Nonlinear Model Equations and Variational Principles

  • Lokenath Debnath


This chapter deals with the basic ideas and many major nonlinear model equations which arise in a wide variety of physical problems. Included are one-dimensional wave, Klein–Gordon (KG), sine–Gordon (SG), Burgers, Fisher, Korteweg–de Vries (KdV), Boussinesq, modified KdV, nonlinear Schrödinger (NLS), Benjamin–Ono (BO), Benjamin–Bona–Mahony (BBM), Ginzburg–Landau (GL), Burgers–Huxley (BH), KP, concentric KdV, Whitham, Davey–Stewartson, Toda lattice, Camassa–Holm (CH), and Degasperis–Procesi (DP) equations. This is followed by variational principles and the Euler–Lagrange equations. Also included are Plateau’s problem, Hamilton’s principle, Lagrange’s equations, Hamilton’s equations, the variational principle for nonlinear Klein–Gordon equations, and the variational principle for nonlinear water waves. Special attention is given to the Euler equation of motion, the continuity equation, the associated energy equation and energy flux, linear water wave problems and their solutions, nonlinear finite amplitude waves (the Stokes waves), gravity waves, gravity-capillary waves, and linear and nonlinear dispersion relations. Finally, the modern theory of nonlinear water waves is formulated.


Variational Principle Euler Equation Water Wave Lagrange Equation Nonlinear Partial Differential Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Ablowitz, M.J. and Haberman, R. (1975). Nonlinear evolution equations-two and three dimensions, Phys. Rev. Lett. 35, 1185–1188. MathSciNetCrossRefGoogle Scholar
  2. Ablowitz, M.J. and Ladik, J.F. (1976a). Nonlinear differential-difference equations and Fourier analysis, J. Math. Phys. 17, 1011–1018. MathSciNetMATHCrossRefGoogle Scholar
  3. Ablowitz, M.J. and Ladik, J.F. (1976b). A nonlinear difference scheme and inverse scattering, Stud. Appl. Math. 55, 213–229. MathSciNetGoogle Scholar
  4. Ablowitz, M.J. and Segur, H. (1979). On the evolution of packets of water waves, J. Fluid Mech. 92, 691–715. MathSciNetMATHCrossRefGoogle Scholar
  5. Amick, C.J., Fraenkel, L.E., and Toland, J.F. (1982). On the Stokes Conjecture for the wave of extreme form, Acta Math. Stockh. 148, 193–214. MathSciNetMATHGoogle Scholar
  6. Ammerman, A.J. and Cavalli-Sforva, L.L. (1971). Measuring the rate of spread of early farming, Man 6, 674–688. CrossRefGoogle Scholar
  7. Aoki, K. (1987). Gene-culture waves of advance, J. Math. Biol. 25, 453–464. MathSciNetMATHCrossRefGoogle Scholar
  8. Arecchi, F.T., Masserini, G.L., and Schwendimann, P. (1969). Electromagnetic propagation in a resonant medium, Rev. Nuovo Cimento 1, 181–192. CrossRefGoogle Scholar
  9. Aris, R. (1975). The Mathematical Theory of Diffusion and Reaction in Permeable Catalyst, Oxford University Press, Oxford. Google Scholar
  10. Arnold, R., Showalter, K., and Tyson, J.J. (1987). Propagation of chemical reactions in space, J. Chem. Educ. 64, 740–742. Google Scholar
  11. Asano, N., Taniuti, T., and Yajima, N. (1969). A perturbation method for a nonlinear wave modulation II, J. Math. Phys. 10, 2020–2024. CrossRefGoogle Scholar
  12. Balmforth, N.J. (1995). Solitary waves and homoclinic orbits, Annu. Rev. Fluid Mech. 27, 335–373. MathSciNetCrossRefGoogle Scholar
  13. Barone, A., Esposito, F., Magee, C.J., and Scott, A.C. (1971). Theory and applications of sine-Gordon equation, Rev. Nuovo Cimento 1(2), 227–267. CrossRefGoogle Scholar
