Orbital Stability: Analysis Meets Geometry

  • Stephan De BièvreEmail author
  • François Genoud
  • Simona Rota Nodari
Part of the Lecture Notes in Mathematics book series (LNM, volume 2146)


We present an introduction to the orbital stability of relative equilibria of Hamiltonian dynamical systems on (finite and infinite dimensional) Banach spaces. A convenient formulation of the theory of Hamiltonian dynamics with symmetry and the corresponding momentum maps is proposed that allows us to highlight the interplay between (symplectic) geometry and (functional) analysis in the proofs of orbital stability of relative equilibria via the so-called energy-momentum method. The theory is illustrated with examples from finite dimensional systems, as well as from Hamiltonian PDE’s, such as solitons, standing and plane waves for the nonlinear Schrödinger equation, for the wave equation, and for the Manakov system.


  1. 1.
    M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering. London Mathematical Society Lecture Note Series, vol. 149 (Cambridge University Press, Cambridge, 1991). doi:10.1017/CBO9780511623998.
  2. 2.
    R. Abraham, J.E. Marsden, Foundations of Mechanics, 2nd edn. (Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, Reading, 1978) [Revised and enlarged, With the assistance of Tudor Raţiu and Richard Cushman]Google Scholar
  3. 3.
    R. Adami, D. Noja, Stability and symmetry-breaking bifurcation for the ground states of a NLS with a δ′ interaction. Commun. Math. Phys. 318(1), 247–289 (2013). doi:10.1007/s00220-012-1597-6.
  4. 4.
    R. Adami, C. Cacciapuoti, D. Finco, D. Noja, Variational properties and orbital stability of standing waves for NLS equation on a star graph. J. Differ. Equ. 257(10), 3738–3777 (2014). doi:10.1016/j.jde.2014.07.008.
  5. 5.
    J. Angulo Pava, Nonlinear Dispersive Equations. Mathematical Surveys and Monographs, vol. 156 (American Mathematical Society, Providence, 2009). doi:10.1090/surv/156. [Existence and stability of solitary and periodic travelling wave solutions]
  6. 6.
    V.I. Arnold, Mathematical methods of classical mechanics, in Graduate Texts in Mathematics, vol. 60 (Springer, New York, 1978) [Translated from the 1974 Russian original by K. Vogtmann and A. Weinstein, Corrected reprint of the second (1989) edition]CrossRefGoogle Scholar
  7. 7.
    M.S. Baouendi, P. Ebenfelt, L.P. Rothschild, Real Submanifolds in Complex Space and Their Mappings. Princeton Mathematical Series (Princeton University Press, Princeton, 1999)zbMATHGoogle Scholar
  8. 8.
    T.B. Benjamin, The stability of solitary waves. Proc. R. Soc. Lond. Ser. A 328, 153–183 (1972)MathSciNetCrossRefGoogle Scholar
  9. 9.
    H. Berestycki, P.L. Lions, Nonlinear scalar field equations. I. Existence of a ground state. Arch. Ration. Mech. Anal. 82(4), 313–345 (1983). doi:10.1007/BF00250555.
  10. 10.
    J. Bona, On the stability theory of solitary waves. Proc. R. Soc. Lond. Ser. A 344(1638), 363–374 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    J.L. Bona, P.E. Souganidis, W.A. Strauss, Stability and instability of solitary waves of Korteweg-de Vries type. Proc. R. Soc. Lond. Ser. A 411(1841), 395–412 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations. Geom. Funct. Anal. 3(2), 107–156 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    J. Boussinesq, Essai sur la Théorie des Eaux Courantes (Imprimerie National, Paris, 1877)zbMATHGoogle Scholar
  14. 14.
    B. Buffoni, Existence and conditional energetic stability of capillary-gravity solitary water waves by minimisation. Arch. Ration. Mech. Anal. 173(1), 25–68 (2004). doi:10.1007/s00205-004-0310-0.
  15. 15.
