Solitary waves induced by boundary motion

Chu, C. K.; Li, Xiang; Baransky, Yurij Andrew

doi:10.1002/cpa.3160360407

Cited by 50 publications

(27 citation statements)

References 5 publications

(1 reference statement)

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“…Two boundary conditions are then needed at x = 0, as stated in (1.2). The same conclusions regarding the appropriate number of boundary conditions for a wellposed problem for the linear KdV equation are drawn by Fokas (2000) (see theorem (3.1) on p. 4201) and by Chu et al (1983), the latter using an energy conservation argument.…”

Section: Solutions Of the Initial-boundary-value Problemmentioning

confidence: 54%

“…A number of physical applications exist for (1.1), such as the generation of waves in a shallow channel by a wave-making device or the critical withdrawal of a stratified fluid from a reservoir (Clarke & Imberger 1994). Chu et al (1983) considered the positive quarter-plane problem (1.1) numerically. The energy conservation law for the KdV equation was used to deduce that one boundary condition should be applied at x = 0, with the other two being boundedness conditions on the solution as x → ∞.…”

Section: Introductionmentioning

confidence: 99%

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The initial boundary problem for the Korteweg-de Vries equation on the negative quarter-plane

Marchant

Smyth

2002

Proc. R. Soc. Lond. A

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The initial-boundary-value problem for the Korteweg-de Vries equation on the negative quarter-plane, x < 0 and t > 0, is considered. The formulation of this problem is different to the usual initial-boundary-value problem on the positive quarter-plane, for which x > 0 and t > 0. Two boundary conditions are required at x = 0 for the negative quarter-plane problem, in contrast to the one boundary condition needed at x = 0 for the positive quarter-plane problem. Solutions of the Korteweg-de Vries equation on the infinite line, such as the soliton, cnoidal wave, mean height variation and undular bore solution, are used to find approximate solutions to the negative quarter-plane problem. Five qualitatively different types of solution are found, depending on the relation between the initial and boundary values. Excellent comparisons are obtained between these solutions and full numerical solutions of the KdV equation.

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Section: Solutions Of the Initial-boundary-value Problemmentioning

confidence: 54%

Section: Introductionmentioning

confidence: 99%

The initial boundary problem for the Korteweg-de Vries equation on the negative quarter-plane

Marchant

Smyth

2002

Proc. R. Soc. Lond. A

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“…The square pulse (in time) at the boundary generates a cascade of solitary waves as time evolves. This interesting phenomenon was first observed by Chu, Xiang, and Baransky in [8] (see also [12]). …”

Section: Kdv Equationmentioning

confidence: 56%

A New Dual-Petrov-Galerkin Method for Third and Higher Odd-Order Differential Equations: Application to the KDV Equation

Shen¹

2003

SIAM J. Numer. Anal.

131

127

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Abstract.A new dual-Petrov-Galerkin method is proposed, analyzed, and implemented for third and higher odd-order equations using a spectral discretization. The key idea is to use trial functions satisfying the underlying boundary conditions of the differential equations and test functions satisfying the "dual" boundary conditions. The method leads to linear systems which are sparse for problems with constant coefficients and well conditioned for problems with variable coefficients. Our theoretical analysis and numerical results indicate that the proposed method is extremely accurate and efficient and most suitable for the study of complex dynamics of higher odd-order equations. 1. Introduction. Over the last thirty years, spectral methods have been playing an increasingly important role in scientific and engineering computations. Most work on spectral methods is concerned with elliptic and parabolic-type equations; there has also been active research on spectral methods for hyperbolic problems (see, for instance, [11,7,14] and the references therein). However, there is only a limited body of literature on spectral methods for dispersive, namely, third and higher odd-order, equations. In particular, relatively few studies are devoted to third and higher oddorder equations in finite intervals. This is partly due to the fact that direct collocation methods for higher odd-order boundary problems lead to very much higher condition numbers-more precisely, of order N 2k , where N is the number of modes and k is the order of the equation-and often exhibit unstable modes if the collocation points are not properly chosen (see, for instance, [17,21]).In a sequence of papers [22,23,25,26], the author constructed efficient spectralGalerkin algorithms for elliptic equations in various situations. In this paper, we extend the main idea for constructing efficient spectral-Galerkin algorithms-using compact combinations of orthogonal polynomials, which satisfy essentially all the underlying homogeneous boundary conditions, as basis functions-to third and higher odd-order equations. Since the main differential operators in these equations are not symmetric, it is quite natural to employ a Petrov-Galerkin method.The key idea of the new spectral dual-Petrov-Galerkin method is the innovative choice of the test and trial functional spaces. More precisely, we choose the trial functions to satisfy the underlying boundary conditions of the differential equations, and we choose the test functions to satisfy the "dual" boundary conditions.Recently, Ma and Sun [19,20] studied an interesting Legendre-Petrov-Galerkin method for third-order equations. The main difference between this paper and [19,20]

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“…According to the theory for a uniform channel, the first soliton at the front is the biggest and is twice as large as the amplitude of the initial perturbation (in the absence of dissipation) (Karpman, 1973). Note that even an infinitely small Reynolds dissipation reduces the maximum amplitude of the perturbation (undular bore) which cannot exceed 1.5 times the amplitude of the initial perturbation in the case of a uniform channel (Chu et al, 1983;Tsuji, 1991).…”

Section: Numerical Modellingmentioning

confidence: 99%

Bore formation, evolution and disintegration into solitons in shallow inhomogeneous channels

Caputo

Stepanyants

2003

Nonlin. Processes Geophys.

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Abstract. The propagation of nonlinear surface waves in channels of smoothly variable in space cross section is studied theoretically and by means of numerical computations. The mathematical model describing wave evolution is based on the generalized Korteweg-de Vries equation with additional terms due to spatial inhomogeneity and energy dissipation. Specifically we consider channels of variable depth and width. The breaking of Riemann waves and the disintegration of hydraulic jumps into trains of solitons have been examined. The results obtained can be useful in particular for the understanding some peculiarities of bore (mascaret) formation, viscous evolution and disintegration into solitons in inhomogeneous channels or rivers.

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Solitary waves induced by boundary motion

Cited by 50 publications

References 5 publications

The initial boundary problem for the Korteweg-de Vries equation on the negative quarter-plane

The initial boundary problem for the Korteweg-de Vries equation on the negative quarter-plane

A New Dual-Petrov-Galerkin Method for Third and Higher Odd-Order Differential Equations: Application to the KDV Equation

Bore formation, evolution and disintegration into solitons in shallow inhomogeneous channels

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