## Kink dynamics in the $\phi ^4$ model: Asymptotic stability for odd perturbations in the energy space

HTML articles powered by AMS MathViewer

- by Michał Kowalczyk, Yvan Martel and Claudio Muñoz
- J. Amer. Math. Soc.
**30**(2017), 769-798 - DOI: https://doi.org/10.1090/jams/870
- Published electronically: September 27, 2016
- PDF | Request permission

## Abstract:

We consider a classical equation known as the $\phi ^4$ model in one space dimension. The kink, defined by $H(x)=\tanh (x/{\sqrt {2}})$, is an explicit stationary solution of this model. From a result of Henry, Perez and Wreszinski it is known that the kink is orbitally stable with respect to small perturbations of the initial data in the energy space. In this paper we show asymptotic stability of the kink for odd perturbations in the energy space. The proof is based on Virial-type estimates partly inspired from previous works of Martel and Merle on asymptotic stability of solitons for the generalized Korteweg-de Vries equations. However, this approach has to be adapted to additional difficulties, pointed out by Soffer and Weinstein in the case of general Klein-Gordon equations with potential: the interactions of the so-called internal oscillation mode with the radiation, and the different rates of decay of these two components of the solution in large time.## References

- Juan A. Barceló, Alberto Ruiz, and Luis Vega,
*Some dispersive estimates for Schrödinger equations with repulsive potentials*, J. Funct. Anal.**236**(2006), no. 1, 1–24. MR**2227127**, DOI 10.1016/j.jfa.2006.03.012 - Dario Bambusi and Scipio Cuccagna,
*On dispersion of small energy solutions to the nonlinear Klein Gordon equation with a potential*, Amer. J. Math.**133**(2011), no. 5, 1421–1468. MR**2843104**, DOI 10.1353/ajm.2011.0034 - Marius Beceanu,
*A centre-stable manifold for the focussing cubic NLS in $\Bbb R^{1+3}$*, Comm. Math. Phys.**280**(2008), no. 1, 145–205. MR**2391193**, DOI 10.1007/s00220-008-0427-3 - Fabrice Bethuel, Philippe Gravejat, and Didier Smets,
*Asymptotic stability in the energy space for dark solitons of the Gross-Pitaevskii equation*, Ann. Sci. Éc. Norm. Supér. (4)**48**(2015), no. 6, 1327–1381 (English, with English and French summaries). MR**3429470**, DOI 10.24033/asens.2271 - Vladimir Buslaev and Galina Perelman,
*Scattering for the nonlinear Schrödinger equations: States close to a soliton*, St. Petersburgh Math. J.**4**(1993), no. 6, 1111–1142. - Vladimir Buslaev and Galina Perelman,
*On the stability of solitary waves for nonlinear Schrödinger equations*, Nonlinear evolution equations, 75–98, Amer. Math. Soc. Transl. Ser. 2, 164, Amer. Math. Soc., Providence, RI, 1995. - Vladimir S. Buslaev and Catherine Sulem,
*On asymptotic stability of solitary waves for nonlinear Schrödinger equations*, Ann. Inst. H. Poincaré C Anal. Non Linéaire**20**(2003), no. 3, 419–475 (English, with English and French summaries). MR**1972870**, DOI 10.1016/S0294-1449(02)00018-5 - Scipio Cuccagna,
*On asymptotic stability in 3D of kinks for the $\phi ^4$ model*, Trans. Amer. Math. Soc.**360**(2008), no. 5, 2581–2614. MR**2373326**, DOI 10.1090/S0002-9947-07-04356-5 - Scipio Cuccagna,
*The Hamiltonian structure of the nonlinear Schrödinger equation and the asymptotic stability of its ground states*, Comm. Math. Phys.**305**(2011), no. 2, 279–331. MR**2805462**, DOI 10.1007/s00220-011-1265-2 - Scipio Cuccagna,
*On asymptotic stability of moving ground states of the nonlinear Schrödinger equation*, Trans. Amer. Math. Soc.**366**(2014), no. 6, 2827–2888. MR**3180733**, DOI 10.1090/S0002-9947-2014-05770-X - S. Cuccagna and R. Jenkins,
*On asymptotic stability of $N$-solitons of the Gross-Pitaevskii equation*, preprint 2012, arXiv:1410.6887. - Scipio Cuccagna and Dmitry E. Pelinovsky,
*The asymptotic stability of solitons in the cubic NLS equation on the line*, Appl. Anal.**93**(2014), no. 4, 791–822. MR**3180019**, DOI 10.1080/00036811.2013.866227 - Sara Cuenda, Niurka R. Quintero, and Angel Sánchez,
*Sine-Gordon wobbles through Bäcklund transformations*, Discrete Contin. Dyn. Syst. Ser. S**4**(2011), no. 5, 1047–1056. MR**2754104**, DOI 10.3934/dcdss.2011.4.1047 - Jean-Marc Delort,
*Existence globale et comportement asymptotique pour l’équation de Klein-Gordon quasi linéaire à données petites en dimension 1*, Ann. Sci. École Norm. Sup. (4)**34**(2001), no. 1, 1–61 (French, with English and French summaries). MR**1833089**, DOI 10.1016/S0012-9593(00)01059-4 - Jean-Marc Delort,