  14. Benjamin, T.B. (1967). Instability of periodic wave trains in nonlinear dispersive systems, Proc. R. Soc. Lond. A299, 59–75. Google Scholar
  15. Benjamin, T.B. and Olver, P.J. (1982). Hamiltonian structure, symmetrics and conservation laws for water waves, J. Fluid Mech. 125, 137–185. MathSciNetMATHCrossRefGoogle Scholar
  16. Benjamin, T.B., Bona, J.L., and Mahony, J.J. (1972). Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R. Soc. A272, 47–78. MathSciNetGoogle Scholar
  17. Benney, D.J. (1966). Long waves on liquid films, J. Math. Phys. 45, 150–155. MathSciNetMATHGoogle Scholar
  18. Benney, D.J. and Roskes, G. (1969). Wave instabilities, Stud. Appl. Math. 48, 377–385. MATHGoogle Scholar
  19. Ben-Jacob, E., Brand, H., Dee, G., Kramer, L., and Langer, J.S. (1985). Pattern propagation in nonlinear dissipative systems, Physica D 14, 348–364. MathSciNetMATHCrossRefGoogle Scholar
  20. Bespalov, V.I. and Talanov, V.I. (1966). Filamentary structure of light beams in nonlinear liquids, JETP Lett. 3, 307–310. Google Scholar
  21. Born, M. and Infeld, L. (1934). Foundations of a new field theory, Proc. R. Soc. Lond. A144, 425–451. Google Scholar
  22. Burgers, J.M. (1948). A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech. 1, 171–199. MathSciNetCrossRefGoogle Scholar
  23. Calogero, F. and Degasperis, A. (1978). Solution by the spectral-transform method of a nonlinear evolution equation including as a special case the cylindrical KdV equation, Lett. Nuovo Cimento 19, 150–154. MathSciNetCrossRefGoogle Scholar
  24. Camassa, R. and Holm, D.D. (1993). An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71, 1661–1664. MathSciNetMATHCrossRefGoogle Scholar
  25. Canosa, J. (1969). Diffusion in nonlinear multiplicative media, J. Math. Phys. 10, 1862–1868. CrossRefGoogle Scholar
  26. Canosa, J. (1973). On a nonlinear diffusion equation describing population growth, IBM J. Res. Dev. 17, 307–313. MathSciNetMATHCrossRefGoogle Scholar
  27. Chen, Y. and Liu, P.L.-F. (1995a). The unified Kadomtsev–Petviashvili equation for interfacial waves, J. Fluid Mech. 288, 383–408. MathSciNetMATHCrossRefGoogle Scholar
  28. Chen, Y. and Liu, P.L.-F. (1995b). Modified Boussinesq equations and associated parabolic models for water wave propagation, J. Fluid Mech. 228, 351–381. CrossRefGoogle Scholar
  29. Chu, V.H. and Mei, C.C. (1970). On slowly varying Stokes waves, J. Fluid Mech. 41, 873–887. MathSciNetMATHCrossRefGoogle Scholar
  30. Davey, A. (1972). The propagation of a weak nonlinear wave, J. Fluid Mech. 53, 769–781. MATHCrossRefGoogle Scholar
  31. Davey, A. and Stewartson, K. (1974). On three dimensional packets of surface waves, Proc. R. Soc. Lond. A338, 101–110. MathSciNetGoogle Scholar
  32. Davis, R.E. and Acrivos, A. (1967). Solitary internal waves in deep water, J. Fluid Mech. 29, 593–607. MATHCrossRefGoogle Scholar
  33. Debnath, L. (1994). Nonlinear Water Waves, Academic Press, Boston. MATHGoogle Scholar
  34. Degasperis, A. and Procesi, M. (1999). Asymptotic Integrability, in Symmetry and Perturbation Theory (eds. A. Degasperis and G. Gaeta), World Scientific, Singapore. Google Scholar
  35. Enz, U. (1963). Discrete mass, elementary length, and a topological invariant as a consequence of relativistic invariant principles, Phys. Rev. 131, 1392–1394. MathSciNetMATHCrossRefGoogle Scholar