    B. Buffoni, M.D. Groves, S.M. Sun, E. Wahlén, Existence and conditional energetic stability of three-dimensional fully localised solitary gravity-capillary water waves. J. Differ. Equ. 254(3), 1006–1096 (2013). doi:10.1016/j.jde.2012.10.007.
  16. 16.
    T. Cazenave, Stable solutions of the logarithmic Schrödinger equation. Nonlinear Anal. 7(10), 1127–1140 (1983). doi:10.1016/0362-546X(83)90022-6.
  17. 17.
    T. Cazenave, Semilinear Schrödinger Equations. Courant Lecture Notes (American Mathematical Society, Providence, 2003)zbMATHGoogle Scholar
  18. 18.
    T. Cazenave, P.L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys. 85, 549–561 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    P.R. Chernoff, J.E. Marsden, Properties of Infinite Dimensional Hamiltonian Systems. Lecture Notes in Mathematics, vol. 425 (Springer, Berlin, 1974)Google Scholar
  20. 20.
    E.A. Coddington, N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955)zbMATHGoogle Scholar
  21. 21.
    M. Colin, L. Jeanjean, M. Squassina, Stability and instability results for standing waves of quasi-linear Schrödinger equations. Nonlinearity 23(6), 1353–1385 (2010). doi:10.1088/0951-7715/23/6/006.
  22. 22.
    A. Comech, D. Pelinovsky, Purely nonlinear instability of standing waves with minimal energy. Commun. Pure Appl. Math. 56(11), 1565–1607 (2003). doi:10.1002/cpa.10104.
  23. 23.
    A. Constantin, L. Molinet, Orbital stability of solitary waves for a shallow water equation. Phys. D 157(1–2), 75–89 (2001). doi:10.1016/S0167-2789(01)00298-6.
  24. 24.
    A. Constantin, W.A. Strauss, Stability of peakons. Commun. Pure Appl. Math. 53(5), 603–610 (2000). doi:10.1002/(SICI)1097-0312(200005)53:5¡603::AID-CPA3¿3.3.CO;2-C.<603::AID-CPA3>3.3.CO;2-C
  25. 25.
    A. Constantin, W.A. Strauss, Stability properties of steady water waves with vorticity. Commun. Pure Appl. Math. 60(6), 911–950 (2007). doi:10.1002/cpa.20165.
  26. 26.
    S. Cuccagna, A survey on asymptotic stability of ground states of nonlinear Schrödinger equations, in Dispersive Nonlinear Problems in Mathematical Physics. Quad. Mat., vol. 15 (Seconda Univ. Napoli, Caserta, 2004), pp. 21–57Google Scholar
  27. 27.
    S. Cuccagna, The Hamiltonian structure of the nonlinear Schrödinger equation and the asymptotic stability of its ground states. Commun. Math. Phys. 305(2), 279–331 (2011). doi:10.1007/s00220-011-1265-2.
  28. 28.
    S. Cuccagna, D.E. Pelinovsky, The asymptotic stability of solitons in the cubic NLS equation on the line. Appl. Anal. 93(4), 791–822 (2014). doi:10.1080/00036811.2013.866227.
  29. 29.
    S. De Bièvre, S. Rota Nodari, Orbital stability of plane wave solutions of periodic nonlinear Schrödinger and Manakov equations (in preparation)Google Scholar
  30. 30.
    A. De Bouard, R. Fukuizumi, Stability of standing waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities. Ann. Henri Poincaré 6(6), 1157–1177 (2005). doi:10.1007/s00023-005-0236-6.
  31. 31.
    M. Duflo, M. Vergne, Une propriété de la représentation coadjointe d’une algèbre de Lie. C. R. Acad. Sci. Paris 268(A), 583–585 (1969)MathSciNetzbMATHGoogle Scholar
  32. 32.
    N. Duruk Mutlubaş, A. Geyer, Orbital stability of solitary waves of moderate amplitude in shallow water. J. Differ. Equ. 255(2), 254–263 (2013). doi:10.1016/j.jde.2013.04.010.
  33. 33.
