A global characterization of gap solitary-wave solutions to a coupled KdV system

Champneys, Alan R; Groves, Mark D.; Woods, P.D.

doi:10.1016/s0375-9601(00)00355-8

Cited by 10 publications

(3 citation statements)

References 23 publications

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“…In particular the theory of localized buckling of long struts bears a substantial similarity to the work reported above [6,7]. This in turn is closely related to the study of solitary waves in the fifth order Kortewegde Vries (KdV) equation arising in the theory of long wavelength water waves [40,41] and related systems [42]. In particular the solitary waves studied in the context of the fifth order KdV equation correspond precisely to the localized states of the Swift-Hohenberg equation (1) in the special case g = 0, with the parameter r related to the speed of the waves.…”

Section: Discussionsupporting

confidence: 54%

Localized states in the generalized Swift-Hohenberg equation

2006

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The Swift-Hohenberg equation with quadratic and cubic nonlinearities exhibits a remarkable wealth of stable spatially localized states. The presence of these states is related to a phenomenon called homoclinic snaking. Numerical computations are used to illustrate the changes in the localized solution as it grows in spatial extent and to determine the stability properties of the resulting states. The evolution of the localized states once they lose stability is illustrated using direct simulations in time.

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Section: Discussionsupporting

confidence: 54%

Localized states in the generalized Swift-Hohenberg equation

2006

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“…Then an equation for the amplitudeÃ 1 (X) as a function of the slow variableX is derived using the method of multiple scales and perturbation analysis [20]. For the scaling given in (2.5) we get a stationary equation forÃ 1 (X) of the following Ginzburg-Landau form, 6) whereν is a constant related to the parameters of the original equation which approaches ν (1.7) as r → 0. Solving (2.6), we get an expression forεÃ 1 (X) which describes a small amplitude homoclinic solution for A [21],…”

Section: Overview and Description Of The Amplitude Equationmentioning

confidence: 99%

Localized periodic patterns for the non-symmetric generalized Swift-Hohenberg equation

Budd

Kuske

2005

Physica D: Nonlinear Phenomena

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A new asymptotic multiple scale expansion is used to derive envelope equations for localized spatially periodic patterns in the context of the generalized Swift-Hohenberg equation. An analysis of this envelope equation results in parametric conditions for localized patterns. Furthermore, it yields corrections for wave number selection which are an order of magnitude larger for asymmetric nonlinearities than for the symmetric case. The analytical results are compared with numerical computations which demonstrate that the condition for localized patterns coincides with vanishing Hamiltonian and Lagrangian for periodic solutions. One striking feature of the choice of scaling parameters is that the derived condition for localized patterns agrees with the numerical results for a significant range of parameters which are an O(1) distance from the bifurcation, thus providing a novel approach for studying these localized patterns.

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“…In addition to serving as a canonical model for fourth order systems, such as those listed in paragraph below equation (1.3), many of the phenomenon observed in this model are also observed in coupled second order systems. The Swift-Hohenberg equation has been used as a "toy" problem for studying pattern formation in coupled reaction-diffusion patterns [10], and localized and multi-bump patterns, which have been observed in models of elastic buckling [4], have been observed through similar mechanisms in coupled KdVtype models in optics [5]. Therefore, understanding the construction and asymptotic behavior of solutions to (1.1) opens the door for similar investigations of multi-bump patterns in these related systems.…”

Section: Introductionmentioning

confidence: 99%

A singular perturbation problem for a fourth order ordinary differential equation

Kuske

Peletier

2005

Nonlinearity

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In higher order model equations such as the Swift-Hohenberg equation and the nonlinear beam equation, different length scales may be distinguished, depending on the parameters in the equation. In this paper we discuss this phenomenon for stationary solutions of the Swift-Hohenberg equation and show that when the scales are very different a multi-scale analysis can be used to yield asymptotic expressions for multi-bump periodic solutions and the bifurcation diagram of such solutions with prescribed qualitative properties, such as the number of bumps.

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A global characterization of gap solitary-wave solutions to a coupled KdV system

Cited by 10 publications

References 23 publications

Localized states in the generalized Swift-Hohenberg equation

Localized states in the generalized Swift-Hohenberg equation

Localized periodic patterns for the non-symmetric generalized Swift-Hohenberg equation

A singular perturbation problem for a fourth order ordinary differential equation

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