Hong Guang Sun ◽

Marzieh Azadi

Keyword(s):

Conservation Laws ◽

Fractional Order ◽

Lie Symmetry ◽

Theoretical Physics ◽

Lie Symmetry Analysis ◽

Kdv Equations ◽

Ode System ◽

Coupled Kdv Equations ◽

Scientific Phenomena ◽

New System

AbstractLie symmetry analysis is achieved on a new system of coupled KdV equations with fractional order, which arise in the analysis of several problems in theoretical physics and numerous scientific phenomena. We determine the reduced fractional ODE system corresponding to the governing factional PDE system.In addition, we develop the conservation laws for the system of fractional order coupled KdV equations.

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