  36. Fife, P.C. (1979). Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics, Vol. 28, Springer, Berlin. MATHGoogle Scholar
  37. Fisher, R.A. (1936). The wave of advance of advantageous genes, Ann. Eugen. 7, 335–369. Google Scholar
  38. FitzHugh, R. (1961). Impulses and physiological states in theoretical models of nerve membrane, Biophys. J. 1, 445–466. CrossRefGoogle Scholar
  39. Freeman, N.C. (1980). Soliton interactions in two dimensions, Adv. Appl. Mech. 20, 1–37. MathSciNetMATHCrossRefGoogle Scholar
  40. Fulton, T.A. (1972). Aspects of vortices in long Josephson junctions, Bull. Am. Phys. Soc. 17, 46–49. Google Scholar
  41. Gibbon, J.D. (1985). A survey of the origins and physical importance of soliton equations, Philos. Trans. R. Soc. Lond. A315, 335–365. MathSciNetGoogle Scholar
  42. Gibbon, J.D., James, I.N. and Moroz, I.M. (1979). An example of soliton behaviour in a rotating fluid, Proc. R. Soc. Lond. A367, 185–219. MathSciNetGoogle Scholar
  43. Hasegawa, A. (1990). Optical Solitons in Fibers, 2nd edition, Springer, Berlin. Google Scholar
  44. Hasegawa, A. and Tappert, F. (1973). Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers I: Anomalous dispersion, Appl. Phys. Lett. 23, 142–144. CrossRefGoogle Scholar
  45. Hasimoto, H. and Ono, H. (1972). Nonlinear modulation of gravity waves, J. Phys. Soc. Jpn. 33, 805–811. CrossRefGoogle Scholar
  46. Hodgkin, A.L. and Huxley, A.F. (1952). A qualitative description of membrane current and its applications to conduction and excitation in nerve, J. Physiol. (Lond.) 117, 500–544. Google Scholar
  47. Ichikawa, Y.H. (1979). Topics on solitons in plasmas, Phys. Scr. 20, 296–305. MathSciNetMATHCrossRefGoogle Scholar
  48. Ichikawa, V.H., Imamura, T., and Taniuti, T. (1972). Nonlinear wave modulation in collisionless plasma, J. Phys. Soc. Jpn. 33, 189–197. CrossRefGoogle Scholar
  49. Infeld, E., Ziemkiewicz, J., and Rowlands, G. (1987). The two surface dimensional stability of periodic and solitary wavetrains on a water surface over arbitrary depth, Phys. Fluids 30, 2330–2338. MATHCrossRefGoogle Scholar
  50. Johnson, R.S. (1980). Water waves and Korteweg–de Vries equations, J. Fluid Mech. 97, 701–719. MathSciNetMATHCrossRefGoogle Scholar
  51. Johnson, R.S. (1997). A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press, Cambridge. MATHCrossRefGoogle Scholar
  52. Josephson, B.D. (1965). Supercurrents through the barriers, Adv. Phys. 14, 419–451. CrossRefGoogle Scholar
  53. Kadomtsev, B.B. and Petviashvili, V.I. (1970). On stability of solitary waves in weakly dispersive media, Dokl. Akad. Nauk SSSR 192, 753–756; Sov. Phys. Dokl. 15, 539–541. Google Scholar
  54. Kako, F. and Yajima, N. (1980). Interaction of ion-acoustic solitons in multidimensional space, J. Phys. Soc. Jpn. 49, 2063–2071. MathSciNetCrossRefGoogle Scholar
  55. Kakutani, T. and Ono, H. (1969). Weak nonlinear hydromagnetic waves in a cold collision-free plasma, J. Phys. Soc. Jpn. 26, 1305–1318. CrossRefGoogle Scholar
  56. Karpman, V.I. and Krushkal, E.M. (1969). Modulated waves in nonlinear dispersive media, Sov. Phys. JETP 28, 277–281. Google Scholar
  57. Kelley, P.L. (1965). Self-focusing of laser beams, Phys. Rev. Lett. 15, 1005–1008. CrossRefGoogle Scholar
  58. Kolmogorov, A., Petrovsky, I., and Piscunov, N. (1937). A study of the equation of diffusion with increase in the quantity of matter and its application to a biological problem, Bull. Univ. Moscow, Ser. Int. Sec. A1, 1–25. Google Scholar
