We consider classical scalar fields in dimension with a self-interaction potential being a symmetric double-well. Such a model admits non-trivial static solutions called kinks and antikinks. A kink cluster is a solution approaching, for large positive times, a superposition of alternating kinks and antikinks whose velocities converge to 0 and mutual distances grow to infinity.
The aim of this note is to expose some developments on asymptotic behaviour of kink clusters obtained in our recent preprint [23]. The results are partially inspired by the notion of “parabolic motions” in the Newtonian -body problem. We present this analogy and mention its limitations. We also explain the role of kink clusters as universal profiles for formation of multi-kink configurations.
@incollection{JEDP_2023____A7_0, author = {Jacek Jendrej and Andrew Lawrie}, title = {Dynamics of kink clusters for scalar fields in dimension~$1+1$}, booktitle = {}, series = {Journ\'ees \'equations aux d\'eriv\'ees partielles}, note = {talk:7}, pages = {1--12}, publisher = {R\'eseau th\'ematique AEDP du CNRS}, year = {2023}, doi = {10.5802/jedp.678}, language = {en}, url = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.678/} }
TY - JOUR AU - Jacek Jendrej AU - Andrew Lawrie TI - Dynamics of kink clusters for scalar fields in dimension $1+1$ JO - Journées équations aux dérivées partielles N1 - talk:7 PY - 2023 SP - 1 EP - 12 PB - Réseau thématique AEDP du CNRS UR - https://proceedings.centre-mersenne.org/articles/10.5802/jedp.678/ DO - 10.5802/jedp.678 LA - en ID - JEDP_2023____A7_0 ER -
%0 Journal Article %A Jacek Jendrej %A Andrew Lawrie %T Dynamics of kink clusters for scalar fields in dimension $1+1$ %J Journées équations aux dérivées partielles %Z talk:7 %D 2023 %P 1-12 %I Réseau thématique AEDP du CNRS %U https://proceedings.centre-mersenne.org/articles/10.5802/jedp.678/ %R 10.5802/jedp.678 %G en %F JEDP_2023____A7_0
Jacek Jendrej; Andrew Lawrie. Dynamics of kink clusters for scalar fields in dimension $1+1$. Journées équations aux dérivées partielles (2023), Talk no. 7, 12 p. doi : 10.5802/jedp.678. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.678/
[1] Valeria Banica; Evelyne Miot Global existence and collisions for symmetric configurations of nearly parallel vortex filaments, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 29 (2012) no. 5, pp. 813-832 | DOI | Numdam | MR | Zbl
[2] Fabrice Bethuel; Giandomenico Orlandi; Didier Smets Dynamics of Multiple Degree Ginzburg-Landau Vortices, Commun. Math. Phys., Volume 272 (2007) no. 1, pp. 229-261 | DOI | MR | Zbl
[3] Gong Chen; Jacek Jendrej Kink networks for scalar fields in dimension , Nonlinear Anal., Theory Methods Appl., Volume 215 (2022), 112643, 23 pages | MR | Zbl
[4] Gong Chen; Jiaqi Liu; Bingying Lu Long-time asymptotics and stability for the sine-Gordon equation (2020) (https://arxiv.org/abs/2009.04260)
[5] Raphaël Côte; Yvan Martel; Frank Merle Construction of multi-soliton solutions for the -supercritical gKdV and NLS equations, Rev. Mat. Iberoam., Volume 27 (2011) no. 1, pp. 273-302 | DOI | MR | Zbl
[6] Raphaël Côte; Claudio Muñoz Multi-solitons for nonlinear Klein–Gordon equations, Forum Math. Sigma, Volume 2 (2014), e15, 38 pages | MR | Zbl
[7] Raphaël Côte; Hatem Zaag Construction of a Multisoliton Blowup Solution to the Semilinear Wave Equation in One Space Dimension, Commun. Pure Appl. Math., Volume 66 (2013) no. 10, pp. 1541-1581 | DOI | MR | Zbl
[8] Jean-Marc Delort; Nader Masmoudi Long-Time Dispersive Estimates for Perturbations of a Kink Solution of One-Dimensional Cubic Wave Equations, Memoirs of the European Mathematical Society, 1, EMS Press, 2022 | DOI | MR
