## Asymptotic stability for odd perturbations of the stationary kink in the variable-speed $\phi ^4$ model

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- by Stanley Snelson PDF
- Trans. Amer. Math. Soc.
**370**(2018), 7437-7460 Request permission

## Abstract:

We consider the $\phi ^4$ model in one space dimension with propagation speeds that are small deviations from a constant function. In the constant-speed case, a stationary solution called the kink is known explicitly, and the recent work of Kowalczyk, Martel, and MuĂ±oz established the asymptotic stability of the kink with respect to odd perturbations in the natural energy space. We show that a stationary kink solution exists also for our class of nonconstant propagation speeds, and extend the asymptotic stability result by taking a perturbative approach to the method of Kowalczyk, Martel, and MuĂ±oz. This requires an understanding of the spectrum of the linearization around the variable-speed kink.## References

- V. S. Buslaev and G. S. PerelâČman,
*Scattering for the nonlinear SchrĂ¶dinger equation: states that are close to a soliton*, Algebra i Analiz**4**(1992), no.Â 6, 63â102 (Russian, with Russian summary); English transl., St. Petersburg Math. J.**4**(1993), no.Â 6, 1111â1142. MR**1199635** - V. S. Buslaev and G. S. PerelâČman,
*On the stability of solitary waves for nonlinear SchrĂ¶dinger equations*, Nonlinear evolution equations, Amer. Math. Soc. Transl. Ser. 2, vol. 164, Amer. Math. Soc., Providence, RI, 1995, pp.Â 75â98. MR**1334139**, DOI 10.1090/trans2/164/04 - Vladimir S. Buslaev and Catherine Sulem,
*On asymptotic stability of solitary waves for nonlinear SchrĂ¶dinger equations*, Ann. Inst. H. PoincarĂ© C Anal. Non LinĂ©aire**20**(2003), no.Â 3, 419â475 (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.Â 5, 2581â2614. MR**2373326**, DOI 10.1090/S0002-9947-07-04356-5 - Scipio Cuccagna,
*The Hamiltonian structure of the nonlinear SchrĂ¶dinger equation and the asymptotic stability of its ground states*, Comm. Math. Phys.**305**(2011), no.Â 2, 279â331. MR**2805462**, DOI 10.1007/s00220-011-1265-2 - Sara Cuenda, Niurka R. Quintero, and Angel SĂĄnchez,
*Sine-Gordon wobbles through BĂ€cklund transformations*, Discrete Contin. Dyn. Syst. Ser. S**4**(2011), no.Â 5, 1047â1056. MR**2754104**, DOI 10.3934/dcdss.2011.4.1047 - M. Goldberg and W. Schlag,
*Dispersive estimates for SchrĂ¶dinger operators in dimensions one and three*, Comm. Math. Phys.**251**(2004), no.Â 1, 157â178. MR**2096737**, DOI 10.1007/s00220-004-1140-5 - 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.Â 3, 351â361. MR**678151**, DOI 10.1007/BF01208719 - P. D. Hislop and I. M. Sigal,
*Introduction to spectral theory*, Applied Mathematical Sciences, vol. 113, Springer-Verlag, New York, 1996. With applications to SchrĂ¶dinger operators. MR**1361167**, DOI 10.1007/978-1-4612-0741-2 - E. Kopylova and A. I. Komech,
*On asymptotic stability of kink for relativistic Ginzburg-Landau equations*, Arch. Ration. Mech. Anal.**202**(2011), no.Â 1, 213â245. 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.Â 1, 225â252. MR**2770013**, DOI 10.1007/s00220-010-1184-7 - MichaĆKowalczyk, Yvan Martel, and Claudio MuĂ±oz,
*Kink dynamics in the $\phi ^4$ model: asymptotic stability for odd perturbations in the energy space*, J. Amer. Math. Soc.**30**(2017), no.Â 3, 769â798. MR**3630087**, DOI 10.1090/jams/870 - 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.Â 4, 339â425. 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.Â 3, 219â254. MR**1826966**, DOI 10.1007/s002050100138 - Frank Merle and Pierre Raphael,
*The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear SchrĂ¶dinger equation*, Ann. of Math. (2)**161**(2005), no.Â 1, 157â222. MR**2150386**, DOI 10.4007/annals.2005.161.157 - Arnold F. Nikiforov and Vasilii B. Uvarov,
*Special functions of mathematical physics*, BirkhĂ€user 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. SamarskiÄ. MR**922041**, DOI 10.1007/978-1-4757-1595-8 - 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** - R. Rajaraman,
*Solitons and instantons*, North-Holland Publishing Co., Amsterdam, 1982. An introduction to solitons and instantons in quantum field theory. MR**719693** - Michael Reed and Barry Simon,
*Methods of modern mathematical physics. IV. Analysis of operators*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR**0493421** - Harvey Segur,
*Wobbling kinks in $\varphi ^{4}$ and sine-Gordon theory*, J. Math. Phys.**24**(1983), no.Â 6, 1439â1443. MR**708660**, DOI 10.1063/1.525867 - Harvey Segur and Martin D. Kruskal,
*Nonexistence of small-amplitude breather solutions in $\phi ^4$ theory*, Phys. Rev. Lett.**58**(1987), no.Â 8, 747â750. MR**879720**, DOI 10.1103/PhysRevLett.58.747 - I. M. Sigal,
*Nonlinear wave and SchrĂ¶dinger equations. I. Instability of periodic and quasiperiodic solutions*, Comm. Math. Phys.**153**(1993), no.Â 2, 297â320. 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), 247â274. MR**353896**, DOI 10.2307/1970847 - A. Soffer and M. I. Weinstein,
*Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations*, Invent. Math.**136**(1999), no.Â 1, 9â74. MR**1681113**, DOI 10.1007/s002220050303 - 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 - 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 SchrĂ¶dinger wave operators on the line*, Comm. Math. Phys.**208**(1999), no.Â 2, 507â520. MR**1729096**, DOI 10.1007/s002200050767 - Ricardo Weder,
*$L^p$-$L^{\dot p}$ estimates for the SchrĂ¶dinger equation on the line and inverse scattering for the nonlinear SchrĂ¶dinger equation with a potential*, J. Funct. Anal.**170**(2000), no.Â 1, 37â68. MR**1736195**, DOI 10.1006/jfan.1999.3507 - Edward Witten,
*From superconductors and four-manifolds to weak interactions*, Bull. Amer. Math. Soc. (N.S.)**44**(2007), no.Â 3, 361â391. MR**2318156**, DOI 10.1090/S0273-0979-07-01167-6

## Additional Information

**Stanley Snelson**- Affiliation: Department of Mathematics, University of Chicago, 5734 South University Avenue, Chicago, Illinois 60637
- Address at time of publication: Department of Mathematical Sciences, Florida Institute of Technology, 150 West University Boulevard, Melbourne, Florida 32901
- MR Author ID: 966432
- Email: ssnelson@fit.edu
- Received by editor(s): April 24, 2017
- Published electronically: June 20, 2018
- Additional Notes: The author was partially supported by NSF grant DMS-1246999.
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**370**(2018), 7437-7460 - MSC (2010): Primary 35L71
- DOI: https://doi.org/10.1090/tran/7300
- MathSciNet review: 3841854