Approximate conservation laws in the KdV equation

Israwi, Samer; Kalisch, Henrik

doi:10.1016/j.physleta.2018.12.009

Cited by 15 publications

(15 citation statements)

References 19 publications

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“…Now given initial data (ξ 0 , u 0 ) ∈ H s × H s+1 and the solution (ξ, u) ∈ C (0, T , H s × H s+1 ) of the system (9) we define initial data for (1) by η 0 = ξ 0 , and using the solution η ∈ C (0, T , H s ) guaranteed by Theorem 8 we define v K dV by (12). Then following the proof laid out in [11], we can obtain the following estimate.…”

Section: Definitionmentioning

confidence: 99%

“…Finally using (14), and the definition of v K dV (12) along with the fact that η ∈ C 1 (0, T /ε, H s−3 ) (cf. [11]) proves the required result.…”

Section: Convergence Of Momentum Densitymentioning

confidence: 99%

“…The second and third integral are sometimes called momentum and energy, but this terminology may be misleading since these integrals are not readily interpreted as approximations of the physical momentum and energy appearing in the context of the Euler equations. Indeed, the authors of [1] already state clearly that they do not believe these integrals to be approximations of the physical momentum and density, and further doubt was cast on this interpretation in more recent work [2,[12][13][14].…”

Section: Introductionmentioning

confidence: 99%

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A Mathematical Justification of the Momentum Density Function Associated to the KdV Equation

Israwi¹,

Kalisch²

2021

Comptes Rendus. Mathématique

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Consideration is given to the KdV equation as an approximate model for long waves of small amplitude at the free surface of an inviscid fluid. It is shown that there is an approximate momentum density associated to the KdV equation, and the difference between this density and the physical momentum density derived in the context of the full Euler equations can be estimated in terms of the long-wave parameter. Résumé. L'équation de KdV est considérée comme un modèle approximatif pour des ondes longues de faible amplitude à la surface libre d'un fluide non visqueux. On montre qu'il y a une densité de moment approximative associée à l'équation de KdV, et que la différence entre cette densité et la densité de de moment physique dérivée dans le contexte du système d'Euler peut être estimée en fonction du paramètre d'onde longue.

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Section: Definitionmentioning

confidence: 99%

“…Finally using (14), and the definition of v K dV (12) along with the fact that η ∈ C 1 (0, T /ε, H s−3 ) (cf. [11]) proves the required result.…”

Section: Convergence Of Momentum Densitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Mathematical Justification of the Momentum Density Function Associated to the KdV Equation

Israwi¹,

Kalisch²

2021

Comptes Rendus. Mathématique

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“…However, the charges of (3.15) and (3.20) do not have a direct interpretation and their relationships to the relevant physical quantities have recently been considered (see e.g. [27]). Third, in order to achieve the trivial charges q (−n) , n = 3, 5, 7 we have removed the non-homogeneous terms, which were dubbed as "anomalies" in [9], and conveniently rewritten the relevant equations as exact conservation laws.…”

Section: Kdv-type Asymptotically Conserved Chargesmentioning

confidence: 99%

Quasi-integrable KdV models, towers of infinite number of anomalous charges and soliton collisions

Blas

Ochoa

Suárez

2020

J. High Energ. Phys.

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We found, through analytical and numerical methods, new towers of infinite number of asymptotically conserved charges for deformations of the Korteweg-de Vries equation (KdV). It is shown analytically that the standard KdV also exhibits some towers of infinite number of anomalous charges, and that their relevant anomalies vanish for N −soliton solution. Some deformations of the KdV model are performed through the Riccati-type pseudo-potential approach, and infinite number of exact non-local conservation laws is provided using a linear formulation of the deformed model. In order to check the degrees of modifications of the charges around the soliton interaction regions, we compute numerically some representative anomalies, associated to the lowest order quasi-conservation laws, depending on the deformation parameters {ǫ1, ǫ2}, which include the standard KdV (ǫ1 = ǫ2 = 0), the regularized long-wave (RLW) (ǫ1 = 1, ǫ2 = 0), the modified regularized long-wave (mRLW) (ǫ1 = ǫ2 = 1) and the KdV-RLW (KdV-BBM) type (ǫ2 = 0, ǫ = {0, 1}) equations, respectively. Our numerical simulations show the elastic scattering of two and three solitons for a wide range of values of the set {ǫ1, ǫ2}, for a variety of amplitudes and relative velocities. The KdV-type equations are quite ubiquitous in several areas of non-linear science, and they find relevant applications in the study of General Relativity on AdS3, Bose-Einstein condensates, superconductivity and soliton gas and turbulence in fluid dynamics.

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“…Some of them are the Boussinesq-type approximations that stimulated many wave motion issues in coastal engineering and that may handle most of wave phenomena occurring in near-shore zones (such as refraction, diffraction, shoaling, frequency dispersion and nonlinear interaction). (See [2,16,18,19,29] for a justification of such approximations.) Nevertheless these models cannot predict either where and when a wave breaks or, particularly, the hydrodynamic features of a breaking wave.…”

mentioning

confidence: 99%