Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-25T17:08:41.687Z Has data issue: false hasContentIssue false
  • English
  • Français

The resonant nonlinear Schrödinger equation in cold plasma physics. Application of Bäcklund–Darboux transformations and superposition principles

Published online by Cambridge University Press:  01 April 2007

J.-H. LEE ,
O.K PASHAEV ,
C. ROGERS  and
W.K. SCHIEF
J.-H. LEE
Affiliation:
Institute of Mathematics, Academia Sinica, Taiwan (leejh@math.sinica.edu.tw)
O.K PASHAEV
Affiliation:
Department of Mathematics, Izmir Institute of Technology, Turkey (pashaev@math.sinica.edu.tw)
C. ROGERS
Affiliation:
School of Mathematics, University of New South Wales, Sydney, Australia (colinr@maths.unsw.edu.au) Australian Research Council Centre of Excellence for Mathematics and Statistics of Complex Systems (schief@maths.unsw.edu.au)
W.K. SCHIEF
Affiliation:
School of Mathematics, University of New South Wales, Sydney, Australia (colinr@maths.unsw.edu.au) Australian Research Council Centre of Excellence for Mathematics and Statistics of Complex Systems (schief@maths.unsw.edu.au)
Get access Rights & Permissions [Opens in a new window]

Abstract.

A system of nonlinear equations governing the transmission of uni-axial waves in a cold collisionless plasma subject to a transverse magnetic field is reduced to the recently proposed resonant nonlinear Schrödinger (RNLS) equation. This integrable variant of the standard nonlinear Schrödinger equation admits novel nonlinear superposition principles associated with Bäcklund–Darboux transformations. These are used here, in particular, to construct analytic descriptions of the interaction of solitonic magnetoacoustic waves propagating through the plasma.

