## Why are solitons stable?

HTML articles powered by AMS MathViewer

- by Terence Tao PDF
- Bull. Amer. Math. Soc.
**46**(2009), 1-33 Request permission

## Abstract:

The theory of linear dispersive equations predicts that waves should spread out and disperse over time. However, it is a remarkable phenomenon, observed both in theory and practice, that once nonlinear effects are taken into account,*solitary wave*or

*soliton*solutions can be created, which can be stable enough to persist indefinitely. The construction of such solutions can be relatively straightforward, but the fact that they are

*stable*requires some significant amounts of analysis to establish, in part due to symmetries in the equation (such as translation invariance) which create degeneracy in the stability analysis. The theory is particularly difficult in the

*critical*case in which the nonlinearity is at exactly the right power to potentially allow for a self-similar blowup. In this article we survey some of the highlights of this theory, from the more classical orbital stability analysis of Weinstein and Grillakis-Shatah-Strauss, to the more recent asymptotic stability and blowup analysis of Martel-Merle and Merle-Raphael, as well as current developments in using this theory to rigorously demonstrate controlled blowup for several key equations.

## References

- Mark J. Ablowitz, David J. Kaup, Alan C. Newell, and Harvey Segur,
*The inverse scattering transform-Fourier analysis for nonlinear problems*, Studies in Appl. Math.**53**(1974), no. 4, 249–315. MR**450815**, DOI 10.1002/sapm1974534249 - T. B. Benjamin,
*The stability of solitary waves*, Proc. Roy. Soc. London Ser. A**328**(1972), 153–183. MR**338584**, DOI 10.1098/rspa.1972.0074 - H. Berestycki and P.-L. Lions,
*Existence of a ground state in nonlinear equations of the Klein-Gordon type*, Variational inequalities and complementarity problems (Proc. Internat. School, Erice, 1978) Wiley, Chichester, 1980, pp. 35–51. MR**578738** - H. Berestycki, P.-L. Lions, and L. A. Peletier,
*An ODE approach to the existence of positive solutions for semilinear problems in $\textbf {R}^{N}$*, Indiana Univ. Math. J.**30**(1981), no. 1, 141–157. MR**600039**, DOI 10.1512/iumj.1981.30.30012 - J. Bona,
*On the stability theory of solitary waves*, Proc. Roy. Soc. London Ser. A**344**(1975), no. 1638, 363–374. MR**386438**, DOI 10.1098/rspa.1975.0106 - J. L. Bona, V. A. Dougalis, O. A. Karakashian, and W. R. McKinney,
*Conservative, high-order numerical schemes for the generalized Korteweg-de Vries equation*, Philos. Trans. Roy. Soc. London Ser. A**351**(1995), no. 1695, 107–164. MR**1336983**, DOI 10.1098/rsta.1995.0027 - J. L. Bona, P. E. Souganidis, and W. A. Strauss,
*Stability and instability of solitary waves of Korteweg-de Vries type*, Proc. Roy. Soc. London Ser. A**411**(1987), no. 1841, 395–412. MR**897729** - V. S. Buslaev and G. S. Perel′man,
*On the stability of solitary waves for nonlinear Schrödinger equations*, Nonlinear evolution equations, Amer. Math. Soc. Transl. Ser. 2, vol. 164, Amer. Math. Soc., Providence, RI, 1995, pp. 75–98. MR**1334139**, DOI 10.1090/trans2/164/04 - Thierry Cazenave,
*Semilinear Schrödinger equations*, Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. MR**2002047**, DOI 10.1090/cln/010 - T. Cazenave and P.-L. Lions,
*Orbital stability of standing waves for some nonlinear Schrödinger equations*, Comm. Math. Phys.**85**(1982), no. 4, 549–561. MR**677997** - Charles V. Coffman,
*Uniqueness of the ground state solution for $\Delta u-u+u^{3}=0$ and a variational characterization of other solutions*, Arch. Rational Mech. Anal.**46**(1972), 81–95. MR**333489**, DOI 10.1007/BF00250684 - Raphaël Côte,
*Large data wave operator for the generalized Korteweg-de Vries equations*, Differential Integral Equations**19**(2006), no. 2, 163–188. MR**2194502** - Raphaël Côte,
