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Book Information:
Authors:
M. J. Ablowitz,
B. Prinari and
A. D. Trubatch
Title:
Discrete and continuous nonlinear Schrödinger systems
Additional book information:
Cambridge Univ. Press,
Cambridge,
2004,
ix+257 pp.,
ISBN 0-521-53437-2,
£38.00
Mark J. Ablowitz and Harvey Segur, Solitons and the inverse scattering transform, SIAM Studies in Applied Mathematics, vol. 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1981. MR 642018
M. J. Ablowitz, B. Prinari, and A. D. Trubatch, Discrete and continuous nonlinear Schrödinger systems, London Mathematical Society Lecture Note Series, vol. 302, Cambridge University Press, Cambridge, 2004. MR 2040621
Catherine Sulem and Pierre-Louis Sulem, The nonlinear Schrödinger equation, Applied Mathematical Sciences, vol. 139, Springer-Verlag, New York, 1999. Self-focusing and wave collapse. MR 1696311
4. A. Hasegawa, Solitons in Optical Communications, Clarendon Press (Oxford, NY, 1995).
5. B.A. Malomed, Variational methods in nonlinear fiber optics and related fields, Progress in Optics 43 (2002) 69-191.
6. V.E. Zakharov, Collapse and Self-focusing of Langmuir Waves,
Handbook of Plasma Physics, (M.N. Rosenbluth and R.Z. Sagdeev, eds.), vol. 2 (A.A. Galeev and R.N. Sudan, eds.) 81-121, Elsevier (1984).
7. V.E. Zakharov, Collapse of Langmuir waves, Sov. Phys. JETP 35 (1972) 908-914.
8. T.B. Benjamin and J.E. Feir, The disintegration of wavetrains in deep water, Part 1, J. Fluid Mech. 27 (1967) 417-430.
9. M. Onorato, A.R. Osborne, M. Serio, and S. Bertone, Freak waves in random oceanic sea states, Phys. Rev. Lett. 86 (2001) 5831-5834.
10. H. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd and J.S. Aitchinson, Discrete Spatial Optical Solitons in Waveguide Arrays, Phys. Rev. Lett. 81 (1998) 3383-3386; H. Eisenberg, Y. Silberberg, R. Morandotti and J.S. Aitchinson, Diffraction Management 85 (2000) 1863-1866.
11. J. Meier, G.I. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Nonlinear Optical Beam Interactions in Waveguide Arrays, Phys. Rev. Lett. 93 (2004) 093903.
12. F. Dalfovo, S. Giorgini, L.P. Pitaveskii and S. Stringari, Rev. Mod. Phys. 71 (1999) 463-512.
13. P.G. Kevrekidis and D.J. Frantzeskakis, Pattern forming dynamical instabilities of Bose-Einstein condensates, Mod. Phys. Lett. B 18 (2004) 173-202; P.G. Kevrekidis, R. Carretero-González, D.J. Frantzeskakis, I.G. Kevrekidis, Vortices in Bose-Einstein condensates: some recent developments, Mod. Phys. Lett. B 18 (2004) 1481-1505.
14. V.A. Brazhnyi and V.V. Konotop, Theory of nonlinear matter waves in optical lattices, Mod. Phys. Lett. B 18 (2004) 627-651.
G. L. Alfimov, P. G. Kevrekidis, V. V. Konotop, and M. Salerno, Wannier functions analysis of the nonlinear Schrödinger equation with a periodic potential, Phys. Rev. E (3) 66 (2002), no. 4, 046608, 6. MR 1935204, DOI 10.1103/PhysRevE.66.046608
Mark J. Ablowitz and Ziad H. Musslimani, Discrete spatial solitons in a diffraction-managed nonlinear waveguide array: a unified approach, Phys. D 184 (2003), no. 1-4, 276–303. Complexity and nonlinearity in physical systems (Tucson, AZ, 2001). MR 2030690, DOI 10.1016/S0167-2789(03)00226-4
17. B.P. Anderson and M.A. Kasevich, Macroscopic quantum interference from atomic tunnel arrays, Science 282 (1998) 1686-1689.
