Nonlinear Science at the Dawn of the 21st Century

by Christiansen, Peter L.; Sorensen, Mads Peter; Scott, Alwyn

• ISBN13: 9783540669180
• ISBN10: 3540669183
• Format: Hardcover
• Copyright: 2000-04-01
• Publisher: Springer Verlag

Summary

Nonlinear science is by now a well established field of research at the interface of many traditional disciplines and draws on the theoretical concepts developed in physics and mathematics. The present volume gathers the contributions of leading scientists to give the state of the art in many areas strongly influenced by nonlinear research, such as superconduction, optics, lattice dynamics, biology and biomolecular dynamics. While this volume is primarily intended for researchers working in the field care, has been taken that it will also be of benefit to graduate students or nonexpert scientist wishing to familiarize themselves with the current status of research.

Table of Contents

I Nonlinear Science	1	(66)
Nonlinear Coherent Phenomena in Continuous Media	3	(44)
E.A. Kuznetsov
V.E. Zakharov
Introduction	3	(1)
Phase randomization in nonlinear media	4	(5)
Nonlinear Schrodinger equation	9	(2)
Solitons in the focusing NSLE	11	(4)
Collapses in the NLSE	15	(3)
Weak, strong and superstrong collapses	18	(3)
Anisotropic black holes	21	(5)
Structure in media with weak dispersion	26	(6)
Singularities on a fluid surface	32	(2)
Solitons and collapses in the generalized KP equation	34	(4)
Self-focusing in the boundary layer	38	(4)
References	42	(5)
Perturbation Theories for Nonlinear Waves	47	(20)
L. Ostrovsky
K. Gorshkov
Introduction	47	(2)
Modulated waves	49	(2)
Direct perturbation method	51	(1)
Perturbation theories for solitary waves	52	(8)
Direct perturbation method for solitons: quasistationary approach	52	(2)
Nonstationary approach	54	(1)
Inverse-scattering perturbation scheme	55	(2)
``Equivalence principle''	57	(1)
Example: soliton interaction in Lagrangian systems	58	(1)
Radiation from solitons	59	(1)
Asymptotic reduction of nonlinear wave equations	60	(1)
Conclusions	61	(1)
References	62	(5)
II Superconductivity and Magnetism	67	(104)
Josephson Devices	69	(18)
A. Barone
S. Pagano
Introduction	69	(1)
Elements of the Josephson effect	70	(2)
SQUIDs	72	(3)
Digital devices	75	(2)
Detectors	77	(1)
Voltage standard	78	(2)
Microwave oscillators	80	(3)
Conclusions	83	(1)
References	83	(4)
Josephson Flux-Flow Oscillators in Microwave Fields	87	(16)
M. Salerno
M. Samuelsen
Introduction	87	(1)
Flux-flow oscillators in uniform microwave fields	88	(3)
Flux-flow oscillators in non-uniform microwave fields	91	(3)
Numerical experiment	94	(4)
Conclusions	98	(1)
Appendix	98	(2)
References	100	(3)
Coupled Structures of Long Josephson Junctions	103	(18)
G. Carapella
G. Costabile
Stacks of two long Josephson junctions	103	(6)
The physical system and its model	103	(3)
Experiments on stacks of two long Josephson junctions	106	(3)
Parallel arrays of Josephson junctions	109	(4)
The physical system and its model	109	(3)
Numerical and experimental results on five-junctions parallel arrays	112	(1)
Triangular cells of long Josephson junctions	113	(5)
The model	114	(1)
Numerical and experimental results	115	(3)
References	118	(3)
Stacked Josephson Junctions	121	(16)
N.F. Pedersen
Introduction	121	(1)
Short summary of fluxon properties	121	(2)
Stacked junctions	123	(2)
Fluxon solutions, selected examples	125	(3)
The coherent 2-fluxon mode	125	(2)
The two modes of the two fluxon case	127	(1)
Stacked junction plasma oscillation solutions	128	(7)
Conclusion	135	(1)
References	135	(2)
Dynamics of Vortices in Two-Dimensional Magnets	137	(34)
F.G. Mertens
A.R. Bishop
Introduction	137	(4)