*Construction of solutions to the subcritical gKdV equations with a given asymptotical behavior*, J. Funct. Anal.**241**(2006), no. 1, 143–211. MR**2264249**, DOI 10.1016/j.jfa.2006.04.007
cote3 R. Côte, - Scipio Cuccagna,
*A survey on asymptotic stability of ground states of nonlinear Schrödinger equations*, Dispersive nonlinear problems in mathematical physics, Quad. Mat., vol. 15, Dept. Math., Seconda Univ. Napoli, Caserta, 2004, pp. 21–57. MR**2231327** - P. Deift and E. Trubowitz,
*Inverse scattering on the line*, Comm. Pure Appl. Math.**32**(1979), no. 2, 121–251. MR**512420**, DOI 10.1002/cpa.3160320202 - Khaled El Dika,
*Stabilité asymptotique des ondes solitaires de l’équation de Benjamin-Bona-Mahony*, C. R. Math. Acad. Sci. Paris**337**(2003), no. 10, 649–652 (French, with English and French summaries). MR**2030105**, DOI 10.1016/j.crma.2003.10.003 - Khaled El Dika,
*Asymptotic stability of solitary waves for the Benjamin-Bona-Mahony equation*, Discrete Contin. Dyn. Syst.**13**(2005), no. 3, 583–622. MR**2152333**, DOI 10.3934/dcds.2005.13.583 - Alan C. Newell (ed.),
*Nonlinear wave motion*, Lectures in Applied Mathematics, Vol. 15, American Mathematical Society, Providence, R.I., 1974. MR**0336014**
ggkm C.S. Gardner, C.S. Greene, M.D. Kruskal, R.M. Miura, - B. Gidas, Wei Ming Ni, and L. Nirenberg,
*Symmetry of positive solutions of nonlinear elliptic equations in $\textbf {R}^{n}$*, Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York-London, 1981, pp. 369–402. MR**634248** - R. T. Glassey,
*On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations*, J. Math. Phys.**18**(1977), no. 9, 1794–1797. MR**460850**, DOI 10.1063/1.523491 - Manoussos Grillakis, Jalal Shatah, and Walter Strauss,
*Stability theory of solitary waves in the presence of symmetry. I*, J. Funct. Anal.**74**(1987), no. 1, 160–197. MR**901236**, DOI 10.1016/0022-1236(87)90044-9
hirota R. Hirota, - N. J. Hitchin, G. B. Segal, and R. S. Ward,
*Integrable systems*, Oxford Graduate Texts in Mathematics, vol. 4, The Clarendon Press, Oxford University Press, New York, 1999. Twistors, loop groups, and Riemann surfaces; Lectures from the Instructional Conference held at the University of Oxford, Oxford, September 1997. MR**1723384** - Justin Holmer, Jeremy Marzuola, and Maciej Zworski,
*Fast soliton scattering by delta impurities*, Comm. Math. Phys.**274**(2007), no. 1, 187–216. MR**2318852**, DOI 10.1007/s00220-007-0261-z
hz J. Holmer, M. Zworski, - Carlos E. Kenig and Frank Merle,
*Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case*, Invent. Math.**166**(2006), no. 3, 645–675. MR**2257393**, DOI 10.1007/s00222-006-0011-4 - Carlos E. Kenig, Gustavo Ponce, and Luis Vega,
*Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle*, Comm. Pure Appl. Math.**46**(1993), no. 4, 527–620. MR**1211741**, DOI 10.1002/cpa.3160460405
KTV R. Killip, T. Tao, and M. Visan, - 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
krieger2 J. Krieger, W. Schlag, - J. Krieger, W. Schlag, and D. Tataru,
*Renormalization and blow up for charge one equivariant critical wave maps*, Invent. Math.**171**(2008), no. 3, 543–615. MR**2372807**, DOI 10.1007/s00222-007-0089-3
kst J. Krieger, W. Schlag, D. Tataru, - Sergei B. Kuksin,
*Analysis of Hamiltonian PDEs*, Oxford Lecture Series in Mathematics and its Applications, vol. 19, Oxford University Press, Oxford, 2000. MR**1857574** - Man Kam Kwong,
*Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\textbf {R}^n$*, Arch. Rational Mech. Anal.**105**(1989), no. 3, 243–266. MR**969899**, DOI 10.1007/BF00251502 - Celine Laurent and Yvan Martel,
*Smoothness and exponential decay of $L^2$-compact solutions of the generalized KdV equations*, Comm. Partial Differential Equations**29**(2004), no. 1-2, 157–171. MR**2038148**, DOI 10.1081/PDE-120028848 - Peter D. Lax,
