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 model in dimension . We construct a sequence of approximate solutions 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 model.
M.Abdon, Dynamics of two interacting kinks for the model, Communications in Mathematical Physics (2023).
2.
M.Abdon, On the kink-kink collision problem for the model, 2023, arXiv:2211.09749.
3.
A.R.Bishop and T.Schneider, Solitons and Condensed Matter Physics, Springer-Verlag, 1978.
4.
E.Coddington and N.Levinson, Theory of Ordinary Differential Equations, McGraw Hill Education, 1955.
5.
P.Dorey, K.Mersh, T.Romanczukiewicz and Y.Shnir, Kink-antikink collisions in the model, Physical Review Letters107 (2011).
6.
V.A.Gani, A.E.Kudryavtsev and M.A.Lizunova, Kink interactions in the (1 + 1)-dimensional model, Physical Review D: Particles and fields89 (2014).
7.
P.Germain and F.Pusateri, Quadratic Klein-Gordon equations with a potential in one dimension, Forum of Mathematics, Pi10 (2022), 1–172. doi:10.1017/fmp.2022.9.
8.
J.T.Giblin, L.Hui, E.A.Lim and I.Yang, How to run through walls: Dynamics of bubble and soliton collisions, Physical Review D: Particles and fields82 (2010).
9.
S.W.Hawking, I.G.Moss and J.M.Stewart, Bubble collisions in the very early universe, Physical Review D26 (1982), 2681–2713. doi:10.1103/PhysRevD.26.2681.
10.
J.Jendrej and G.Chen, Kink networks for scalar fields in dimension , Nonlinear Analysis215 (2022).
11.
M.Kowalczyk, Y.Martel and C.Muñoz, Kink dynamics in the model: Asymptotic stability for odd perturbations in the energy space, Journal of the American Mathematical Society30 (2017), 769–798. doi:10.1090/jams/870.
12.
M.Kowalczyk, Y.Martel, C.Muñoz and H.Van Den Bosch, A sufficient condition for asymptotic stability of kinks in general (1 + 1)-scalar field models, Annals of PDE7 (2021), 1–98. doi:10.1007/s40818-021-00098-y.
13.
J.Krieger, K.Nakanishi and W.Schlag, Global dynamics above the ground state energy for the one-dimensional NLKG equation, Mathematische Zeitschrift272 (2012), 297–316. doi:10.1007/s00209-011-0934-3.
14.
A.Lawrie, J.Jendrej and M.Kowalczyk, Dynamics of strongly interacting kink-antikink pairs for scalar fields on a line, Duke Mathematical Journal (2019).
15.
N.Manton and P.Sutcliffe, Topological Solitons. Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2004.
16.
Y.Martel and F.Merle, Stability of two soliton colision for non-integrable gKdV equations, Communications in Mathematical Physics286 (2009), 39–79. doi:10.1007/s00220-008-0685-0.
17.
Y.Martel and F.Merle, Inelastic interaction of nearly equal solitons for the quartic gKdV equation, Inventiones Mathematicae183(3) (2011), 563–648. doi:10.1007/s00222-010-0283-6.
18.
C.Munoz, On the inelastic 2-soliton collision for generalized KdV equations, International Mathematical Research Notices9 (2010), 1624–1719.
19.
C.Munoz, Inelastic character of solitons of slowly varying gKdV equations, Communications in Mathematical Physics314(3) (2012), 817–852. doi:10.1007/s00220-012-1463-6.
20.
G.Perelman, Two soliton collision for nonlinear Schrödinger equations in dimension 1, Annales de l’IHP Anal. Non Linéaire28(3) (2011), 357–384.
21.
W.Schlag and J.Lühmann, Asymptotic stability of the sine-Gordon kink under odd perturbations, 2021, arXiv:2106.09605.
22.
A.Vilekin and E.P.S.Shellard, Cosmic Strings and Other Topological Defects, Cambridge University Press, 1994.
