Skip to main content
SpringerLink
Log in

The Three-Wave Resonant Interaction Equations: Spectral and Numerical Methods

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

The spectral theory of the integrable partial differential equations which model the resonant interaction of three waves is considered with the purpose of numerically solving the direct spectral problem for both vanishing and non vanishing boundary values. Methods of computing both the continuum spectrum data and the discrete spectrum eigenvalues are given together with examples of such computations. The explicit spectral representation of the Manley-Rowe invariants is also displayed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ablowitz M.J., Kaup D.J., Newell A.C., Segur H.: Method for Solving the Sine–Gordon Equation. Phys. Rev. Lett. 30, 1262 (1973)

    Article  ADS  MathSciNet  Google Scholar 

  2. Gardner C.S., Greene J.M., Kruskal M.D., Miura R.M.: Method for solving the Korteweg-de Vries equation. Phys. Rev. Lett. 19, 1095 (1967)

    Article  MATH  ADS  Google Scholar 

  3. Zakharov, V.E., Shabat, A.B.: Exact theory of two-dimensional self-focusing and one- dimensional self-modulation of waves in non linear media. Zh. Exp. Teor. Fiz., 61, 118 (1971) (Russian); English transl. in Sov. Phys. JEPT, 34, 62 (1972)

  4. Ablowitz M.J., Segur H.: Solitons and the Inverse Scattering Transform. SIAM, Philadelphia (1981)

    Book  MATH  Google Scholar 

  5. Wadati M.: The exact solution of the modified Korteweg-de Vries equation. J. Phys. Soc. Japan 32, 1681 (1972)

    Article  ADS  Google Scholar 

  6. Calogero F., Degasperis A.: Solution by spectral-transform method of a non-linear evolution equation including as a special case cylindrical KDV equation. Lett. Nuovo Cim. 23, 150 (1978)

    Article  MathSciNet  Google Scholar 

  7. Mikhailov A.V.: On the integrability of two-dimensional generalization of the toda Lattice. Sov. Phys. JEPT Lett. 30, 414 (1979)

    ADS  Google Scholar 

  8. Fordy A.P., Gibbons J.: Integrable non-linear Klein–Gordon equations and toda- lattices. Commun. Math. Phys. 77, 21 (1980)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. Pohlmeyer K.: Integrable hamiltonian systems and interactions through quadratic constraints. Commun. Math. Phys. 46, 207 (1976)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  10. Lund F., Regge T.: Unified approach to strings and vortices with soliton solutions. Phys. Rev. D14, 1524 (1976)

    ADS  MathSciNet  Google Scholar 

  11. Gibbon J.D., Caudrey P.J., Bullough R.K., Eilbeck J.C.: AnN-soliton solution of a nonlinear optics equation derived by a general inverse method. Lett. Nuovo Cim. 8, 775 (1973)

    Article  Google Scholar 

  12. Lamb G.L.: Phase variation in coherent-optical-pulse propagation. Phys. Rev. Lett. 31, 196 (1973)

    Article  ADS  Google Scholar 

  13. Ablowitz M.J., Kaup D.J., Newell A.C.: Coherent pulse propagation a dispersive irreversible phenomenon. J. Math. Phys. 15, 1852 (1974)

    Article  ADS  Google Scholar 

  14. Manakov S.V.: Complete integrability and stochastization of discrete dynamical systems. Sov. Phys. JEPT 40, 269 (1975)

    ADS  MathSciNet  Google Scholar 

  15. Mikhailov A.V.: Integrability of 2-dimensional thirring model. JEPT Lett. 23, 320 (1976)

    ADS  Google Scholar 

  16. Villarroel J.: The DBAR problem and the thirring model. Stud. Appl. Math. 84, 207 (1991)

    MATH  MathSciNet  Google Scholar 

  17. Kruskal M.: The Korteweg-de Vries equation and related evolution equations. Lect. Appl. Math. 15, 61 (1974)

    MathSciNet  Google Scholar 

  18. Zakharov V.E., Manakov S.V.: Resonance interaction of wave packets. Sov. Phys. JEPT Lett. 18, 243 (1973)

    ADS  Google Scholar 

  19. Ablowitz M.J., Haberman R.: Resonantly coupled nonlinear evolution equations. J. Math. Phys. 16, 2301 (1975)

    Article  ADS  MathSciNet  Google Scholar 

  20. Kaup D.J.: The three-wave interaction—a nondispersive phenomenon. Stud. Appl. Math 55, 9–44 (1976)

    MathSciNet  Google Scholar 

  21. Ablowitz M.J., Clarkson P.A.: Solitons, nonlinear evolution equations and inverse scattering. Cambridge University Press, Cambridge (1991)