Type
Papers
Information
Journal of Plasma Physics , Volume 73 , Issue 2 , April 2007 , pp. 257 - 272
Copyright
Copyright © Cambridge University Press 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Pashaev, O. K. and Lee, J.-H. 2002 Resonance solitons as black holes in Madelung fluid. Mod. Phys. Lett. A 17, 1601–1619.CrossRefGoogle Scholar
[2]Hasegawa, A. and Kodama, Y. 1995 Solitons in Optical Communications. Oxford: Clarendon Press.CrossRefGoogle Scholar
[3]deBroglie, L. Broglie, L. 1926 Sur le possibilité de relier les phénomènes d'interfère et de diffraction á la théorie des quanta de lumière. C. R. Acad. Sci.(Paris) 183, 447–448.Google Scholar
[4]Bohm, D. 1952 A suggested interpretation of the quantum theory in terms of ‘hidden variables’ I. Phys. Rev. 85, 166–179.CrossRefGoogle Scholar
[5]Akhmanov, A., Sukhorukov, A. P. and Khokhlov, R. V. 1968 Self-focusing and diffraction of light in a nonlinear medium. Sov. Phys. Uspékhi 93, 609–636.CrossRefGoogle Scholar
[6]Nelson, E. 1966 Derivation of the Schrödinger equation from Newtonian mechanics. Phys. Rev. 150, 1079–1085.CrossRefGoogle Scholar
[7]Salesi, G. 1996 Spin and Madelung fluid. Mod. Phys. Lett. A 11, 1815–1823.CrossRefGoogle Scholar
[8]Guerra, F. and Pusterla, M. 1982 A nonlinear Schrödinger equation and its relativistic generalization from basic principles. Lett. Nuovo Cimento 34, 351–356.CrossRefGoogle Scholar
[9]Smolin, L. 1986 Quantum fluctuations and inertia. Phys. Lett. A 113, 408–412.CrossRefGoogle Scholar
[10]Bertolami, O. 1991 Nonlinear connections to quantum mechanics from quantum gravity. Phys. Lett. A 154, 225–229.CrossRefGoogle Scholar
[11]Antonovskii, L. K., Rogers, C. and Schief, W. K. 1997 Note on a capillary model and the nonlinear Schrödinger equation. J. Phys. A: Math. Gen. 30, L555–L557.CrossRefGoogle Scholar
[12]Rogers, C. and Schief, W. K. 1999 The resonant nonlinear Schrödinger equation via an integrable capillarity model. Nuovo Cimento 114, 1409–1412.Google Scholar
[13]Karpman, V. I. 1975 Nonlinear Waves in Dispersive Media. Oxford: Pergamon.Google Scholar
[14]Akhiezer, A. I. 1975 Plasma Electrodynamics. Oxford: PergamonGoogle Scholar
[15]Adlam, J. H. and Allen, J. E. 1958 The structure of strong collision-free hydromagnetic waves. Philos. Mag. 3, 448–455.CrossRefGoogle Scholar
[16]Whitham, G. B. 1965 Nonlinear dispersive waves. Proc. R. Soc. London, Ser. A 283, 238–261.Google Scholar
[17]Gurevich, A. V. and Meshcherkin, A. P. 1984 Expanding self–similar discontinuities and shock waves in dispersive hydrodynamics. Sov. Phys.–JETP 60, 732–740.Google Scholar
[18]El, G. A., Khodorovskii, V. V. and Tyurina, A. V. 2004 Hypersonic flow past slender bodies in dispersive hydrodynamics. Phys. Lett. A 333, 334–340.CrossRefGoogle Scholar
[19]Gurevich, A. V. and Krylov, A. L. 1988 A shock wave in dispersive hydrodynamics. Sov. Phys. Dokl. 32, 73–74.Google Scholar
[20]Hirota, R. 1971 Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–1194.CrossRefGoogle Scholar
[21]Ablowitz, M. J., Kaup, D. J., Newell, A. C. and Segur, H. 1973 Nonlinear evolution equations of physical significance. Phys. Rev. Lett. 31, 125–127.CrossRefGoogle Scholar
[22]Hasimoto, H. 1972 A soliton on a vortex filament. J. Fluid. Mech. 51, 477–485.CrossRefGoogle Scholar
[23]DaRios, L. S. Rios, L. S. 1906 Sul moto d'un liquido indefinito con un filetto vorticoso. Rend. Circ. Mat. Palermo 22, 117–135.Google Scholar
[24]Ablowitz, M. J. and Segur, H. 1981 Solitons and the Inverse Scattering Transform. Philadelphia, PA: SIAM.CrossRefGoogle Scholar
[25]Rogers, C. and Schief, W. K. 2002 Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
[26]Gu, C., Hu, H. and Zhou, Z. 2005 Darboux Transformations in Integrable Systems: Theory and their Applications to Geometry. Dordrecht: Springer.CrossRefGoogle Scholar
[27]Rogers, C. and Shadwick, W. F. 1989 Bäcklund Transformations and their Applications. New York: Academic Press.Google Scholar
[28]Konopelchenko, B. G. 1979 The group structure of Bäcklund transformations. Phys. Lett. A 74, 189–192.CrossRefGoogle Scholar
[29]Konopelchenko, B. G. 1981 Transformation properties of the integrable evolution equations. Phys. Lett. B 100, 254–260.CrossRefGoogle Scholar
[30]Konopelchenko, B. G. 1982 Elementary Bäcklund transformations, nonlinear superposition principles and solutions of integrable equations. Phys. Lett. A 87, 445–448.CrossRefGoogle Scholar
[31]Calogero, F. and Degasperis, A. 1984 Elementary Bäcklund transformations, nonlinear superposition formulae and algebraic construction of solutions for the nonlinear evolution equations solvable by the Zakharov–Shabat spectral transform. Physica. D 14, 103–116.CrossRefGoogle Scholar
[32]Konopelchenko, B. G. and Rogers, C. 1992 Bäcklund and reciprocal transformations: gauge connections. In: Nonlinear Equations in the Applied Sciences (ed. Ames, W. F. and Rogers, C.). New York: Academic Press, pp. 317–362.Google Scholar
[33]Neugebauer, G. and Meinel, R. 1984 General N-soliton solution of the AKNS class on arbitrary background. Phys. Lett. A 100, 467–470.CrossRefGoogle Scholar
[34]Schief, W. K. Discrete Chebyshev nets and a universal permutability theorem. (In preparation.)Google Scholar
[35]Cieśliński, J., Gragert, P. K. H. and Sym, A. 1986 Exact solutions to localised induction-approximation equation modelling smoke-ring motion. Phys. Rev. Lett. 57, 1507–151.CrossRefGoogle ScholarPubMed
[36]Rogers, C. and Schief, W. K. 1998 Intrinsic geometry of the NLS equation and its auto-Bäcklund transformation. Stud. Appl. Math. 101, 267–287.CrossRefGoogle Scholar