An asymptotic approach to solitary wave instability and critical collapse in long-wave KdV-type evolution equations

Pelinovsky, Dmitry E.; Grimshaw, Roger

doi:10.1016/0167-2789(96)00093-0

Cited by 39 publications

(40 citation statements)

References 18 publications

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“…For example, this has been done for the case of longitudinal instabilities (i.e. β = 0) by Pelinovsky & Grimshaw (1996) and Skryabin (2000). These asymptotic expansions should also extend to the case β = 0.…”

Section: Discussionmentioning

confidence: 99%

“…For the case of longitudinal instability of KdV-type equations results are known even for p = 4. Pelinovsky & Grimshaw (1996) expand the Evans-type function for solitary waves to higher order and show that when p = 4 the KdV-type solitary wave is unstable. It is not difficult to show that this result carries over to the case of transverse instabilities.…”

Section: A Universal Geometric Condition For Transverse Instabilitymentioning

confidence: 99%

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Transverse instability of solitary-wave states of the water-wave problem

Bridges

2001

J. Fluid Mech.

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Transverse stability and instability of solitary waves correspond to a class of perturbations that are travelling in a direction transverse to the direction of the basic solitary wave. In this paper we consider the problem of transverse instability of solitary waves for the water-wave problem, from both the model equation point of view and the full water-wave equations. A new universal geometric condition for transverse instability forms the backbone of the analysis. The theory is first illustrated by application to model PDEs for water waves such as the KP equation, and then it is applied to the full water-wave problem. This is the first theory proposed for transverse instability of solitary waves of the full water-wave problem. The theory suggests the introduction of a new functional for water waves, whose importance is suggested by the mathematical structure. Without explicit calculation, the theory is used to argue that the basic class of solitary waves of the water-wave problem, which bifurcate at Froude number unity, are likely to be stable to transverse perturbations, even at large amplitude.

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Section: Discussionmentioning

confidence: 99%

Section: A Universal Geometric Condition For Transverse Instabilitymentioning

confidence: 99%

Transverse instability of solitary-wave states of the water-wave problem

Bridges

2001

J. Fluid Mech.

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“…Thus, these oscillatory tails always appear ahead of the embedded soliton. Behind the embedded soliton, there is the possibility that a flat shelf may develop as in the perturbed KdV equation [23,24] (see also [18]). If a shelf develops, it moves to the region x −1 at the velocity −c ES in the moving coordinate system (3.1).…”

Section: Dynamics Of Embedded Solitons Under Perturbationsmentioning

confidence: 99%

“…We will use below the internal perturbation analysis described in [14] (see also [18]). The idea is to recognize that under small perturbations, the eigenfunctions U ES (ξ ) and ∂U/∂c(ξ ; δ) for the double embedded eigenvalue λ = 0 of the linearized problem renormalize the location and velocity of the embedded soliton.…”

Section: Dynamics Of Embedded Solitons Under Perturbationsmentioning

confidence: 99%

Semi-stability of embedded solitons in the general fifth-order KdV equation

2002

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Evolution of perturbed embedded solitons in the general Hamiltonian fifth-order Korteweg-de Vries (KdV) equation is studied. When an embedded soliton is perturbed, it sheds a one-directional continuous-wave radiation. It is shown that the radiation amplitude is not minimal in general. A dynamical equation for velocity of the perturbed embedded soliton is derived. This equation shows that a neutrally stable embedded soliton is in fact semi-stable. When the perturbation increases the momentum of the embedded soliton, the perturbed state approaches asymptotically the embedded soliton, while when the perturbation reduces the momentum of the embedded soliton, the perturbed state decays into radiation. Classes of initial conditions to induce soliton decay or persistence are also determined. Our analytical results are confirmed by direct numerical simulations of the fifth-order KdV equation.

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“…学家的广泛关注。许多数学工具和方法被用来求解非线性偏微分方程的行波解。如反散色法 [1]，Backlund 法 [2]，Darboux 变换法 [3]，Hirota 双线性法 [4]，延拓法 [5]，Painleve 分析法 [6]，有限差分法 [7]，Tanh 法 …”

Section: 自然科学中的定律及原理是用偏微分方程来表达的，因而偏微分方程是联结数学与自然科学的关键性纽带，求解偏微分方程的显示解，unclassified