*Semiclassical microlocal normal forms and global solutions of modified one-dimensional KG equations*, Ann. Inst. Fourier (Grenoble)**66**(2016), no. 4, 1451–1528 (English, with English and French summaries). MR**3494176** - Jean-Marc Delort, Daoyuan Fang, and Ruying Xue,
*Global existence of small solutions for quadratic quasilinear Klein-Gordon systems in two space dimensions*, J. Funct. Anal.**211**(2004), no. 2, 288–323. MR**2056833**, DOI 10.1016/j.jfa.2004.01.008 - Jochen Denzler,
*Nonpersistence of breather families for the perturbed sine Gordon equation*, Comm. Math. Phys.**158**(1993), no. 2, 397–430. MR**1249601** - M. Goldberg and W. Schlag,
*Dispersive estimates for Schrödinger operators in dimensions one and three*, Comm. Math. Phys.**251**(2004), no. 1, 157–178. MR**2096737**, DOI 10.1007/s00220-004-1140-5 - Philippe Gravejat and Didier Smets,
*Asymptotic stability of the black soliton for the Gross-Pitaevskii equation*, Proc. Lond. Math. Soc. (3)**111**(2015), no. 2, 305–353. MR**3384514**, DOI 10.1112/plms/pdv025 - Nakao Hayashi and Pavel I. Naumkin,
*The initial value problem for the cubic nonlinear Klein-Gordon equation*, Z. Angew. Math. Phys.**59**(2008), no. 6, 1002–1028. MR**2457221**, DOI 10.1007/s00033-007-7008-8 - Nakao Hayashi and Pavel I. Naumkin,
*Quadratic nonlinear Klein-Gordon equation in one dimension*, J. Math. Phys.**53**(2012), no. 10, 103711, 36. MR**3050628**, DOI 10.1063/1.4759156 - Daniel B. Henry, J. Fernando Perez, and Walter F. Wreszinski,
*Stability theory for solitary-wave solutions of scalar field equations*, Comm. Math. Phys.**85**(1982), no. 3, 351–361. MR**678151** - Sergiu Klainerman,
*Global existence for nonlinear wave equations*, Comm. Pure Appl. Math.**33**(1980), no. 1, 43–101. MR**544044**, DOI 10.1002/cpa.3160330104 - Sergiu Klainerman,
*Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions*, Comm. Pure Appl. Math.**38**(1985), no. 5, 631–641. MR**803252**, DOI 10.1002/cpa.3160380512 - E. Kopylova and A. I. Komech,
*On asymptotic stability of kink for relativistic Ginzburg-Landau equations*, Arch. Ration. Mech. Anal.**202**(2011), no. 1, 213–245. MR**2835867**, DOI 10.1007/s00205-011-0415-1 - E. A. Kopylova and A. I. Komech,
*On asymptotic stability of moving kink for relativistic Ginzburg-Landau equation*, Comm. Math. Phys.**302**(2011), no. 1, 225–252. MR**2770013**, DOI 10.1007/s00220-010-1184-7 - M. Kowalczyk, Y. Martel and C. Muñoz,
*Nonexistence of small odd breathers for a class of nonlinear wave equations*, preprint 2016. - J. Krieger and W. Schlag,
*Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension*, J. Amer. Math. Soc.**19**(2006), no. 4, 815–920. MR**2219305**, DOI 10.1090/S0894-0347-06-00524-8 - Harvey Segur and Martin D. Kruskal,
*Nonexistence of small-amplitude breather solutions in $\phi ^4$ theory*, Phys. Rev. Lett.**58**(1987), no. 8, 747–750. MR**879720**, DOI 10.1103/PhysRevLett.58.747 - Hans Lindblad and Avy Soffer,
*A remark on long range scattering for the nonlinear Klein-Gordon equation*, J. Hyperbolic Differ. Equ.**2**(2005), no. 1, 77–89. MR**2134954**, DOI 10.1142/S0219891605000385 - Hans Lindblad and Avy Soffer,
*A remark on asymptotic completeness for the critical nonlinear Klein-Gordon equation*, Lett. Math. Phys.**73**(2005), no. 3, 249–258. MR**2188297**, DOI 10.1007/s11005-005-0021-y - Hans Lindblad and Avy Soffer,
*Scattering for the Klein-Gordon equation with quadratic and variable coefficient cubic nonlinearities*, (2013) preprint arXiv:1307.5882. - Nicholas Manton and Paul Sutcliffe,
*Topological solitons*, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2004. MR**2068924**, DOI 10.1017/CBO9780511617034 - Yvan Martel and Frank Merle,
*A Liouville theorem for the critical generalized Korteweg-de Vries equation*, J. Math. Pures Appl. (9)**79**(2000), no. 4, 339–425. MR**1753061**, DOI 10.1016/S0021-7824(00)00159-8 - Yvan Martel and Frank Merle,
*Asymptotic stability of solitons for subcritical generalized KdV equations*, Arch. Ration. Mech. Anal.**157**(2001), no. 3, 219–254. MR**1826966**, DOI 10.1007/s002050100138 - Yvan Martel and Frank Merle,
*Asymptotic stability of solitons of the subcritical gKdV equations revisited*, Nonlinearity**18**(2005), no. 1, 55–80. MR**2109467**, DOI 10.1088/0951-7715/18/1/004 - F. Merle and L. Vega,