    M. Ehrnström, M.D. Groves, E. Wahlén, On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type. Nonlinearity 25(10), 2903–2936 (2012). doi:10.1088/0951-7715/25/10/2903.
  34. 34.
    E. Faou, L. Gauckler, C. Lubich, Sobolev stability of plane wave solutions to the cubic nonlinear Schrödinger equation on a torus. Commun. Partial Differ. Equ. 38(7), 1123–1140 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    E. Fermi, J. Pasta, S. Ulam, Studies of nonlinear problems. Los Alamos Scientific Laboratory Report No. LA-1940 (1955)Google Scholar
  36. 36.
    G. Fibich, X.P. Wang, Stability of solitary waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities. Phys. D 175(1–2), 96–108 (2003). doi:10.1016/S0167-2789(02)00626-7.
  37. 37.
    G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques. Ann. Sci. École Norm. Sup. [2] 12, 47–88 (1883)Google Scholar
  38. 38.
    R. Fukuizumi, Stability of standing waves for nonlinear Schrödinger equations with critical power nonlinearity and potentials. Adv. Differ. Equ. 10(3), 259–276 (2005)MathSciNetzbMATHGoogle Scholar
  39. 39.
    R. Fukuizumi, M. Ohta, Stability of standing waves for nonlinear Schrödinger equations with potentials. Differ. Integr. Equ. 16(1), 111–128 (2003)MathSciNetzbMATHGoogle Scholar
  40. 40.
    T. Gallay, M. Hărăgus, Orbital stability of periodic waves for the nonlinear Schrödinger equation. J. Dyn. Differ. Equ. 19(4), 825–865 (2007)CrossRefzbMATHGoogle Scholar
  41. 41.
    T. Gallay, M. Hărăgus, Stability of small periodic waves for the nonlinear Schrödinger equation. J. Differ. Equ. 234(2), 544–581 (2007)CrossRefzbMATHGoogle Scholar
  42. 42.
    C.S. Gardner, J.M. Greene, M.D. Kruskal, R.M. Miura, Method for solving the Korteweg-de Vries equation. Phys. Rev. Lett. 19(19), 1095 (1967)Google Scholar
  43. 43.
    M. Gazeau, Analyse de modèles mathématiques pour la propagation de la lumière dans les fibres optiques en présence de biréfringence aléatoire. Ph.D. thesis, École Polytechnique (2012)Google Scholar
  44. 44.
    F. Genoud, Existence and orbital stability of standing waves for some nonlinear Schrödinger equations, perturbation of a model case. J. Differ. Equ. 246, 1921–1943 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    F. Genoud, Bifurcation and stability of travelling waves in self-focusing planar waveguides. Adv. Nonlinear Stud. 10, 357–400 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    F. Genoud, A smooth global branch of solutions for a semilinear elliptic equation on \(\mathbb{R}^{n}\). Calc. Var. Partial Differ. Equ. 38, 207–232 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    F. Genoud, Bifurcation from infinity for an asymptotically linear problem on the half-line. Nonlinear Anal. 74, 4533–4543 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    F. Genoud, Orbitally stable standing waves for the asymptotically linear one-dimensional NLS. Evol. Equ. Control Theory 2, 81–100 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    F. Genoud, C.A. Stuart, Schrödinger equations with a spatially decaying nonlinearity: existence and stability of standing waves. Discrete Contin. Dyn. Syst. 21, 137–186 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    H. Goldstein, Classical Mechanics. Addison-Wesley Series in Physics, 2nd edn. (Addison-Wesley, Reading, 1980)Google Scholar
  51. 51.
    M. Grillakis, Linearized instability for nonlinear Schrödinger and Klein-Gordon equations. Commun. Pure Appl. Math. 41(6), 747–774 (1988). doi:10.1002/cpa.3160410602.
  52. 52.
    M. Grillakis, Analysis of the linearization around a critical point of an infinite-dimensional Hamiltonian system. Commun. Pure Appl. Math. 43(3), 299–333 (1990). doi:10.1002/cpa.3160430302.
  53. 53.
