Inversion of the linearized Korteweg-de Vries equation at the multisoliton solutions

Hărăguş-Courcelle, Mariana; Sattinger, D. H.

doi:10.1007/s000000050101

Cited by 8 publications

(14 citation statements)

References 15 publications

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“…Since they are linearized inhomogeneous KdV equations, linearized about a KdV solution, they are in principle explicitly solvable [18]. However, the form of the solution that results is quite complicated (see [19] and [11]), and thus it requires some effort to show that these solutions remain uniformly bounded in the norms which we use to bound the errors. As we noted above, the functions S 1 and S 2 do not actually form a part of the approximation at O( 4 ); however, we will show that they remain bounded over the time scales of interest as a part of controlling the error in our approximation.…”

Section: Formal Derivation Of the Modulation Equationsmentioning

confidence: 99%

“…We shall solve the Boussinesq equation (1.3) numerically. Though techniques are known for finding explicit solutions to the linearized KdV equation (see [11] and [18]), the resulting expressions are quite complicated, and so we also solve (1.6) numerically.…”

Section: Some Numericsmentioning

confidence: 99%

“…The terms of O(ǫ 7 ) in (11) give rise to two sets of linear evolution equations, one for F and G and one for S 1 and S 2 . Both are inhomogeneous systems of equations due to the presence of terms involving U , V , A, B and their derivatives.…”

Section: Formal Derivation Of the Modulation Equationsmentioning

confidence: 99%

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Higher Order Modulation Equations for a Boussinesq Equation

Wayne¹,

Wright²

2002

SIAM J. Appl. Dyn. Syst.

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Abstract. In order to investigate corrections to the common KdV approximation to long waves, we derive modulation equations for the evolution of long wavelength initial data for a Boussinesq equation. The equations governing the corrections to the KdV approximation are explicitly solvable, and we prove estimates showing that they do indeed give a significantly better approximation than the KdV equation alone. We also present the results of numerical experiments which show that the error estimates we derive are essentially optimal.Key words. Boussinesq equation, KdV approximation, modulation equations, water wave problem AMS subject classifications. 76B15, 35Q51, 35Q53PII. S11111111024112981. Introduction. Modulation, or amplitude, equations are approximate, often explicitly solvable, model equations derived-usually through asymptotic analysis and the method of multiple time scales-to model more complicated physical situations. Although these equations have been used for over a century, only lately has there been an attempt to rigorously relate solutions of the modulation equations to the original physical problem. [23], the validity of Korteweg-de Vries (KdV) equations as a leading order approximation to the evolution of long wavelength water waves and to a number of other dispersive partial differential equations has been established.While the KdV approximation is extremely useful due to its simplicity and the fact that the KdV equation can be explicitly solved by the inverse scattering transform, both experimentally and numerically one observes departures from the predictions of the KdV equation. Our goal in this paper is to derive modulation equations which govern corrections to the KdV model. In the present paper, we will not work with the full water wave problem but rather will study modulation equations for long wavelength solutions of the Boussinesq equation:

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Section: Formal Derivation Of the Modulation Equationsmentioning

confidence: 99%

Section: Some Numericsmentioning

confidence: 99%

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Higher Order Modulation Equations for a Boussinesq Equation

Wayne¹,

Wright²

2002

SIAM J. Appl. Dyn. Syst.

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“…The exact formulation of the KdV equation comes from Korteweg and de Vries (1895) [15]. This equation was re-discovered in a numerical work by N. Zabusky and M. Kruskal in 1965 [34].After this work, a great amount of literature has emerged, physical, numerical and mathematical, for the study of this equation, see for example [5,14,30,18,9,28,27]. Although under different points of view, among the main topics treated are the following: existence of explicit solutions and their stability, local and global well posedness, long time behavior properties and, of course, related generalized models, hierarchies and their properties.This continuous, focused research on the KdV equation can be in part explained by some striking algebraic properties.…”

mentioning

confidence: 99%

On the Inelastic Two-Soliton Collision for gKdV Equations with General Nonlinearity

Muñoz¹

2009

International Mathematics Research Notices

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We study the problem of 2-soliton collision for the generalized Korteweg-de Vries equations, completing some recent works of Y. Martel and F. Merle [24,25]. We classify the nonlinearities for which collisions are elastic or inelastic. Our main result states that in the case of small solitons, with one soliton smaller than the other one, the unique nonlinearities allowing a perfectly elastic collision are precisely the integrable cases, namely the quadratic (KdV), cubic (mKdV) and Gardner nonlinearities.Here u = u(t, x) is a real-valued function, and f : R → R a nonlinear function, often refered as the nonlinearity of (1). This equation represents a mathematical generalization of the Korteweg-de Vries equation (KdV), namely the case f (s) = s 2 ,(2) other physically important cases are the cubic one f (s) = s 3 , and the quadratic-cubic nonlinearity, namely f (s) = s 2 −µs 3 , µ ∈ R. In the former case, the equation (1) is often refered as the (focusing) modified KdV equation (mKdV), and in the latter, it is known as the Gardner equation. Concerning the KdV equation, it arises in Physics as a model of propagation of dispersive long waves, as was pointed out by J. S. Russel in 1834 [27]. The exact formulation of the KdV equation comes from Korteweg and de Vries (1895) [15]. This equation was re-discovered in a numerical work by N. Zabusky and M. Kruskal in 1965 [34].After this work, a great amount of literature has emerged, physical, numerical and mathematical, for the study of this equation, see for example [5,14,30,18,9,28,27]. Although under different points of view, among the main topics treated are the following: existence of explicit solutions and their stability, local and global well posedness, long time behavior properties and, of course, related generalized models, hierarchies and their properties.This continuous, focused research on the KdV equation can be in part explained by some striking algebraic properties. One of the first properties is the existence of localized, rapidly decaying, stable and smooth solutions called solitons. Given three real numbers t 0 , x 0 and c > 0, solitons are solutions of (2) of the form u(t, x) := Q c (x − x 0 − c(t − t 0 )), Q c (s) := cQ(c 1/2 s),and where Q satisfies the second order nonlinear differential equation Q ′′ − Q + Q 2 = 0, Q(x) = 3 2 cosh 2 ( 1 2 x).

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“…For explicit forms of these driving terms see the equations (22). Linearized KdV equations are explicitly solvable, though this is a complicated matter (see [27] and [15]). However, solutions are simple to compute numerically.…”

Section: Introductionmentioning

confidence: 99%

Corrections to the KdV Approximation for Water Waves

Wright¹

2005

SIAM J. Math. Anal.

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Abstract. In order to investigate corrections to the common KdV approximation for surface water waves in a canal, we derive modulation equations for the evolution of long wavelength initial data. We work in Lagrangian coordinates. The equations which govern corrections to the KdV approximation consist of linearized and inhomogeneous KdV equations plus an inhomogeneous wave equation. These equations are explicitly solvable and we prove estimates showing that they do indeed give a significantly better approximation than the KdV equation alone.

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Inversion of the linearized Korteweg-de Vries equation at the multisoliton solutions

Cited by 8 publications

References 15 publications

Higher Order Modulation Equations for a Boussinesq Equation

Higher Order Modulation Equations for a Boussinesq Equation

On the Inelastic Two-Soliton Collision for gKdV Equations with General Nonlinearity

Corrections to the KdV Approximation for Water Waves

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