    Book  MATH  Google Scholar 

  22. Newell A.C.: Solitons in Mathematics and Physics. SIAM, Philadelphia (1985)

    Book  Google Scholar 

  23. Dodd R.K., Eilbeck J.C., Gibbon J.D., Morris H.C.: Solitons and Nonlinear Wave Equations. Academic Press, London (1982)

    MATH  Google Scholar 

  24. Calogero F., Degasperis A.: Spectral Transform and Solitons, vol. 1. North Holland Publishing, Amsterdam (1982)

    Google Scholar 

  25. Picozzi A., Haelterman M.: Spontaneous formation of symbiotic solitary waves in a backward quasi-phase-matched parametric oscillator. Opt. Lett. 23, 1808 (1998)

    Article  ADS  Google Scholar 

  26. Malkin V.M., Shvets G., Fish N.J.: Detuned Raman amplification of short laser pulses in plasma. Phys. Rev. Lett. 84, 1208 (2000)

    Article  ADS  Google Scholar 

  27. Dodin I.Y., Fish N.J.: Storing, retrieving, and processing optical information by Raman backscattering in plasmas. Phys. Rev. Lett. 88, 165001 (2002)

    Article  ADS  Google Scholar 

  28. Calogero F., Degasperis A.: Novel solution of the system describing the resonant interaction of three waves. Phys. D Nonlinear Phenom. 200(3), 242 (2005)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  29. Degasperis A., Conforti M., Baronio F., Wabnitz S.: Stable control of pulse speed in parametric three-wave solitons. Phys. Rev. Lett. 97, 093901 (2006)

    Article  ADS  Google Scholar 

  30. Conforti M., Degasperis A., Baronio F., Wabnitz S.: Inelastic scattering and interactions of three-wave parametric solitons. Phys. Rev. E 74, 065602(R) (2006)

    Article  ADS  Google Scholar 

  31. Conforti M., Baronio F., Degasperis A., Wabnitz S.: Parametric frequency conversion of short optical pulses controlled by a CW background. Opt. Express 15, 12246 (2007)

    Article  ADS  Google Scholar 

  32. Baronio F., Conforti M., Degasperis A., Wabnitz S.: Three-wave trapponic solitons for tunable high-repetition rate pulse train generation. IEEE J. Quantum Electron. 44, 542 (2008)

    Article  ADS  Google Scholar 

  33. Baronio F., Conforti M., Andreana M., Couderc V., De Angelis C., Wabnitz S., Barthelemy A., Degasperis A.: Frequency generation and solitonic decay in three-wave interactions. Opt. Express 17, 12889 (2009)

    Article  Google Scholar 

  34. Baronio F., Conforti M., De Angelis C., Degasperis A., Andreana M., Couderc V., Barthelemy A.: Velocity-locked solitary waves in quadratic media. Phys. Rev. Lett. 104, 113902 (2010)

    Article  ADS  Google Scholar 

  35. Calogero F.: Why are certain nonlinear PDEs both widely applicable and integrable?. In: Zakharov, V.E. (eds) What is Integrability?, pp. 1–62. Springer, Berlin (1990)

    Google Scholar 

  36. Kaup D.J., Reiman A., Bers A.: Space–time evolution of nonlinear three-wave interactions. I. Interaction in a homogeneous medium. Rev. Mod. Phys. 51, 277 (1979)

    ADS  MathSciNet  Google Scholar 

  37. Degasperis A., Lombardo S.: Exact solutions of the 3-wave resonant interaction equation. Phys. D Nonlinear Phenom. 214, 157 (2006)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  38. Degasperis A., Lombardo S.: Multicomponent integrable wave equations I. The Darboux–Dressing transformations. J. Phys. A: Math. Theor. 40, 961 (2007)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  39. Degasperis A., Lombardo S.: Multicomponent integrable wave equations. II. Soliton solutions. J. Phys. A: Math. Theor. 42, 385206 (2009)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CNISM and Dipartimento di Ingegneria dell’Informazione, Università di Brescia, Brescia, Italy

    Matteo Conforti, Fabio Baronio & Stefan Wabnitz

  2. Dipartimento di Fisica, Università di Roma “La Sapienza”, Rome, Italy

    Antonio Degasperis

  3. Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Rome, Italy

    Antonio Degasperis

  4. Department of Mathematics, Vrije Universiteit, Amsterdam, The Netherlands

    Sara Lombardo

  5. School of Mathematics, University of Manchester, Alan Turing Building, Manchester, UK

    Sara Lombardo

Authors
  1. Antonio Degasperis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Matteo Conforti
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fabio Baronio
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Stefan Wabnitz
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Sara Lombardo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matteo Conforti.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Degasperis, A., Conforti, M., Baronio, F. et al. The Three-Wave Resonant Interaction Equations: Spectral and Numerical Methods. Lett Math Phys 96, 367–403 (2011). https://doi.org/10.1007/s11005-010-0430-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11005-010-0430-4

Mathematics Subject Classification (2000)

Keywords

Search

Navigation