Asymptotic Methods and Nonlinear Evolution Equations

  • Lokenath Debnath


Many physical systems involving nonlinear wave propagation include the effects of dispersion, dissipation, and/or the inhomogeneous property of the medium. The governing equations are usually derived from conservation laws. In simple cases, these equations are hyperbolic. However, in general, the physical processes involved are so complex that the governing equations are very complicated, and hence, are not integrable by analytic methods. So, special attention is given to seeking mathematical methods which lead to a less complicated problem, yet retain all of the important physical features. In recent years, several asymptotic methods have been developed for the derivation of the evolution equations which describe how some dynamical variables evolve in time and space.


Dispersion Relation Multiple Scale Burger Equation Nonlinear Evolution Equation Boussinesq 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. Asano, N. (1970). Reductive perturbation method for nonlinear wave propagation in inhomogeneous media III, J. Phys. Soc. Jpn. 29, 220–224. MathSciNetCrossRefGoogle Scholar
  2. Asano, N. (1974). Modulation for nonlinear wave in dissipative or unstable media, J. Phys. Soc. Jpn. 36, 861–868. CrossRefGoogle Scholar
  3. Asano, N. and Ono, H. (1971). Nonlinear dispersive or dissipative waves, J. Phys. Soc. Jpn. 31, 1830–1836. MathSciNetCrossRefGoogle Scholar
  4. Asano, N. and Taniuti, T. (1970). Reductive perturbation method for nonlinear wave propagation in inhomogeneous media II, J. Phys. Soc. Jpn. 29, 209–214. MathSciNetCrossRefGoogle Scholar
  5. 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
  6. Benjamin, T.B. and Feir, J.E. (1967). The disintegration of wavetrains on deep water, Part 1, Theory, J. Fluid Mech. 27, 417–430. MATHCrossRefGoogle Scholar
  7. Bhatnagar, P.L. (1979). Nonlinear Waves in One-Dimensional Dispersive Systems, Oxford University Press, Oxford. MATHGoogle Scholar
  8. Bhatnagar, P.L. and Prasad, P. (1971). Study of the self-similar and steady flows near singularities, Part II, Proc. Roy Soc. London A322, 45–62. Google Scholar
  9. Cramer, M.S. and Sen, R. (1992). A general scheme for the derivation of evolution equations describing mixed nonlinearity, Wave Motion 15, 333–355. MathSciNetMATHCrossRefGoogle Scholar
  10. Davey, A. and Stewartson, K. (1974). On three dimensional packets of surface waves, Proc. R. Soc. Lond. A338, 101–110. MathSciNetGoogle Scholar
  11. Frieman, E.A. (1963). On a new method in the theory of irreversible process, J. Math. Phys. 4, 410–418. MathSciNetMATHCrossRefGoogle Scholar
  12. Gardner, C.S. and Morikawa, G.K. (1960). Similarity in the asymptotic behavior of collision-free hydromagnetic waves and water waves, Courant Inst. Math. Sci. Rep. NYO-9082, 1–30. Google Scholar
  13. Gorschkov, K.A., Ostrovsky, L.A., and Pelinovsky, E.N. (1974). Some problems of asymptotic theory of nonlinear waves, Proc. IEEE 62, 1511–1517. CrossRefGoogle Scholar
  14. Hasimoto, H. and Ono, H. (1972). Nonlinear modulation of gravity waves, J. Phys. Soc. Jpn. 33, 805–811. CrossRefGoogle Scholar
  15. Inoue, Y. and Matsumoto, Y. (1974). Nonlinear wave modulation in dispersive media, J. Phys. Soc. Jpn. 36, 1446–1455. CrossRefGoogle Scholar
  16. Jeffrey, A. and Engelbrecht, J. (1994). Nonlinear Waves in Solids, Springer, New York. MATHGoogle Scholar
  17. Jeffrey, A. and Kakutani, T. (1972). Weak nonlinear dispersive waves: A discussion centered around the KdV equation, SIAM Rev. 14, 582–643. MathSciNetMATHCrossRefGoogle Scholar
  18. Jeffrey, A. and Kawahara, T. (1982). Asymptotic Methods in Nonlinear Wave Theory, Pitman Advance Publishing Program, Boston. MATHGoogle Scholar
  19. Johnson, R.S. (1997). A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press, Cambridge. MATHCrossRefGoogle Scholar
  20. Kakutani, T. (1971a). Effect of an uneven bottom on gravity waves, J. Phys. Soc. Jpn. 30, 272–276. CrossRefGoogle Scholar
  21. Kakutani, T. (1971b). Weak nonlinear magneto-acoustic waves in an inhomogeneous plasma, J. Phys. Soc. Jpn. 31, 1246–1248. CrossRefGoogle Scholar
  22. Kakutani, T., Ono, H., Taniuti, T., and Wei, C.C. (1968). Reductive perturbation method in nonlinear wave propagation II. Application to hydromagnetic waves in cold plasma, J. Phys. Soc. Jpn. 24, 1159–1166. CrossRefGoogle Scholar
  23. Kawahara, T. (1973). The derivative-expansion method and nonlinear dispersive waves, J. Phys. Soc. Jpn. 35, 1537–1544. CrossRefGoogle Scholar
  24. Kawahara, T. (1975a). Nonlinear self-modulation of capillary-gravity waves on liquid layer, J. Phys. Soc. Jpn. 38, 265–270. CrossRefGoogle Scholar
  25. Kawahara, T. (1975b). Derivative-expansion method for nonlinear waves on a liquid layer of slowly varying depth, J. Phys. Soc. Jpn. 38, 1200–1206. CrossRefGoogle Scholar