[9] Maciej Dunajski; Nicholas Manton Reduced dynamics of Ward solitons, Nonlinearity, Volume 18 (2005) no. 4, pp. 1677-1689 | DOI | MR | Zbl
[10] Thomas Duyckaerts; Frank Merle Dynamics of Threshold Solutions for Energy-Critical Wave Equation, IMRP, Int. Math. Res. Pap., Volume 2008 (2008), rpn002, 67 pages | MR | Zbl
[11] Pierre Germain; Fabio Pusateri Quadratic Klein–Gordon equations with a potential in one dimension, Forum Math. Pi, Volume 10 (2022), e17, 172 pages | MR | Zbl
[12] Stephen Gustafson; Israel M. Sigal Effective dynamics of magnetic vortices, Adv. Math., Volume 199 (2006) no. 2, pp. 448-498 | DOI | MR | Zbl
[13] Nakao Hayashi; Pavel I. Naumkin Quadratic nonlinear Klein–Gordon equation in one dimension, J. Math. Phys., Volume 53 (2012) no. 10, 103711, 36 pages | MR | Zbl
[14] Michel Hénon Integrals of the Toda lattice, Phys. Rev. B, Volume 9 (1974) no. 4, pp. 1921-1923 | DOI | MR | Zbl
[15] Daniel B. Henry; J. Fernando Perez; Walter F. Wreszinski Stability theory for solitary-wave solutions of scalar field equations, Commun. Math. Phys., Volume 85 (1982) no. 3, pp. 351-361 | DOI | MR | Zbl
[16] Jacek Jendrej Construction of two-bubble solutions for the energy-critical NLS, Anal. PDE, Volume 10 (2017) no. 8, pp. 1923-1959 | DOI | MR | Zbl
[17] Jacek Jendrej Dynamics of strongly interacting unstable two-solitons for generalized Korteweg–de Vries equations (2018) (https://arxiv.org/abs/1802.06294)
[18] Jacek Jendrej Nonexistence of radial two-bubbles with opposite signs for the energy-critical wave equation, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 18 (2018) no. 2, pp. 735-778 | MR | Zbl
[19] Jacek Jendrej Construction of two-bubble solutions for energy-critical wave equations, Am. J. Math., Volume 141 (2019) no. 1, pp. 55-118 | DOI | MR | Zbl
[20] Jacek Jendrej; Michał Kowalczyk; Andrew Lawrie Dynamics of strongly interacting kink-antikink pairs for scalar fields on a line, Duke Math. J., Volume 171 (2022) no. 18, pp. 3643-3705 | MR | Zbl
[21] Jacek Jendrej; Andrew Lawrie Two-bubble dynamics for threshold solutions to the wave maps equation, Invent. Math., Volume 213 (2018) no. 3, pp. 1249-1325 | DOI | MR | Zbl
[22] Jacek Jendrej; Andrew Lawrie Uniqueness of two-bubble wave maps in high equivariance classes, Commun. Pure Appl. Math., Volume 76 (2022) no. 8, pp. 1608-1656 | DOI | MR
[23] Jacek Jendrej; Andrew Lawrie Dynamics of kink clusters for scalar fields in dimension 1+1 (2023) (https://arxiv.org/abs/2303.11297)
[24] Robert L. Jerrard; Didier Smets Vortex dynamics for the two-dimensional non-homogeneous Gross–Pitaevskii equation, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 14 (2015) no. 3, pp. 729-766 | MR | Zbl
[25] Robert L. Jerrard; Halil Mete Soner Dynamics of Ginzburg‐-Landau Vortices, Arch. Ration. Mech. Anal., Volume 142 (1998) no. 2, pp. 99-125 | DOI | MR | Zbl
[26] Robert L. Jerrard; Daniel Spirn Refined Jacobian Estimates and Gross–Pitaevsky Vortex Dynamics, Arch. Ration. Mech. Anal., Volume 190 (2008) no. 3, pp. 425-475 | DOI | MR | Zbl
[27] A Dynamical Perspective on the Model (Panayotis G. Kevrekidis; Jesús Cuevas-Maraver, eds.), Nonlinear Systems and Complexity, 26, Springer, 2019 | DOI | MR
[28] Michał Kowalczyk; Yvan Martel; Claudio Muñoz Kink dynamics in the model: asymptotic stability for odd perturbations in the energy space, J. Am. Math. Soc., Volume 30 (2017) no. 3, pp. 769-798 | DOI | MR | Zbl
[29] Joachim Krieger; Kenji Nakanishi; Wilhelm Schlag Center-stable manifold of the ground state in the energy space for the critical wave equation, Math. Ann., Volume 361 (2015) no. 1–2, pp. 1-50 | DOI | MR | Zbl