    M. Grillakis, J. Shatah, W. Strauss, Stability theory of solitary waves in the presence of symmetry. I. J. Funct. Anal. 74(1), 160–197 (1987). doi:10.1016/0022-1236(87)90044-9.
  54. 54.
    M. Grillakis, J. Shatah, W. Strauss, Stability theory of solitary waves in the presence of symmetry. II. J. Funct. Anal. 94(2), 308–348 (1990). doi:10.1016/0022-1236(90)90016-E.
  55. 55.
    H. Hajaiej, C.A. Stuart, On the variational approach to the stability of standing waves for the nonlinear Schrödinger equation. Adv. Nonlinear Stud. 4, 469–501 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    L. Jeanjean, S. Le Coz, An existence and stability result for standing waves of nonlinear Schrödinger equations. Adv. Differ. Equ. 11(7), 813–840 (2006)zbMATHGoogle Scholar
  57. 57.
    C.K.R.T. Jones, Instability of standing waves for nonlinear Schrödinger-type equations. Ergodic Theory Dyn. Syst. 8 (Charles Conley Memorial Issue), 119–138 (1988). doi:10.1017/S014338570000938X.
  58. 58.
    C.K.R.T. Jones, J.V. Moloney, Instability of standing waves in nonlinear optical waveguides. Phys. Lett. A 117(4), 175–180 (1986). doi:
  59. 59.
    E. Kirr, A. Zarnescu, Asymptotic stability of ground states in 2D nonlinear Schrödinger equation including subcritical cases. J. Differ. Equ. 247(3), 710–735 (2009). doi:10.1016/j.jde.2009.04.015.
  60. 60.
    C. Klein, J.C. Saut, IST versus PDE, a comparative study (2014).
  61. 61.
    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. 39, 422–443 (1895)CrossRefzbMATHGoogle Scholar
  62. 62.
    P.D. Lax, Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 21(5), 467–490 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    S. Le Coz, Standing waves in nonlinear Schrödinger equations, in Analytical and Numerical Aspects of Partial Differential Equations (Walter de Gruyter, Berlin, 2009), pp. 151–192zbMATHGoogle Scholar
  64. 64.
    S. Le Coz, R. Fukuizumi, G. Fibich, B. Ksherim, Y. Sivan, Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential. Phys. D 237(8), 1103–1128 (2008). doi:10.1016/j.physd.2007.12.004.
  65. 65.
    M. Lemou, F. Méhats, P. Raphaël, Orbital stability of spherical galactic models. Invent. Math. 187(1), 145–194 (2012). doi:10.1007/s00222-011-0332-9.
  66. 66.
    E.M. Lerman, S.F. Singer, Stability and persistence of relative equilibria at singular values of the moment map. Nonlinearity 11(6), 1637–1649 (1998). doi:10.1088/0951-7715/11/6/012.
  67. 67.
    P. Libermann, C.M. Marle, Symplectic Geometry and Analytical Mechanics. Mathematics and Its Applications, vol. 35 (D. Reidel Publishing Co., Dordrecht, 1987). doi:10.1007/978-94-009-3807-6. [Translated from the French by Bertram Eugene Schwarzbach]
  68. 68.
    P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I. Ann. Inst. H. Poincaré Anal. Non Linéaire 1(2), 109–145 (1984).
  69. 69.
    A.M. Lyapunov, Problème Général de la Stabilité du Mouvement (Princeton University Press, Princeton, 1952) [French translation of the original manuscript published in Russian by the Mathematical Society of Kharkov in 1892]Google Scholar
  70. 70.
    M. Maeda, Stability and instability of standing waves for 1-dimensional nonlinear Schrödinger equation with multiple-power nonlinearity. Kodai Math. J. 31(2), 263–271 (2008). doi:10.2996/kmj/1214442798.
  71. 71.
    M. Maeda, Stability of bound states of Hamiltonian PDEs in the degenerate cases. J. Funct. Anal. 263(2), 511–528 (2012). doi:10.1016/j.jfa.2012.04.006.