*Integrals of nonlinear equations of evolution and solitary waves*, Comm. Pure Appl. Math.**21**(1968), 467–490. MR**235310**, DOI 10.1002/cpa.3160210503 - Yvan Martel,
*Asymptotic $N$-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations*, Amer. J. Math.**127**(2005), no. 5, 1103–1140. MR**2170139** - Yvan Martel,
*Linear problems related to asymptotic stability of solitons of the generalized KdV equations*, SIAM J. Math. Anal.**38**(2006), no. 3, 759–781. MR**2262941**, DOI 10.1137/050637510 - 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,
*Blow up in finite time and dynamics of blow up solutions for the $L^2$-critical generalized KdV equation*, J. Amer. Math. Soc.**15**(2002), no. 3, 617–664. MR**1896235**, DOI 10.1090/S0894-0347-02-00392-2 - Yvan Martel and Frank Merle,
*Nonexistence of blow-up solution with minimal $L^2$-mass for the critical gKdV equation*, Duke Math. J.**115**(2002), no. 2, 385–408. MR**1944576**, DOI 10.1215/S0012-7094-02-11526-9 - Yvan Martel and Frank Merle,
*Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation*, Ann. of Math. (2)**155**(2002), no. 1, 235–280. MR**1888800**, DOI 10.2307/3062156 - 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 - Yvan Martel and Frank Merle,
*Refined asymptotics around solitons for gKdV equations*, Discrete Contin. Dyn. Syst.**20**(2008), no. 2, 177–218. MR**2358258**, DOI 10.3934/dcds.2008.20.177 - Yvan Martel and Frank Merle,
*Asymptotic stability of solitons of the gKdV equations with general nonlinearity*, Math. Ann.**341**(2008), no. 2, 391–427. MR**2385662**, DOI 10.1007/s00208-007-0194-z
martel-collision Y. Martel, F. Merle, - Yvan Martel and Frank Merle,
*Review on blow up and asymptotic dynamics for critical and subcritical gKdV equations*, Noncompact problems at the intersection of geometry, analysis, and topology, Contemp. Math., vol. 350, Amer. Math. Soc., Providence, RI, 2004, pp. 157–177. MR**2082397**, DOI 10.1090/conm/350/06344 - Yvan Martel and Frank Merle,
*Qualitative results on the generalized critical KdV equation*, Lectures on partial differential equations, New Stud. Adv. Math., vol. 2, Int. Press, Somerville, MA, 2003, pp. 175–179. MR**2055847** - Yvan Martel, Frank Merle, and Tai-Peng Tsai,
*Stability and asymptotic stability in the energy space of the sum of $N$ solitons for subcritical gKdV equations*, Comm. Math. Phys.**231**(2002), no. 2, 347–373. MR**1946336**, DOI 10.1007/s00220-002-0723-2 - Yvan Martel, Frank Merle, and Tai-Peng Tsai,
*Stability in $H^1$ of the sum of $K$ solitary waves for some nonlinear Schrödinger equations*, Duke Math. J.**133**(2006), no. 3, 405–466. MR**2228459**, DOI 10.1215/S0012-7094-06-13331-8 - F. Merle,
*Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power*, Duke Math. J.**69**(1993), no. 2, 427–454. MR**1203233**, DOI 10.1215/S0012-7094-93-06919-0 - Frank Merle,
*Existence of blow-up solutions in the energy space for the critical generalized KdV equation*, J. Amer. Math. Soc.**14**(2001), no. 3, 555–578. MR**1824989**, DOI 10.1090/S0894-0347-01-00369-1 - F. Merle and P. Raphael,
*Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation*, Geom. Funct. Anal.**13**(2003), no. 3, 591–642. MR**1995801**, DOI 10.1007/s00039-003-0424-9 - Frank Merle and Pierre Raphael,
*On universality of blow-up profile for $L^2$ critical nonlinear Schrödinger equation*, Invent. Math.**156**(2004), no. 3, 565–672. MR**2061329**, DOI 10.1007/s00222-003-0346-z - 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 - Frank Merle and Hatem Zaag,
*A Liouville theorem for vector-valued nonlinear heat equations and applications*, Math. Ann.**316**(2000), no. 1, 103–137. MR**1735081**, DOI 10.1007/s002080050006 - Robert M. Miura,
*Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation*, J. Mathematical Phys.**9**(1968), 1202–1204. MR**252825**, DOI 10.1063/1.1664700 - Robert M. Miura,
*The Korteweg-de Vries equation: a survey of results*, SIAM Rev.**18**(1976), no. 3, 412–459. MR**404890**, DOI 10.1137/1018076 - Tetsu Mizumachi,
*Asymptotic stability of solitary wave solutions to the regularized long-wave equation*, J. Differential Equations**200**(2004), no. 2, 312–341. MR**2052617**, DOI 10.1016/j.jde.2004.01.006 - V. Ju. Novokšenov,
*Asymptotic behavior as $t\rightarrow \infty$ of the solution of the Cauchy problem for a nonlinear Schrödinger equation*, Dokl. Akad. Nauk SSSR**251**(1980), no. 4, 799–802 (Russian). MR**568535** - Robert L. Pego and Michael I. Weinstein,
*Asymptotic stability of solitary waves*, Comm. Math. Phys.**164**(1994), no. 2, 305–349. MR**1289328** - Galina Perelman,
*Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations*, Comm. Partial Differential Equations**29**(2004), no. 7-8, 1051–1095. MR**2097576**, DOI 10.1081/PDE-200033754 - Igor Rodnianski, Wilhelm Schlag, and Avraham Soffer,
*Dispersive analysis of charge transfer models*, Comm. Pure Appl. Math.**58**(2005), no. 2, 149–216. MR**2094850**, DOI 10.1002/cpa.20066
rs I. Rodnianski, J. Sterbenz, - Harvey Segur and Mark J. Ablowitz,
*Asymptotic solutions and conservation laws for the nonlinear Schrödinger equation. I*, J. Mathematical Phys.**17**(1976), no. 5, 710–713. MR**450822**, DOI 10.1063/1.522967 - Herbert Spohn,
*Kinetic equations from Hamiltonian dynamics: Markovian limits*, Rev. Modern Phys.**52**(1980), no. 3, 569–615. MR**578142**, DOI 10.1103/RevModPhys.52.569 - Elias M. Stein,
*Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals*, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR**1232192** - Avy Soffer,
*Soliton dynamics and scattering*, International Congress of Mathematicians. Vol. III, Eur. Math. Soc., Zürich, 2006, pp. 459–471. MR**2275691** - Terence Tao,
*On the asymptotic behavior of large radial data for a focusing non-linear Schrödinger equation*, Dyn. Partial Differ. Equ.**1**(2004), no. 1, 1–48. MR**2091393**, DOI 10.4310/DPDE.2004.v1.n1.a1 - Terence Tao,
*A (concentration-)compact attractor for high-dimensional non-linear Schrödinger equations*, Dyn. Partial Differ. Equ.**4**(2007), no. 1, 1–53. MR**2304091**, DOI 10.4310/DPDE.2007.v4.n1.a1 - Terence Tao,
*Nonlinear dispersive equations*, CBMS Regional Conference Series in Mathematics, vol. 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. Local and global analysis. MR**2233925**, DOI 10.1090/cbms/106 - Terence Tao,
*Scattering for the quartic generalised Korteweg-de Vries equation*, J. Differential Equations**232**(2007), no. 2, 623–651. MR**2286393**, DOI 10.1016/j.jde.2006.07.019 - Terence Tao, Monica Visan, and Xiaoyi Zhang,
*Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions*, Duke Math. J.**140**(2007), no. 1, 165–202. MR**2355070**, DOI 10.1215/S0012-7094-07-14015-8 - Nikolay Tzvetkov,
*On the long time behavior of KdV type equations [after Martel-Merle]*, Astérisque**299**(2005), Exp. No. 933, viii, 219–248. Séminaire Bourbaki. Vol. 2003/2004. MR**2167208** - Michael I. Weinstein,
*Nonlinear Schrödinger equations and sharp interpolation estimates*, Comm. Math. Phys.**87**(1982/83), no. 4, 567–576. MR**691044** - Michael I. Weinstein,
*Modulational stability of ground states of nonlinear Schrödinger equations*, SIAM J. Math. Anal.**16**(1985), no. 3, 472–491. MR**783974**, DOI 10.1137/0516034 - 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
zab N. J. Zabusky, M. D. Kruskal, - V. E. Zakharov and S. V. Manakov,
*Asymptotic behavior of non-linear wave systems integrated by the inverse scattering method*, Z. Èksper. Teoret. Fiz.**71**(1976), no. 1, 203–215 (Russian, with English summary); English transl., Soviet Physics JETP**44**(1976), no. 1, 106–112. MR**0673411**

