Stable localized solutions of arbitrary length for the quintic Swift-Hohenberg equation

Sakaguchi, Hidetsugu; Brand, Helmut R.

doi:10.1016/0167-2789(96)00077-2

Cited by 108 publications

(116 citation statements)

References 39 publications

Supporting

Mentioning

111

Contrasting

Order By: Relevance

“…The two remaining undetermined parameters in the ansatz (10) are ϕ and L. The variation of the effective Lagrangian with respect to L and ϕ can be readily calculated from (14). From ∂ ϕ L ef f = 0, there are two different possibilities.…”

mentioning

confidence: 99%

“…This same conclusion was reported in [15] after a lengthy 'beyond all orders' calculation. It is expected that the ansatz (4) only resembles exact [2,4,14]. For localised states in the snaking regime, the envelope solution has a plateau, which becomes longer as the norm N increases.…”

mentioning

confidence: 99%

“…The approach has several advantages: the exponentially small terms responsible for the phase locking arise naturally in the Lagrangian, giving a simple formula (14) from which the snake and ladder localised states can easily be found, along with their stability. We have concentrated on a simple model equation in the small-parameter regime, but the method is very widely applicable and can be expected to open up a new avenue of research into this challenging field.…”

mentioning

confidence: 99%

“…The third term in (14), proportional to −Lδr, indicates a preference for increasing values of L if δr > 0, so that the patterned region grows in this case and shrinks if δr < 0.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Variational approximations to homoclinic snaking

Susanto

Matthews

2011

Phys. Rev. E

View full text Add to dashboard Cite

We investigate the snaking of localised patterns, seen in numerous physical applications, using a variational approximation. This method naturally introduces the exponentially small terms responsible for the snaking structure, that are not accessible via standard multiple-scales asymptotic techniques. We obtain the symmetric snaking solutions and the asymmetric 'ladder' states, and also predict the stability of the localised states. The resulting approximate formulas for the width of the snaking region show good agreement with numerical results.There has been much recent interest in the phenomenon of spatially localised patterns, [2,4,7,11], extending our understanding of earlier work on this topic [1,13]. As discussed in the review article by Dawes [9], this work is motivated by wide-ranging applications in many different areas of physics, including buckling of struts and cylinders [6,10,15], nonlinear optics [7], convection patterns [1,12], gas discharge systems and granular media. Most theoretical work concentrates on the Swift-Hohenberg equation, which is the simplest model equation that illustrates the effect [1,9,11].These localised patterns arise as a result of bistability between a uniform state and regular patterned state. A front linking these two states might be expected to move in one direction or the other, except at a specific parameter value, referred to as the Maxwell point. However, due to a pinning effect first described qualitatively by Pomeau [13], the front can lock to the pattern, resulting in a finite range of parameter values around the Maxwell point where a stationary front can exist. Combining two such fronts leads to a pinned localised state. The pinning range is seen in numerical simulations [3-5, 14, 18] and leads to a 'snaking' bifurcation diagram in which the control parameter oscillates about the Maxwell point as the size of the localised pattern increases.The localised states can also been seen from the viewpoint of spatial dynamics as a homoclinic connection from the uniform state to itself, leading to a geometrical argument for the existence of such states over a finite range of parameter values [6,8,10].It has been appreciated for some time [13] that the pinning effect cannot be described by conventional multiplescales asymptotics, since this method treats the scale of the pattern and the scale of its envelope as independent variables, leading to an arbitrary phase in the envelope function. Hence, the pinning range is 'beyond all orders', or is exponentially small in the small parameter corresponding to the pattern amplitude [1]. A very thorough analysis of the exponential asymptotics of this problem has recently been carried out by Kozyreff and Chapman [7,11], extending earlier work [15,19]. The calculation of the pinning range is extremely complicated and unfortunately requires two fitting parameters. An ansatz based on the weakly nonlinear solution is substituted into the Lagrangian of the system, and then this is minimised over the unknown parameters. This approach has a...

show abstract

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

“…The third term in (14), proportional to −Lδr, indicates a preference for increasing values of L if δr > 0, so that the patterned region grows in this case and shrinks if δr < 0.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Variational approximations to homoclinic snaking

Susanto

Matthews

2011

Phys. Rev. E

View full text Add to dashboard Cite

show abstract

“…Lets consider a prototype model that exhibits localized patterns and hole solutions in one-dimension extended system, the subcritical Swift-Hohenberg equation [11]:…”

mentioning

confidence: 99%

Localized patterns and hole solutions in one-dimensional extended systems

Clerc

Falcón

2005

Physica A: Statistical Mechanics and its Applications

View full text Add to dashboard Cite

The existence, stability properties, and bifurcation diagrams of localized patterns and hole solutions in one-dimensional extended systems is studied from the point of view of front interactions. An adequate envelope equation is derived from a prototype model that exhibits these particle-type solutions. This equation allow us to obtain an analytical expression for the front interaction, which is in good agreement with numerical simulations. , and nonlinear optics [8,9]. One can understand these localized patterns as patterns extend only over a small portion of the spatial homogeneous systems. From the dynamic point of view, localized patterns in one-dimensional spatial systems are the homoclinic connection for the stationary dynamical system. Recently, from a geometrical point of view, the existence, stability properties, and bifurcation diagrams of localized patterns in one-dimensional extended systems have been studied [10].The aim of this manuscript is to describe how one-dimensional localized patterns and hole solutions arise from front interactions. From a prototype model that exhibits localized patterns and hole solutions, the subcritical Swift-Hohenberg equation, we deduce an adequate equation for the envelope of these particle type solutions. This model has a front solution that connects a stable homogeneous state with an also stable but spatially periodic one. Due to the

show abstract

Localized Structures In Pattern-Forming Systems

Riecke

1999

Pattern Formation in Continuous and Coupled Systems

View full text Add to dashboard Cite

show abstract

Stable localized solutions of arbitrary length for the quintic Swift-Hohenberg equation

Cited by 108 publications

References 39 publications

Variational approximations to homoclinic snaking

Variational approximations to homoclinic snaking

Localized patterns and hole solutions in one-dimensional extended systems

Localized Structures In Pattern-Forming Systems

Contact Info

Product

Resources

About