  59. Korteweg, D.J. and de Vries, G. (1895). On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag. (5) 39, 422–443. Google Scholar
  60. Kynch, G.F. (1952). A theory of sedimentation, Trans. Faraday Soc. 48, 166–176. CrossRefGoogle Scholar
  61. Lamb, G.L. (1971). Analytical descriptions of ultrashort optical pulse propagation in a resonant medium, Rev. Mod. Phys. 49, 99–124. MathSciNetCrossRefGoogle Scholar
  62. Leibovich, S. (1970). Weakly nonlinear waves in rotating fluids, J. Fluids Mech. 42, 803–822. MathSciNetMATHCrossRefGoogle Scholar
  63. Lighthill, M.J. (1956). Viscosity effects in sound waves of finite amplitude, in Surveys in Mechanics (eds. G.K. Batchelor and R.M. Davies), Cambridge University Press, Cambridge, 250–351. Google Scholar
  64. Lighthill, M.J. (1967). Some special cases treated by the Whitham theory, Proc. R. Soc. Lond. A299, 38–53. Google Scholar
  65. Lighthill, M.J. (1978). Waves in Fluids, Cambridge University Press, Cambridge. MATHGoogle Scholar
  66. Lighthill, M.J. and Whitham, G.B. (1955). On kinematic waves: I. Flood movement in long rivers; II. Theory of traffic flow on long crowded roads, Proc. R. Soc. Lond. A229, 281–345. MathSciNetGoogle Scholar
  67. Luke, J.C. (1967). A variational principle for a fluid with a free surface, J. Fluid Mech. 27, 395–397. MathSciNetMATHCrossRefGoogle Scholar
  68. Madsen, P.A., Murray, R. and Sorensen, O.R. (1991). A new form of the Boussinesq equations with improved linear dispersion characteristics, Coast. Eng. 15, 371–388. CrossRefGoogle Scholar
  69. Madsen, P.A. and Sorensen, O.R. (1992). A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2. A slowly varying bathymetry, Coast. Eng. 18, 183–204. CrossRefGoogle Scholar
  70. Madsen, P.A. and Sorensen, O.R. (1993). Bound waves and triad interactions in shallow water, Ocean Eng. 20, 359–388. CrossRefGoogle Scholar
  71. Maxon, S. and Viecelli, J. (1974). Cylindrical solitons, Phys. Fluids 17, 1614–1616. CrossRefGoogle Scholar
  72. Nagumo, J.S., Arimoto, S., and Yoshizawa, S. (1962). An active pulse transmission line simulating nerve axon, Proc. Inst. Radio Eng. 50, 2061–2071. Google Scholar
  73. Nayfeh, A.H. and Saric, W.H. (1971). Nonlinear Kelvin–Helmholtz instability, J. Fluid Mech. 55, 311–327. CrossRefGoogle Scholar
  74. Nwogu, O. (1993). Alternative form of Boussinesq equations for nearshore wave propagation, J. Waterw., Port, Coastal, Ocean Eng., ASCE 119, 618–638. CrossRefGoogle Scholar
  75. Nye, J.F. (1960). The response of glaciers and ice-sheets to seasonal and climatic changes, Proc. R. Soc. Lond. A256, 559–584. MathSciNetGoogle Scholar
  76. Nye, J.F. (1963). The response of a glacier to changes in the rate of nourishment and wastage, Proc. R. Soc. Lond. A275, 87–112. Google Scholar
  77. Olver, P.J. (1984a). Hamiltonian and non-Hamiltonian models for water waves, in Trends and Applications of Pure Mathematics to Mechanics (eds. P.G. Ciarlet, and M. Roseau), Lecture Notes in Physics, Vol. 195, Springer, New York. CrossRefGoogle Scholar
  78. Olver, P.J. (1984b). Hamiltonian perturbation theory and water waves, Contemp. Math. 28, 231–249. MathSciNetMATHGoogle Scholar
  79. Ono, H. (1975). Algebraic solitary waves in stratified fluids, J. Phys. Soc. Jpn. 39, 1082–1091. CrossRefGoogle Scholar
  80. Pawlik, M. and Rowlands, G. (1975). The propagation of solitary waves in piezoelectric semiconductors, J. Phys. C8, 1189–1204. Google Scholar