Collective variable theories at zero temperature	141	(14)
Thiele equation	141	(2)
Vortex mass	143	(3)
Hierarchy of equations of motion	146	(6)
An alternative approach: coupling to magnons	152	(3)
Effects of thermal noise on vortex dynamics	155	(8)
Equilibrium and non-equilibrium situations	155	(1)
Collective variable theory and Langevin dynamics simulations	155	(6)
Noise-induced transitions between opposite polarizations	161	(2)
Dynamics above the Kosterlitz-Thouless transition	163	(3)
Vortex-gas approach	163	(1)
Comparison with simulations and experiments	164	(1)
Vortex motion in Monte Carlo simulations	165	(1)
Conclusion	166	(1)
References	167	(4)
III Nonlinear Optics	171	(92)
Spatial Optical Solitons	173	(22)
Yu. S. Kivshar
Introduction	173	(2)
Spatial vs. temporal solitons	175	(1)
Basic equations	176	(2)
Stability of solitary waves	178	(4)
One-parameter solitary waves	179	(1)
Two-parameter solitary waves	180	(2)
Experiments on self-focusing	182	(2)
Soliton spiralling	184	(2)
Multi-hump solitons and solitonic gluons	186	(3)
Discrete spatial optical solitons	189	(2)
References	191	(4)
Nonlinear Fiber Optics	195	(18)
G.P. Agrawal
Introduction	195	(1)
Fiber characteristics	196	(2)
Single-mode fibers	196	(1)
Fiber nonlinearities	196	(1)
Group-velocity dispersion	197	(1)
Pulse propagation in fibers	198	(1)
Nonlinear Schrodinger equation	198	(1)
Modulation instability	198	(1)
Optical solitons	199	(6)
Bright solitons	200	(1)
Dark solitons	201	(1)
Loss-managed solitons	202	(1)
Dispersion-managed solitons	203	(2)
Nonlinear optical switching	205	(4)
SPM-based optical switching	205	(2)
XPM-based optical switching	207	(2)
Concluding remarks	209	(1)
References	209	(4)
Self-Focusing and Collapse of Light Beams in Nonlinear Dispersive Media	213	(16)
L. Berge
J. Juul Rasmussen
Introduction	213	(1)
General properties of self-focusing with anomalous group velocity dispersion	214	(6)
Basic properties	214	(2)
Self-similar wave collapses	216	(4)
Self-focusing with normal group velocity dispersion	220	(4)
Discussion of the general properties, outlook	224	(2)
References	226	(3)
Coherent Structures in Dissipative Nonlinear Optical Systems	229	(18)
J.V. Moloney
Introduction	229	(5)
Nonlinear waveguide channeling in air	234	(5)
Dynamic spatial replenishment of femtosecond pulses propagating in air	236	(3)
Control of optical turbulence in semiconductor lasers	239	(4)
The control	240	(3)
Summary and conclusions	243	(2)
References	245	(2)
Solitons in Optical Media with Quadratic Nonlinearity	247	(16)
B.A. Malomed
Introduction	247	(2)
The basic theoretical models	249	(4)
The solitons	253	(7)
Conclusion	260	(1)
References	261	(2)
IV Lattice Dynamics and Applications	263	(108)
Nonlinear Models for the Dynamics of Topological Defects in Solids	265	(28)
Yu.S. Kivshar
H. Benner
O.M. Braun
Introduction	265	(1)
The FK model and the SG equation	266	(3)
Physical models	269	(8)
Dislocations in solids	269	(1)
Magnetic chains	270	(1)
Josephson junctions	271	(2)
Hydrogen-bonded chains	273	(2)
Surface physics and adsorbed atomic layers	275	(1)
Models of the DNA dynamics	276	(1)
Properties of kinks	277	(4)
On-site potential of general form	277	(2)
Discreteness effects	279	(1)
Kinks in external fields	280	(1)
Compacton kinks	281	(1)
Experimental verifications	281	(4)
Josephson junctions	281	(2)
Magnetic systems	283	(2)
Concluding remarks	285	(1)
References	286	(7)
2-D Breathers and Applications	293	(14)
J.L. Marin
J.C. Eilbeck
F.M. Russell
Introduction	293	(1)
Deciphering the lines in mica	294	(2)
Numerical and analogue studies	296	(3)
Longitudinal moving breathers in 2D lattices	299	(3)
Breather collisions	302	(1)
Conclusions and further applications	303	(1)
Application to sputtering	303	(1)