*Construction of solutions to the $L^2$-critical KdV equation with a given asymptotic behaviour*, preprint.

*Method for Solving the Korteweg-de Vries Equation*, Phys. Rev. Lett.

**19**(1967), 1095–1097.

*Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons*, Phys. Rev. Lett.

**27**(1971), 1192–1194

*Soliton interaction with slowly varying potentials*, preprint.

*The cubic nonlinear Schrödinger equation in two dimensions with radial data,*preprint math.AP/0707.3188. KVZ R. Killip, M. Visan, X. Zhang,

*The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher,*preprint. KdV D.J. Korteweg, G. de Vries,

*On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves*, Philos. Mag.

**539**(1895), 422–443.

*Non-generic blow-up solutions for the critical focusing NLS in 1-d*, preprint. krieger-stable J. Krieger, W. Schlag,

*On the focusing critical semi-linear wave equation*, preprint.

*Slow blow-up solutions for the $H^1(\mathbf {R}^3)$ critical focusing semi-linear wave equation in $\mathbf {R}^3$*, preprint.

*Description of two soliton collision for the quartic gKdV equation*, preprint. martel-collision2 Y. Martel, F. Merle,

*Stability of two soliton collision for nonintegrable gKdV equations*, preprint.

*On the Formation of Singularities in the Critical $O(3)$ Sigma-Model*, preprint. schlag W. Schlag,

*Spectral Theory and Nonlinear PDE: a Survey*, preprint.

*Interaction of ‘Solitons’ in a Collisionless Plasma and the Recurrence of Initial States.*, Phys. Rev. Lett.

**15**(1965), 240.

## Additional Information

**Terence Tao**- Affiliation: UCLA Department of Mathematics, Los Angeles, California 90095-1596
- MR Author ID: 361755
- ORCID: 0000-0002-0140-7641
- Email: tao@@math.ucla.edu
- Received by editor(s): June 20, 2008
- Published electronically: September 5, 2008
- Additional Notes: The author is supported by NSF grant CCF-0649473 and a grant from the MacArthur Foundation, and also thanks David Hansen, Frank Merle, Robert Miura, Jeff Kimmel, and Jean-Claude Saut for helpful comments and corrections.
- © Copyright 2008 American Mathematical Society
- Journal: Bull. Amer. Math. Soc.
**46**(2009), 1-33 - MSC (2000): Primary 35Q51
- DOI: https://doi.org/10.1090/S0273-0979-08-01228-7
- MathSciNet review: 2457070