*$L^2$ stability of solitons for KdV equation*, Int. Math. Res. Not.**13**(2003), 735–753. MR**1949297**, DOI 10.1155/S1073792803208060 - Frank Merle and Pierre Raphael,
*The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation*, Ann. of Math. (2)**161**(2005), no. 1, 157–222. MR**2150386**, DOI 10.4007/annals.2005.161.157 - Arnold F. Nikiforov and Vasilii B. Uvarov,
*Special functions of mathematical physics*, Birkhäuser Verlag, Basel, 1988. A unified introduction with applications; Translated from the Russian and with a preface by Ralph P. Boas; With a foreword by A. A. Samarskiĭ. MR**922041**, DOI 10.1007/978-1-4757-1595-8 - Robert L. Pego and Michael I. Weinstein,
*Eigenvalues, and instabilities of solitary waves*, Philos. Trans. Roy. Soc. London Ser. A**340**(1992), no. 1656, 47–94. MR**1177566**, DOI 10.1098/rsta.1992.0055 - Michael E. Peskin and Daniel V. Schroeder,
*An introduction to quantum field theory*, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1995. Edited and with a foreword by David Pines. MR**1402248** - Michael Reed and Barry Simon,
*Methods of Modern mathematical Physics IV: Analysis of Operators*, Academic Press, New York, 1978. - I. Rodnianski, W. Schlag, and A. Soffer,
*Asymptotic stability of N-soliton states of NLS*, preprint, arXiv:math/0309114. - Harvey Segur,
*Wobbling kinks in $\varphi ^{4}$ and sine-Gordon theory*, J. Math. Phys.**24**(1983), no. 6, 1439–1443. MR**708660**, DOI 10.1063/1.525867 - Jalal Shatah,
*Normal forms and quadratic nonlinear Klein-Gordon equations*, Comm. Pure Appl. Math.**38**(1985), no. 5, 685–696. MR**803256**, DOI 10.1002/cpa.3160380516 - I. M. Sigal,
*Nonlinear wave and Schrödinger equations. I. Instability of periodic and quasiperiodic solutions*, Comm. Math. Phys.**153**(1993), no. 2, 297–320. MR**1218303** - Barry Simon,
*Resonances in $n$-body quantum systems with dilatation analytic potentials and the foundations of time-dependent perturbation theory*, Ann. of Math. (2)**97**(1973), 247–274. MR**353896**, DOI 10.2307/1970847 - A. Soffer and M. I. Weinstein,
*Time dependent resonance theory*, Geom. Funct. Anal.**8**(1998), no. 6, 1086–1128. MR**1664792**, DOI 10.1007/s000390050124 - A. Soffer and M. I. Weinstein,
*Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations*, Invent. Math.**136**(1999), no. 1, 9–74. MR**1681113**, DOI 10.1007/s002220050303 - Jacob Sterbenz,
*Dispersive decay for the 1D Klein-Gordon equation with variable coefficient nonlinearities*, Trans. Amer. Math. Soc.**368**(2016), no. 3, 2081–2113. MR**3449234**, DOI 10.1090/tran/6478 - E. C. Titchmarsh,
*Eigenfunction Expansions Associated with Second-Order Differential Equations*, Oxford, at the Clarendon Press, 1946 (German). MR**0019765** - Tanmay Vachaspati,
*Kinks and domain walls*, Cambridge University Press, New York, 2006. An introduction to classical and quantum solitons. MR**2282481**, DOI 10.1017/CBO9780511535192 - Tai-Peng Tsai and Horng-Tzer Yau,
*Asymptotic dynamics of nonlinear Schrödinger equations: resonance-dominated and dispersion-dominated solutions*, Comm. Pure Appl. Math.**55**(2002), no. 2, 153–216. MR**1865414**, DOI 10.1002/cpa.3012 - A. H. Vartanian,
*Long-time asymptotics of solutions to the Cauchy problem for the defocusing nonlinear Schrödinger equation with finite-density initial data. II. Dark solitons on continua*, Math. Phys. Anal. Geom.**5**(2002), no. 4, 319–413. MR**1942685**, DOI 10.1023/A:1021179311172 - A. Vilenkin and E. P. S. Shellard,
*Cosmic strings and other topological defects*, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1994. MR**1446491** - Ricardo Weder,
*The $W_{k,p}$-continuity of the Schrödinger wave operators on the line*, Comm. Math. Phys.**208**(1999), no. 2, 507–520. MR**1729096**, DOI 10.1007/s002200050767 - Ricardo Weder,
*$L^p$-$L^{\dot p}$ estimates for the Schrödinger equation on the line and inverse scattering for the nonlinear Schrödinger equation with a potential*, J. Funct. Anal.**170**(2000), no. 1, 37–68. MR**1736195**, DOI 10.1006/jfan.1999.3507 - Michael I. Weinstein,
*Lyapunov stability of ground states of nonlinear dispersive evolution equations*, Comm. Pure Appl. Math.**39**(1986), no. 1, 51–67. MR**820338**, DOI 10.1002/cpa.3160390103 - Edward Witten,
*From superconductors and four-manifolds to weak interactions*, Bull. Amer. Math. Soc. (N.S.)**44**(2007), no. 3, 361–391. MR**2318156**, DOI 10.1090/S0273-0979-07-01167-6