  72. 72.
    A.I. Maimistov, Solitons in nonlinear optics. Quantum Electron. 40(9), 756–781 (2010)CrossRefGoogle Scholar
  73. 73.
    S.V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic waves. Sov. Phys. JETP 38(2), 248–253 (1974)MathSciNetGoogle Scholar
  74. 74.
    J.E. Marsden, T.S. Ratiu, Introduction to Mechanics and Symmetry. Texts in Applied Mathematics, vol. 17 (Springer, New York, 1994). doi:10.1007/978-1-4612-2682-6. [A basic exposition of classical mechanical systems]
  75. 75.
    Y. Martel, F. Merle, Asymptotic stability of solitons for subcritical generalized KdV equations. Arch. Ration. Mech. Anal. 157(3), 219–254 (2001). doi:10.1007/s002050100138.
  76. 76.
    Y. Martel, F. Merle, Asymptotic stability of solitons of the subcritical gKdV equations revisited. Nonlinearity 18(1), 55–80 (2005). doi:10.1088/0951-7715/18/1/004.
  77. 77.
    Y. Martel, F. Merle, Asymptotic stability of solitons of the gKdV equations with general nonlinearity. Math. Ann. 341(2), 391–427 (2008). doi:10.1007/s00208-007-0194-z.
  78. 78.
    Y. Martel, F. Merle, T.P. Tsai, Stability and asymptotic stability in the energy space of the sum of N solitons for subcritical gKdV equations. Commun. Math. Phys. 231(2), 347–373 (2002). doi:10.1007/s00220-002-0723-2.
  79. 79.
    Y. Martel, F. Merle, T.P. Tsai, Stability in H 1 of the sum of K solitary waves for some nonlinear Schrödinger equations. Duke Math. J. 133(3), 405–466 (2006). doi:10.1215/S0012-7094-06-13331-8.
  80. 80.
    J. Montaldi, Persistence and stability of relative equilibria. Nonlinearity 10(2), 449–466 (1997). doi:10.1088/0951-7715/10/2/009.
  81. 81.
    J. Montaldi, M. Rodríguez-Olmos, On the stability of Hamiltonian relative equilibria with non-trivial isotropy. Nonlinearity 24(10), 2777–2783 (2011). doi:10.1088/0951-7715/24/10/007.
  82. 82.
    M. Ohta, Stability and instability of standing waves for one-dimensional nonlinear Schrödinger equations with double power nonlinearity. Kodai Math. J. 18(1), 68–74 (1995). doi:10.2996/kmj/1138043354.
  83. 83.
    J.P. Ortega, T.S. Ratiu, Stability of Hamiltonian relative equilibria. Nonlinearity 12(3), 693–720 (1999). doi:10.1088/0951-7715/12/3/315.
  84. 84.
    G.W. Patrick, Relative equilibria in Hamiltonian systems: the dynamic interpretation of nonlinear stability on a reduced phase space. J. Geom. Phys. 9(2), 111–119 (1992). doi:10.1016/0393-0440(92)90015-S.
  85. 85.
    G.W. Patrick, M. Roberts, C. Wulff, Stability of Poisson equilibria and Hamiltonian relative equilibria by energy methods. Arch. Ration. Mech. Anal. 174(3), 301–344 (2004). doi:10.1007/s00205-004-0322-9.
  86. 86.
    R.L. Pego, M.I. Weinstein, Asymptotic stability of solitary waves. Commun. Math. Phys. 164(2), 305–349 (1994).
  87. 87.
    H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, Tome I (Gauthier-Villars et Fils, Paris, 1892)zbMATHGoogle Scholar
  88. 88.
    M. Roberts, T. Schmah, C. Stoica, Relative equilibria in systems with configuration space isotropy. J. Geom. Phys. 56(5), 762–779 (2006). doi:10.1016/j.geomphys.2005.04.017.
  89. 89.
    A. Shabat, V. Zakharov, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP 34, 62–69 (1972)MathSciNetGoogle Scholar
  90. 90.