  81. Peregrine, D.H. (1967). Long waves on a beach, J. Fluid Mech. 27, 815–827. MATHCrossRefGoogle Scholar
  82. Peregrine, D.H. (1983). Water waves, nonlinear Schrödinger equations and their solutions, J. Aust. Math. Soc. B25, 16–43. MathSciNetGoogle Scholar
  83. Prasad, P. and Ravindran, R. (1977). A theory of nonlinear waves in multi-dimensions: with special reference to surface water waves, J. Inst. Math. Appl. 20, 9–20. MathSciNetMATHCrossRefGoogle Scholar
  84. Ravindran, R. and Prasad, P. (1979). A mathematical analysis of nonlinear waves in a fluid filled viscoelastic tube, Acta Mech. 31, 253–280. MathSciNetMATHCrossRefGoogle Scholar
  85. Redekopp, L.G. and Weidman, P.D. (1968). Solitary Rossby waves in zonal shear flows and their interactions, J. Atmos. Sci. 35, 790–804. CrossRefGoogle Scholar
  86. Richards, P.I. (1956). Shock waves on the highway, Oper. Res. 4, 42–51. MathSciNetCrossRefGoogle Scholar
  87. Satsuma, J. (1987a). Topics in Soliton Theory and Exactly Solvable Nonlinear Equations, World Publishing Company, Singapore (eds. M.A. Ablowitz, et al.). Google Scholar
  88. Satsuma, J. (1987b). Explicit solutions of nonlinear equations with density dependent diffusion, J. Phys. Soc. Jpn. 56, 1947–1950. MathSciNetCrossRefGoogle Scholar
  89. Schiff, L.I. (1951). Nonlinear meson theory of nuclear forces, Phys. Rev. 84, 1–11. CrossRefGoogle Scholar
  90. Schimizu, K. and Ichikawa, V.H. (1972). Automodulation of ion oscillation modes in plasma, J. Philos. Soc. Jpn. 33, 789–792. CrossRefGoogle Scholar
  91. Scott, A.C. (1969). A nonlinear Klein–Gordon equation, Am. J. Phys. 37, 52–61. CrossRefGoogle Scholar
  92. Scott, A.C. (1977). Neurophysics, Wiley, New York. Google Scholar
  93. Scott, A.C., Chu, F.Y.F., and McLaughlin, D.W. (1973). The soliton—A new concept in applied science, Proc. IEEE 61, 1443–1483. MathSciNetCrossRefGoogle Scholar
  94. Skyrme, T.H.R. (1958). A nonlinear theory of strong interactions, Proc. R. Soc. Lond. A247, 260–278. Google Scholar
  95. Skyrme, T.H.R. (1961). Particle states of a quantized meson field, Proc. R. Soc. Lond. A262, 237–245. MathSciNetGoogle Scholar
  96. Sleeman, B.D. (1982). Small amplitude periodic waves for the FitzHugh–Nagumo equation, J. Math. Biol. 14, 309–325. MathSciNetMATHCrossRefGoogle Scholar
  97. Stewartson, K. and Stuart, J.T. (1971). A nonlinear instability theory for wave system in plane Poiseuille flow, J. Fluid Mech. 48, 529–545. MathSciNetMATHCrossRefGoogle Scholar
  98. Stokes, G. (1847). On the theory of oscillatory waves, Trans. Camb. Philos. Soc. 8, 197–229. Google Scholar
  99. Talanov, V.I. (1965). Self-focusing of wave beams in nonlinear media, JETP Lett. 2, 138–141. Google Scholar
  100. Taniuti, T. and Washimi, H. (1968). Self-trapping and instability of hydromagnetic waves along the magnetic field in a cold plasma, Phys. Rev. Lett. 21, 209–212. CrossRefGoogle Scholar
  101. Tappert, F. and Varma, C.M. (1970). Asymptotic theory of self-trapping of heat pulses in solids, Phys. Rev. Lett. 25, 1108–1111. MathSciNetCrossRefGoogle Scholar
  102. Toda, M. (1967a). Vibration of a chain with nonlinear interaction, J. Phys. Soc. Jpn. 22, 431–436. CrossRefGoogle Scholar
  103. Toda, M. (1967b). Wave propagation in anharmonic lattices, J. Phys. Soc. Jpn. 23, 501–506. CrossRefGoogle Scholar