18. K.S. Strecker, G.B. Partridge, A.G. Truscott, and R.G Hulet, Formation and propagation of matter wave soliton trains, Nature 417 (2002) 150-153.
19. H. Ott, J. Fortágh, S. Kraft, A. Günther, D. Komma, and C. Zimmermann, Nonlinear dynamics of a Bose-Einstein condensate in a magnetic waveguide, Phys. Rev. Lett. 91 (2003) 040402.
20. A.S. Desyatnikov, L. Torner and Yu.S. Kivshar, Optical vortices and vortex solitons, nlin.PS/0501026.
R. S. MacKay and S. Aubry, Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators, Nonlinearity 7 (1994), no. 6, 1623–1643. MR 1304442
M. J. Ablowitz and J. F. Ladik, Nonlinear differential-difference equations, J. Mathematical Phys. 16 (1975), 598–603. MR 377223, DOI 10.1063/1.522558
Todd Kapitula and Panayotis Kevrekidis, Stability of waves in discrete systems, Nonlinearity 14 (2001), no. 3, 533–566. MR 1830906, DOI 10.1088/0951-7715/14/3/306
24. S.V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic waves, Sov. Phys. JETP 38 (1974) 248-253.
Mark J. Ablowitz, Yasuhiro Ohta, and A. David Trubatch, On discretizations of the vector nonlinear Schrödinger equation, Phys. Lett. A 253 (1999), no. 5-6, 287–304. MR 1681692, DOI 10.1016/S0375-9601(99)00048-1
26. Yu.S. Kivshar and G.P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Academic Press (San Diego, 2003).
Lev Pitaevskii and Sandro Stringari, Bose-Einstein condensation, International Series of Monographs on Physics, vol. 116, The Clarendon Press, Oxford University Press, Oxford, 2003. MR 2012737
28. C.J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases, Cambridge University Press (Cambridge, 2002).
- 1.
- M.J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM (Philadelphia, 1981). M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press (Cambridge, 1991). MR 0642018 (84a:35251), MR 1149378 (93g:35108)
- 2.
- M.J. Ablowitz, B. Prinari and A.D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems, Cambridge University Press (Cambridge, 2004). MR 2040621 (2005c:37117)
- 3.
- C. Sulem and P.L. Sulem,
The Nonlinear Schrödinger Equation, Springer-Verlag (New York, 1999). MR 1696311 (2000f:35139)
- 4.
- A. Hasegawa, Solitons in Optical Communications, Clarendon Press (Oxford, NY, 1995).
- 5.
- B.A. Malomed, Variational methods in nonlinear fiber optics and related fields, Progress in Optics 43 (2002) 69-191.
- 6.
- V.E. Zakharov, Collapse and Self-focusing of Langmuir Waves,
Handbook of Plasma Physics, (M.N. Rosenbluth and R.Z. Sagdeev, eds.), vol. 2 (A.A. Galeev and R.N. Sudan, eds.) 81-121, Elsevier (1984).
- 7.
- V.E. Zakharov, Collapse of Langmuir waves, Sov. Phys. JETP 35 (1972) 908-914.
- 8.
- T.B. Benjamin and J.E. Feir, The disintegration of wavetrains in deep water, Part 1, J. Fluid Mech. 27 (1967) 417-430.
- 9.
- M. Onorato, A.R. Osborne, M. Serio, and S. Bertone, Freak waves in random oceanic sea states, Phys. Rev. Lett. 86 (2001) 5831-5834.
- 10.
- H. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd and J.S. Aitchinson, Discrete Spatial Optical Solitons in Waveguide Arrays, Phys. Rev. Lett. 81 (1998) 3383-3386; H. Eisenberg, Y. Silberberg, R. Morandotti and J.S. Aitchinson, Diffraction Management 85 (2000) 1863-1866.