    J. Shatah, Stable standing waves of nonlinear Klein-Gordon equations. Commun. Math. Phys. 91(3), 313–327 (1983).
  91. 91.
    J. Shatah, W. Strauss, Instability of nonlinear bound states. Commun. Math. Phys. 100(2), 173–190 (1985).
  92. 92.
    A. Soffer, Soliton dynamics and scattering, in International Congress of Mathematicians, vol. 3 (Eur. Math. Soc., Zürich, 2006), pp. 459–471zbMATHGoogle Scholar
  93. 93.
    A. Soffer, M.I. Weinstein, Multichannel nonlinear scattering for nonintegrable equations. Commun. Math. Phys. 133(1), 119–146 (1990).
  94. 94.
    A. Soffer, M.I. Weinstein, Multichannel nonlinear scattering for nonintegrable equations. II. The case of anisotropic potentials and data. J. Differ. Equ. 98(2), 376–390 (1992). doi:10.1016/0022-0396(92)90098-8.
  95. 95.
    J.M. Souriau, Structure of Dynamical Systems: A Symplectic View of Physics. Progress in Mathematics, vol. 149 (Springer, New York, 1997)Google Scholar
  96. 96.
    M. Spivak, A Comprehensive Introduction to Differential Geometry, vol. 1, 2nd edn. (Publish or Perish Inc., Wilmington, 1979)Google Scholar
  97. 97.
    W.A. Strauss, Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55(2), 149–162 (1977)CrossRefMathSciNetzbMATHGoogle Scholar
  98. 98.
    C.A. Stuart, An introduction to elliptic equations on \(\mathbb{R}^{n}\), in Nonlinear Functional Analysis and Applications to Differential Equations (Trieste, 1997) (World Science, River Edge, 1998), pp. 237–285Google Scholar
  99. 99.
    C.A. Stuart, Lectures on the orbital stability of standing waves and application to the nonlinear Schrödinger equation. Milan J. Math. 76, 329–399 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  100. 100.
    C. Sulem, P.L. Sulem, The Nonlinear Schrödinger Equation. Self-Focusing and Wave Collapse. (Springer, New York, 1999)Google Scholar
  101. 101.
    T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis. CBMS Regional Conf. Ser. Math. (American Mathematical Society, Providence, 2006)Google Scholar
  102. 102.
    T. Tao, Why are solitons stable? Bull. Am. Math. Soc. 46(1), 1–33 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  103. 103.
    N. Vakhitov, A.A. Kolokolov, Stationary solutions of the wave equation in a medium with nonlinearity saturation. Radiophys. Quantum Electron. 16 (1973)Google Scholar
  104. 104.
    M.I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations. Commun. Pure Appl. Math. 39(1), 51–67 (1986)CrossRefMathSciNetzbMATHGoogle Scholar
  105. 105.
    N.J. Zabusky, M.D. Kruskal, Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15(6), 240–243 (1965)CrossRefzbMATHGoogle Scholar
  106. 106.
    P.E. Zhidkov, Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory. Lecture Notes in Mathematics (Springer, Heidelberg, 2001)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Stephan De Bièvre
    • 1
    • 2
    Email author
  • François Genoud
    • 3
    • 4
  • Simona Rota Nodari
    • 5
  1. 1.Laboratoire Paul Painlevé, CNRS, UMR 8524 et UFR de MathématiquesUniversité Lille 1, Sciences et TechnologiesVilleneuve d’Ascq CedexFrance
  2. 2.Equipe-Projet MEPHYSTO, Centre de Recherche INRIA FutursParc Scientifique de la Haute BorneVilleneuve d’Ascq CedexFrance
  3. 3.Faculty of MathematicsUniversity of ViennaViennaAustria
  4. 4.Delft Institute of Applied MathematicsDelft University of TechnologyDelftThe Netherlands
  5. 5.Laboratoire Paul Painlevé, CNRS, UMR 8524 et UFR de MathématiquesUniversité Lille 1, Sciences et TechnologiesVilleneuve d’Ascq CedexFrance

Personalised recommendations