Symmetry of Traveling Wave Solutions to the Allen–Cahn Equation in $${{\mathbb R^{2}}}$$

Gui, Changfeng

doi:10.1007/s00205-011-0480-5

Cited by 32 publications

(11 citation statements)

References 43 publications

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“…We like to mention that axial symmetry has also been shown for traveling wave solutions and saddle solutions of the Allen-Cahn equation in [7] and [8].…”

Section: Theorem 3 Let U Be a Bounded Positive Solution Of (1) Such mentioning

confidence: 78%

Axial symmetry of some steady state solutions to nonlinear Schrödinger equations

Gui

Malchiodi

2011

Proc. Amer. Math. Soc.

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“…We like to mention that axial symmetry has also been shown for traveling wave solutions and saddle solutions of the Allen-Cahn equation in [7] and [8].…”

Section: Theorem 3 Let U Be a Bounded Positive Solution Of (1) Such mentioning

confidence: 78%

Axial symmetry of some steady state solutions to nonlinear Schrödinger equations

Gui

Malchiodi

2011

Proc. Amer. Math. Soc.

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“…These solutions are all invariant with respect to time in a moving frame. Let us also mention that standard traveling fronts ψ(x − cte), with other shapes, still exist when f is balanced, that is, 1 0 f (s)ds = 0, see [11,12,21].…”

Section: Notions Of Transition Fronts and Global Mean Speedmentioning

confidence: 99%

On the mean speed of bistable transition fronts in unbounded domains

Guo

Hamel

Sheng

2020

Journal de Mathématiques Pures et Appliquées

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This paper is concerned with the existence and further properties of propagation speeds of transition fronts for bistable reaction-diffusion equations in exterior domains and in some domains with multiple cylindrical branches. In exterior domains we show that all transition fronts with complete propagation propagate with the same global mean speed, which turns out to be equal to the uniquely defined planar speed. In domains with multiple cylindrical branches, we show that the solutions emanating from some branches and propagating completely are transition fronts propagating with the unique planar speed. We also give some geometrical and scaling conditions on the domain, either exterior or with multiple cylindrical branches, which guarantee that any transition front has a global mean speed.

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“…they are monotone functions of the x and the y variables(see [6] for details and [7] for related results concerning traveling wave solutions of the Allen-Cahn equation). , restricted to Q , consists of a single curve, which is asymptotic to the half of an affine line Λ.…”

Section: Construction Of the Approximate Solutionmentioning

confidence: 99%

End-to-end construction for the Allen–Cahn equation in the plane

et al. 2014

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Abstract. In this paper, we construct a wealth of bounded, entire solutions of the Allen-Cahn equation in the plane. The asymptotic behavior at infinity of these solutions is determined by 2L half affine lines, in the sense that, along each of these half affine lines, the solution is close to a suitable translated and rotated copy of a one dimensional heteroclinic solution. The solutions we construct belong to a smooth 2L-dimensional family of bounded, entire solutions of the Allen-Cahn equation, in agreement with the result of [3] and, in some sense, they provide a description of a collar neighborhood of part of the compactification of the moduli space of 2L-ended solutions for the Allen-Cahn equation. Our construction is inspired by a construction of minimal surfaces by M. Traizet [12].

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Symmetry of Traveling Wave Solutions to the Allen–Cahn Equation in $${{\mathbb R^{2}}}$$

Abstract: Abstract. In this paper, we prove even symmetry of monotone traveling wave solutions to the balanced Allen-Cahn equation in the entire plane. Related results for the unbalanced Allen-Cahn equation are also discussed.

Cited by 32 publications

References 43 publications

Axial symmetry of some steady state solutions to nonlinear Schrödinger equations

Axial symmetry of some steady state solutions to nonlinear Schrödinger equations

On the mean speed of bistable transition fronts in unbounded domains

End-to-end construction for the Allen–Cahn equation in the plane

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