- 11.
- J. Meier, G.I. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Nonlinear Optical Beam Interactions in Waveguide Arrays, Phys. Rev. Lett. 93 (2004) 093903.
- 12.
- F. Dalfovo, S. Giorgini, L.P. Pitaveskii and S. Stringari, Rev. Mod. Phys. 71 (1999) 463-512.
- 13.
- P.G. Kevrekidis and D.J. Frantzeskakis, Pattern forming dynamical instabilities of Bose-Einstein condensates, Mod. Phys. Lett. B 18 (2004) 173-202; P.G. Kevrekidis, R. Carretero-González, D.J. Frantzeskakis, I.G. Kevrekidis, Vortices in Bose-Einstein condensates: some recent developments, Mod. Phys. Lett. B 18 (2004) 1481-1505.
- 14.
- V.A. Brazhnyi and V.V. Konotop, Theory of nonlinear matter waves in optical lattices, Mod. Phys. Lett. B 18 (2004) 627-651.
- 15.
- G. L. Alfimov, P. G. Kevrekidis, V. V. Konotop, and M. Salerno, Wannier functions analysis of the nonlinear Schrödinger equation with a periodic potential, Phys. Rev. E 66 (2002) 046608. MR 1935204 (2003j:81003)
- 16.
- M.J. Ablowitz and Z.H. Musslimani, Discrete spatial solitons in a diffraction-managed nonlinear waveguide array: a unified approach, Physica D 184 (2003) 276-303. MR 2030690 (2004k:78020)
- 17.
- B.P. Anderson and M.A. Kasevich, Macroscopic quantum interference from atomic tunnel arrays, Science 282 (1998) 1686-1689.
- 18.
- K.S. Strecker, G.B. Partridge, A.G. Truscott, and R.G Hulet, Formation and propagation of matter wave soliton trains, Nature 417 (2002) 150-153.
- 19.
- H. Ott, J. Fortágh, S. Kraft, A. Günther, D. Komma, and C. Zimmermann, Nonlinear dynamics of a Bose-Einstein condensate in a magnetic waveguide, Phys. Rev. Lett. 91 (2003) 040402.
- 20.
- A.S. Desyatnikov, L. Torner and Yu.S. Kivshar, Optical vortices and vortex solitons, nlin.PS/0501026.
- 21.
- R.S. MacKay and S. Aubry, Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators,
Nonlinearity 7 (1994) 1623-1643. MR 1304442 (96a:34085)
- 22.
- M.J. Ablowitz and J.F. Ladik, Nonlinear differential-difference equations, J. Math. Phys. 16 (1975) 598-603; M.J. Ablowitz and J.F. Ladik, Nonlinear differential-difference equations and Fourier analysis, J. Math. Phys. 17 (1976) 1011-1018. MR 0377223 (51:13396), MR 0427867 (55:897)
- 23.
- T. Kapitula and P. Kevrekidis, Stability of waves in discrete systems, Nonlinearity 14 (2001) 533-566. MR 1830906 (2002f:37116)
- 24.
- S.V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic waves, Sov. Phys. JETP 38 (1974) 248-253.
- 25.
- M.J. Ablowitz, Y. Ohta and A.D. Trubatch, Phys. Lett. A 253 (1999) 287-304. MR 1681692 (99m:35215)
- 26.
- Yu.S. Kivshar and G.P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Academic Press (San Diego, 2003).
- 27.
- L.P. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Oxford University Press (Oxford, 2003). MR 2012737 (2004i:82038)
- 28.
- C.J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases, Cambridge University Press (Cambridge, 2002).
Review Information:
Reviewer:
P. G. Kevrekidis
Affiliation:
University of Massachusetts
Email:
kevrekid@math.umass.edu
Journal:
Bull. Amer. Math. Soc.
43 (2006), 127-132
Published electronically:
July 8, 2005
Review copyright:
© Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