  104. Toda, M. and Wadati, M. (1973). A soliton and two solitons in an exponential lattice and related equations, J. Phys. Soc. Jpn. 34, 18–25. CrossRefGoogle Scholar
  105. Toland, J.F. (1978). On the existence of a wave of greatest height and Stokes’ conjecture, Proc. R. Soc. Lond. A363, 469–485. MathSciNetGoogle Scholar
  106. Van Saarloos, W. (1989). Front propagation into unstable states: Marginal stability as a dynamical mechanism for velocity selection, Phys. Rev. A 37, 211–229. CrossRefGoogle Scholar
  107. Van Wijngaarden, L. (1968). On the equation of motion for mixtures of liquid and gas bubble, J. Fluid Mech. 33, 465–474. MATHCrossRefGoogle Scholar
  108. Wang, X.Y. (1985). Nerve propagation and wall in liquid crystals, Phys. Lett. 112A, 402–406. Google Scholar
  109. Wang, X.Y. (1986). Brochard–Leger wall in liquid crystals, Phys. Rev. A34, 5179–5182. Google Scholar
  110. Wang, X.Y., Shu, Z.S., and Lu, Y.K. (1990). Solitary wave solutions of the generalized Burgers–Huxley equation, J. Phys. Math. Gen. 23, 271–274. MATHCrossRefGoogle Scholar
  111. Ward, R.S. (1984). The Painlevé property for the self-dual gauge-field equations, Phys. Lett. 102A, 279–282. Google Scholar
  112. Ward, R.S. (1985). Integrable and solvable systems and relations among them, Philos. Trans. R. Soc. Lond. A315, 451–457. Google Scholar
  113. Ward, R.S. (1986). Multi-dimensional integrable systems, in Field Theory, Quantum Gravity, and Strings, II Proceedings, Meudon and Paris, VI, France 1985/1986 (eds. H.J. de Vega and N. Sanchez), Lect. Notes Phys., Vol. 280, Springer, Berlin, 106–110. CrossRefGoogle Scholar
  114. Washimi, H. and Taniuti, T. (1966). Propagation of ion-acoustic solitary waves of small amplitude, Phys. Rev. Lett. 17, 996–998. CrossRefGoogle Scholar
  115. Weiland, J. and Wiljelmsson, H. (1977). Coherent Nonlinear Interaction of Waves in Plasmas, Pergamon Press, Oxford. Google Scholar
  116. Weiland, J., Ichikawa, Y.H., and Wiljelmsson, H. (1978). A perturbation expansion for the NLS equation with application to the influence of nonlinear Landau damping, Phys. Scr. 17, 517–522. CrossRefGoogle Scholar
  117. Whitham, G.B. (1965a). A general approach to linear and nonlinear dispersive waves using a Lagrangian, J. Fluid Mech. 22, 273–283. MathSciNetCrossRefGoogle Scholar
  118. Whitham, G.B. (1965b). Nonlinear dispersive waves, Proc. R. Soc. Lond. A283, 238–261. MathSciNetGoogle Scholar
  119. Whitham, G.B. (1967a). Nonlinear dispersion of water waves, J. Fluid Mech. 27, 399–412. MathSciNetMATHCrossRefGoogle Scholar
  120. Whitham, G.B. (1967b). Variational methods and application to water waves, Proc. R. Soc. Lond. A299, 6–25. Google Scholar
  121. Whitham, G.B. (1974). Linear and Nonlinear Waves, Wiley, New York. MATHGoogle Scholar
  122. Yuen, H.C. and Lake, B.M. (1975). Nonlinear deep water waves: Theory and experiment, Phys. Fluids 18, 956–960. MATHCrossRefGoogle Scholar
  123. Zabusky, N.J. (1967). A synergetic approach to problems of nonlinear dispersive wave propagation and interaction, in Proc. Symp. on Nonlinear Partial Differential Equations (ed. W.F. Ames), Academic Press, Boston. Google Scholar
  124. Zakharov, V.E. (1968a). Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys. 9, 86–94. Google Scholar
  125. Zakharov, V.E. (1968b). Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys. 2, 190–194. Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Texas, Pan AmericanEdinburgUSA

Personalised recommendations