[30] Joachim Krieger; Pierre Raphaël; Yvan Martel Two-soliton solutions to the three-dimensional gravitational Hartree equation, Commun. Pure Appl. Math., Volume 62 (2009) no. 11, pp. 1501-1550 | DOI | MR | Zbl
[31] Jonas Lührmann; Wilhelm Schlag Asymptotic stability of the sine-Gordon kink under odd perturbations, Duke Math. J., Volume 172 (2023) no. 14, pp. 2715-2820 | MR | Zbl
[32] Ezequiel Maderna; Andrea Venturelli Globally Minimizing Parabolic Motions in the Newtonian -body Problem, Arch. Ration. Mech. Anal., Volume 194 (2009) no. 1, pp. 283-313 | DOI | MR | Zbl
[33] Nicholas Manton; Paul Sutcliffe Topological solitons, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2004 | DOI | MR
[34] Yvan Martel Asymptotic -soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations, Am. J. Math., Volume 127 (2005) no. 5, pp. 1103-1140 | DOI | MR | Zbl
[35] Yvan Martel; Pierre Raphael Strongly interacting blow up bubbles for the mass critical NLS, Ann. Sci. Éc. Norm. Supér., Volume 51 (2018) no. 3, pp. 701-737 | DOI | Zbl
[36] Frank Merle Construction of solutions with exactly blow-up points for the Schrödinger equation with critical nonlinearity, Commun. Math. Phys., Volume 129 (1990) no. 2, pp. 223-240 | DOI | MR | Zbl
[37] Frank Merle Determination of minimal blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power, Duke Math. J., Volume 69 (1993) no. 2, pp. 427-454 | MR | Zbl
[38] Frank Merle; Hatem Zaag Existence and classification of characteristic points at blowup for a semi-linear wave equation in one space dimension, Am. J. Math., Volume 134 (2012) no. 3, pp. 581-648 | DOI | MR | Zbl
[39] Frank Merle; Hatem Zaag Isolatedness of characteristic points at blowup for a 1-dimensional semilinear wave equation., Duke Math. J., Volume 161 (2012) no. 15, pp. 2837-2908 | MR | Zbl
[40] Abdon Moutinho On the collision problem of two kinks for the model with low speed (2022) (https://arxiv.org/abs/2211.09749)
[41] Kenji Nakanishi; Wilhelm Schlag Global dynamics above the ground state energy for the focusing nonlinear Klein–Gordon equation, J. Differ. Equations, Volume 250 (2011) no. 5, pp. 2299-2333 | DOI | MR | Zbl
[42] Kenji Nakanishi; Wilhelm Schlag Global Dynamics Above the Ground State for the Nonlinear Klein–Gordon Equation Without a Radial Assumption, Arch. Ration. Mech. Anal., Volume 203 (2011) no. 3, pp. 809-851 | DOI | MR | Zbl
[43] Yuriĭ N. Ovchinnikov; Israel M. Sigal The Ginzburg–Landau equation III. Vortex dynamics, Nonlinearity, Volume 11 (1998) no. 5, pp. 1277-1294 | DOI | MR | Zbl
[44] Harry Pollard Gravitational systems, J. Math. Mech., Volume 17 (1967), pp. 601-612 | Zbl
[45] Pierre Raphaël; Jeremie Szeftel Existence and uniqueness of minimal mass blow up solutions to an inhomogeneous -critical NLS, J. Am. Math. Soc., Volume 24 (2011) no. 2, pp. 471-546 | DOI | MR | Zbl
[46] Tony H. R. Skyrme A unified field theory of mesons and baryons, Nucl. Phys., Volume 31 (1962), pp. 556-569 | DOI | MR
[47] David M. A. Stuart The geodesic approximation for the Yang–Mills–Higgs equations, Commun. Math. Phys., Volume 166 (1994), pp. 149-190 | DOI | Zbl
[48] Tanmay Vachaspati Kinks and Domain Walls: An Introduction to Classical and Quantum Solitons, Cambridge University Press, 2023 | DOI | MR
[49] Nguyen Tien Vinh Strongly interacting multi-solitons with logarithmic relative distance for the gKdV equation, Nonlinearity, Volume 30 (2017) no. 12, pp. 4614-4648 | DOI | MR | Zbl
[50] Miki Wadati; Kenji Ohkuma Multiple-pole solutions of modified Korteweg–de Vries equation, J. Phys. Soc. Japan, Volume 51 (1982), pp. 2029-2035 | DOI | MR
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