Numerical study of the regularized long-wave equation I: Numerical methods

Eilbeck, J. C.; McGuire, George

doi:10.1016/0021-9991(75)90115-1

Cited by 114 publications

(93 citation statements)

References 6 publications

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“…The solution to (11) with initial data as in (20) will be a traveling-wave solution only when γ = 1. Indeed, elementary phase-plane analysis shows the solutions displayed in (6) and their negatives in case p is odd, to be the only bounded, traveling-wave solutions that tend to zero at ±∞. When c > c p , so that the solitary wave is orbitally stable, then if γ is near to 1-corresponding to a small perturbation-the solution emanating from γ φ c (x) proves to resolve into a single, stable solitary wave.…”

Section: Experiments With Stable Solitary Wavesmentioning

confidence: 99%

“…The full solution is composed of the stable solitary-wave plus the hump featured in the remaining plots corresponding to this experiment. As seen in the figures, the secondary structure is not quite symmetric about its center of mass, so it is not itself a traveling wave (recall that the only bounded traveling waves tending to zero at ±∞ are those defined by (6), or minus those in case p is odd). Close examination of the solution revealed that the hump has a speed and amplitude that are constant to six decimal places for a very long time.…”

Section: Experiments With Unstable Solitary Wavesmentioning

confidence: 99%

“…To test the accuracy of the numerical scheme, the solution (14)- (15) was initialized with the projection onto S of the solitary-wave solution in (6), and the result compared with the exact traveling wave. It is helpful to observe the error made in numerical simulations of solitary waves in three parts, as follows.…”

Section: Description Of the Numerical Techniquementioning

confidence: 99%

“…It is also considerably easier to numerically integrate (cf. [6], [7], [8]), but does not feature the inverse-scattering theory nor the infinite collection of polynomial conservation laws that obtain for the KdV equation (see [9]). Equation (1) and its KdV relative are typically derived under the assumption that the nonlinear, dispersive media being modeled features small-amplitude, long-wavelength disturbances.…”

Section: Introductionmentioning

confidence: 99%

“…where γ is a real parameter, c > c p , and φ c is as presented in (6). The positive number γ will be referred to as the amplitude perturbation parameter.…”

Section: Experiments With Stable Solitary Wavesmentioning

confidence: 99%

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Stable and Unstable Solitary-Wave Solutions of the Generalized Regularized Long-Wave Equation

2000

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Summary. Investigated here are interesting aspects of the solitary-wave solutions of the generalized Regularized Long-Wave equationFor p > 5, the equation has both stable and unstable solitary-wave solutions, according to the theory of Souganidis and Strauss. Using a high-order accurate numerical scheme for the approximation of solutions of the equation, the dynamics of suitably perturbed solitary waves are examined. Among other conclusions, we find that unstable solitary waves may evolve into several, stable solitary waves and that positive initial data need not feature solitary waves at all in its long-time asymptotics.

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Section: Experiments With Stable Solitary Wavesmentioning

confidence: 99%

Section: Experiments With Unstable Solitary Wavesmentioning

confidence: 99%

Section: Description Of the Numerical Techniquementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…where γ is a real parameter, c > c p , and φ c is as presented in (6). The positive number γ will be referred to as the amplitude perturbation parameter.…”

Section: Experiments With Stable Solitary Wavesmentioning

confidence: 99%

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