Application to layered HTSC materials	303	(1)
References	304	(3)
Scale Competition in Nonlinear Schrodinger Models	307	(16)
Yu. B. Gaididei
P.L. Christiansen
S.F. Mingaleev
Introduction	307	(1)
Excitations in nonlinear Kronig-Penney models	308	(3)
Discrete NLS models with long-range dispersive interactions	311	(5)
Stabilization of nonlinear excitations by disorder	316	(3)
Summary	319	(1)
References	320	(3)
Demonstration Systems for Kink-Solitons	323	(16)
M. Remoissenet
Introduction	323	(2)
Mechanical chains with double-well potential	325	(6)
Chain with torsion and gravity	325	(4)
Chain with flexion and gravity	329	(1)
Numerical simulations	330	(1)
Experiments	331	(2)
Chain with torsion and gravity	331	(1)
Chain with flexion and gravity	332	(1)
Lattice effects	333	(1)
Conclusion	334	(1)
Appendix	335	(1)
References	336	(3)
Quantum Lattice Solitons	339	(18)
A. C. Scott
Introduction	339	(1)
Local modes in the dihalomethanes	339	(5)
Classical analysis	340	(1)
Quantum analysis	341	(3)
Comparison with experiments	344	(1)
A lattice nonlinear Schrodinger equation	344	(4)
Local modes in crystalline acetanilide	348	(6)
Conclusions	354	(1)
References	355	(2)
Noise in Molecular Systems	357	(14)
G.P. Tsironis
Introduction	357	(1)
Additive correlated ratchets	358	(5)
Current reversal	363	(1)
Synthetic motor protein motion	364	(4)
Targeted energy transfer and nonequilibrium fluctuations in bioenergetics	368	(1)
References	369	(2)
V Biomolecular Dynamics and Biology	371	(76)
Nonlinear Dynamics of DNA	373	(20)
L. V. Yakushevich
Introduction	373	(2)
General description of DNA dynamics. Classification of the internal motions	375	(1)
Mathematical modeling of DNA dynamics. Hierarchy of the models	376	(3)
Principles of modeling	376	(1)
Structural hierarchy	376	(1)
Dynamical hierarchy	377	(2)
Nonlinear mathematical models. Solved and unsolved problems	379	(2)
Ideal models	379	(1)
Nonideal models	380	(1)
Statistics of solitons in DNA	380	(1)
Nonlinear DNA models and experiment	381	(4)
Hydrogen-tritium (or hydrogen-deuterium) exchange	381	(1)
Resonant microwave absorption	382	(1)
Scattering of neutrons and light	383	(1)
Fluorescence depolarization	384	(1)
Nonlinear conception and mechanisms of DNA functioning	385	(2)
Nonlinear mechanism of conformational transitions	385	(1)
Nonlinear conformational waves and long-range effects	385	(1)
Direction of transcription process	386	(1)
References	387	(6)
From the FPU Chain to Biomolecular Dynamics	393	(16)
A.V. Zolotaryuk
A.V. Savin
P.L. Christiansen
Introduction	393	(1)
Helices in two and three dimensions	394	(3)
Equations of motion for a helix backbone	397	(1)
Small-amplitude limit	398	(2)
Three-component soliton solutions	400	(5)
3D case: solitons of longitudinal compression	402	(2)
2D case: other types of solutions	404	(1)
Conclusions	405	(2)
References	407	(2)
Mutual Dynamics of Swimming Microorganisms and Their Fluid Habitat	409	(18)
J.O. Kessler
G.D. Burnett
K.E. Remick
Introduction	409	(2)
Bioconvection (I)	411	(4)
Observations	411	(1)
Continuum theory	412	(3)
Bacteria in constraining environments (II)	415	(2)
Possibilities for computer simulation	417	(4)
Conclusion	421	(2)
Appendix I: Statistical methods	423	(1)
Appendix II	424	(1)
References	425	(2)
Nonlinearities in Biology: The Brain as an Example	427	(20)
H. Haken
Introduction	427	(1)
Some salient features of neurons	427	(2)
The noisy lighthouse model of a neural network	429	(1)
The special case of two neurons	430	(3)
Time-averages	433	(1)
The averaged neural equations	434	(6)
How to make contact with experimental data? Synergetics as a guide	440	(3)
Concluding remarks and outlook	443	(1)
References	444	(3)
Index	447