  26. Kulikovskii, A.G. and Slobodkina, F.A. (1967). Equilibrium of arbitrary steady flows at the transonic points, Prikl. Mat. Meh. 31, 623–630. MathSciNetMATHGoogle Scholar
  27. Leibovich, S. and Seebass, A.R. (1972). Nonlinear Waves, Cornell University Press, Ithaca. Google Scholar
  28. Nayfeh, A.H. (1965a). A perturbation method for treating nonlinear oscillation problems, J. Math. Phys. 44, 368–374. MathSciNetMATHGoogle Scholar
  29. Nayfeh, A.H. (1965b). Nonlinear oscillations in a hot electron plasma, Phys. Fluids 8, 1896–1898. CrossRefGoogle Scholar
  30. Nayfeh, A.H. (1971). Third-harmonic resonance in the interaction of capillary and gravity waves, J. Fluid Mech. 48, 385–395. MathSciNetMATHCrossRefGoogle Scholar
  31. Nayfeh, A.H. (1973). Perturbation Methods, Wiley, New York. MATHGoogle Scholar
  32. Nayfeh, A.H. and Hassan, S.D. (1971). The method of multiple scales and nonlinear dispersive waves, J. Fluid Mech. 48, 463–475. MathSciNetMATHCrossRefGoogle Scholar
  33. Nozaki, K. and Taniuti, T. (1973). Propagation of solitary pulses in interactions of plasma waves, J. Phys. Soc. Jpn. 34, 796–800. CrossRefGoogle Scholar
  34. Oikawa, M. and Yajima, N. (1973). Interactions of solitary waves—A perturbation approach to nonlinear systems, J. Phys. Soc. Jpn. 34, 1093–1099. MathSciNetCrossRefGoogle Scholar
  35. Oikawa, M. and Yajima, N. (1974a). A class of exactly solvable nonlinear evolution equations, Prog. Theor. Phys. Suppl. 54, 1576–1577. MathSciNetGoogle Scholar
  36. Oikawa, M. and Yajima, M. (1974b). A perturbation approach to nonlinear systems II. Interaction of nonlinear modulated waves, J. Phys. Soc. Jpn. 37, 486–496. MathSciNetCrossRefGoogle Scholar
  37. Ostrovsky, L.A. and Pelinovsky, E.N. (1971). Averaging method for nonsinusoidal waves, Sov. Phys. Dokl. 15, 1097–1099. Google Scholar
  38. Ostrovsky, L.A. and Pelinovsky, E.N. (1972). Method of averaging and the generalized variational principle for nonsinusoidal wave, J. Appl. Math. Mech. (Prikl. Math. Mech.) 36, 63–78. Google Scholar
  39. Peregrine, D.H. (1967). Long waves on a beach, J. Fluid Mech. 27, 815–827. MATHCrossRefGoogle Scholar
  40. Prasad, P. (1973). Nonlinear wave propagation on an arbitrary steady transonic flow, J. Fluid Mech. 57, 721–737. MathSciNetMATHCrossRefGoogle Scholar
  41. Prasad, P. (1975). Approximation of perturbation equations of a quasi-linear hyperbolic system in a neighbourhood of a bicharacteristic, J. Math. Anal. Appl. 50, 470–482. MathSciNetMATHCrossRefGoogle Scholar
  42. 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
  43. Sandri, G. (1963). The foundations of nonequilibrium statistical mechanics I, II, Ann. Phys. 24, 332–379. MathSciNetCrossRefGoogle Scholar
  44. Sandri, G. (1965). A new method of expansion in mathematical physics—I, Nuovo Cimento B36, 67–93. MathSciNetGoogle Scholar
  45. Sandri, G. (1967). Uniformization of asymptotic expansions, in Nonlinear Partial Differential Equations: In A Symposium on Methods of Solutions (ed. W.F. Ames), Academic Press, New York, 259–277. Google Scholar
  46. Stuart, J.T. (1960). On the nonlinear mechanics of wave disturbances in stable and unstable parallel flows, Part 1. The basic behaviour in plane Poiseuille flow, J. Fluid Mech. 9, 353–370. MathSciNetMATHCrossRefGoogle Scholar
  47. Sturrock, P.A. (1957). Nonlinear effects in electron plasma, Proc. R. Soc. Lond. A242, 277–299. MathSciNetGoogle Scholar
  48. Su, C.S. and Gardner, C.S. (1969). The Korteweg–de Vries equation and generalizations, III. Derivation of the Korteweg–de Vries equation and Burgers’ equation, J. Math. Phys. 10, 536–539. MathSciNetMATHCrossRefGoogle Scholar
  49. Taniuti, T. (1974). Reductive perturbation method and far fields of wave equations, Prog. Theor. Phys. Suppl. 55, 1–35. CrossRefGoogle Scholar
  50. Taniuti, T. and Nishihara, K. (1983). Nonlinear Waves, Pitman Advanced Publishing Program, Boston. MATHGoogle Scholar
  51. 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
  52. Taniuti, T. and Wei, C.C. (1968). Reductive perturbation method in nonlinear wave propagation-I, J. Phys. Soc. Jpn. 24, 941–946. CrossRefGoogle Scholar
  53. Taniuti, T. and Yajima, N. (1969). Perturbation method for a nonlinear wave modulation I, J. Math. Phys. 10, 1369–1372. MathSciNetCrossRefGoogle Scholar
  54. Taniuti, T. and Yajima, N. (1973). Perturbation method for a nonlinear wave modulation III, J. Math. Phys. 14, 1389–1397. CrossRefGoogle Scholar
  55. Watson, J. (1960). On the nonlinear mechanics of wave disturbances in stable and unstable parallel flows, Part 2. The development of a solution for plane Poiseuille flow and for plane Couette flow, J. Fluid Mech. 9, 371–389. MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

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

Personalised recommendations