## Bibliographic Information

**Michał Kowalczyk**- Affiliation: Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile
- Email: kowalczy@dim.uchile.cl
**Yvan Martel**- Affiliation: CMLS, École polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau Cedex, France
- MR Author ID: 367956
- Email: yvan.martel@polytechnique.edu
**Claudio Muñoz**- Affiliation: CNRS and Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile
- MR Author ID: 806855
- Email: claudio.munoz@math.u-psud.fr, cmunoz@dim.uchile.cl
- Received by editor(s): July 9, 2015
- Received by editor(s) in revised form: July 18, 2016
- Published electronically: September 27, 2016
- Additional Notes: The first author was partially supported by Chilean research grants Fondecyt 1130126, Fondo Basal CMM-Chile and ERC 291214 BLOWDISOL. The author would like to thank Centre de mathématiques Laurent Schwartz at the Ecole Polytechnique and the Université Cergy-Pontoise where part of this work was done.

The second author was partially supported by ERC 291214 BLOWDISOL

The third author would like to thank the Laboratoire de Mathématiques d’Orsay where part of this work was completed. His work was partly funded by ERC 291214 BLOWDISOL, and Chilean research grants FONDECYT 1150202, Fondo Basal CMM-Chile, and Millennium Nucleus Center for Analysis of PDE NC130017 - © Copyright 2016 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**30**(2017), 769-798 - MSC (2010): Primary 35L71; Secondary 35Q51, 37K40
- DOI: https://doi.org/10.1090/jams/870
- MathSciNet review: 3630087