Approximate kink-kink solutions for the ϕ 6 model in the low-speed limit

Abstract

In this paper, we consider the problem of elasticity and stability of the collision of two kinks with low speed v for the nonlinear wave equation known as the $ϕ^{6}$ model in dimension $1 + 1$ . We construct a sequence of approximate solutions ${(ϕ_{k} (v, t, x))}_{k \in N_{⩾ 2}}$ for this model to understand the effects of the collision in the movement of each soliton during a large time interval. The construction uses a new asymptotic method which is not only restricted to the $ϕ^{6}$ model.

Keywords

Kink soliton non-integrable model scalar field partial differential equation ordinary differential equation collision

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References

Abdon , Dynamics of two interacting kinks for the

ϕ^{6}

model, Communications in Mathematical Physics (2023).

Abdon , On the kink-kink collision problem for the

ϕ^{6}

model, 2023, arXiv:2211.09749.

A.R.

Bishop and

Schneider , Solitons and Condensed Matter Physics, Springer-Verlag, 1978.

Coddington and

Levinson , Theory of Ordinary Differential Equations, McGraw Hill Education, 1955.

Dorey ,

Mersh ,

Romanczukiewicz and

Shnir , Kink-antikink collisions in the

ϕ^{6}

model, Physical Review Letters 107 (2011).

V.A.

Gani ,

A.E.

Kudryavtsev and

M.A.

Lizunova , Kink interactions in the (1 + 1)-dimensional

ϕ^{6}

model, Physical Review D: Particles and fields 89 (2014).

Germain and

Pusateri , Quadratic Klein-Gordon equations with a potential in one dimension, Forum of Mathematics, Pi 10 (2022), 1–172. doi:10.1017/fmp.2022.9.

J.T.

Giblin ,

Hui ,

E.A.

Lim and

Yang , How to run through walls: Dynamics of bubble and soliton collisions, Physical Review D: Particles and fields 82 (2010).

S.W.

Hawking ,

I.G.

Moss and

J.M.

Stewart , Bubble collisions in the very early universe, Physical Review D 26 (1982), 2681–2713. doi:10.1103/PhysRevD.26.2681.

10.

Jendrej and

Chen , Kink networks for scalar fields in dimension

1 + 1

, Nonlinear Analysis 215 (2022).

11.

Kowalczyk ,

Martel and

Muñoz , Kink dynamics in the

ϕ^{4}

model: Asymptotic stability for odd perturbations in the energy space, Journal of the American Mathematical Society 30 (2017), 769–798. doi:10.1090/jams/870.

12.

Kowalczyk ,

Martel ,

Muñoz and

Van Den Bosch , A sufficient condition for asymptotic stability of kinks in general (1 + 1)-scalar field models, Annals of PDE 7 (2021), 1–98. doi:10.1007/s40818-021-00098-y.

13.

Krieger ,

Nakanishi and

Schlag , Global dynamics above the ground state energy for the one-dimensional NLKG equation, Mathematische Zeitschrift 272 (2012), 297–316. doi:10.1007/s00209-011-0934-3.

14.

Lawrie ,

Jendrej and

Kowalczyk , Dynamics of strongly interacting kink-antikink pairs for scalar fields on a line, Duke Mathematical Journal (2019).

15.

Manton and

Sutcliffe , Topological Solitons. Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2004.

16.

Martel and

Merle , Stability of two soliton colision for non-integrable gKdV equations, Communications in Mathematical Physics 286 (2009), 39–79. doi:10.1007/s00220-008-0685-0.

17.

Martel and

Merle , Inelastic interaction of nearly equal solitons for the quartic gKdV equation, Inventiones Mathematicae 183(3) (2011), 563–648. doi:10.1007/s00222-010-0283-6.

18.

Munoz , On the inelastic 2-soliton collision for generalized KdV equations, International Mathematical Research Notices 9 (2010), 1624–1719.

19.

Munoz , Inelastic character of solitons of slowly varying gKdV equations, Communications in Mathematical Physics 314(3) (2012), 817–852. doi:10.1007/s00220-012-1463-6.

20.

Perelman , Two soliton collision for nonlinear Schrödinger equations in dimension 1, Annales de l’IHP Anal. Non Linéaire 28(3) (2011), 357–384.

21.

Schlag and

Lühmann , Asymptotic stability of the sine-Gordon kink under odd perturbations, 2021, arXiv:2106.09605.

22.

Vilekin and

E.P.S.

Shellard , Cosmic Strings and Other Topological Defects, Cambridge University Press, 1994.