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| List of Figures | |
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| The Birth of a Paradigm | |
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| From the Great Wave to the Great War | |
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| Hydrodynamics | |
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| Nonlinear diffusion | |
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| Backlund transformation theory | |
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| A theory of matter | |
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| Between the wars | |
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| Nonlinear research from 1945 to 1985 | |
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| Nerve studies | |
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| Autocatalytic chemical reactions | |
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| Solitons | |
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| Local modes in molecules and molecular crystals | |
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| Elementary particle research | |
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| Recent developments | |
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| References | |
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| Linear Wave Theory | |
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| Dispersionless linear equations | |
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| Dispersive linear equations | |
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| The linear diffusion equation | |
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| Driven systems | |
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| Green's method | |
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| Fredholm's theorem | |
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| Stability | |
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| General definitions | |
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| Linear stability | |
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| Signaling problems | |
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| Scattering theory | |
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| Solutions of Schrodinger's equation | |
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| Gel'fand-Levitan theory | |
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| A reflectionless potential | |
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| Problems | |
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| References | |
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| The Classical Soliton Equations | |
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| The Korteweg-de Vries (KdV) equation | |
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| Long water waves | |
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| Solitary wave solutions | |
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| Periodic solutions | |
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| A Backlund transformation for KdV | |
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| N-soliton formulas | |
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| The sine-Gordon (SG) equation | |
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| Long Josephson junctions | |
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| Solitary waves | |
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| Periodic waves | |
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| Nonlinear standing waves | |
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| Two-soliton solutions | |
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| More spatial dimensions | |
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| The nonlinear Schrodinger (NLS) equation | |
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| Nonlinear wave packets | |
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| Modulated traveling-wave solutions of NLS(+) | |
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| Dark solition solutions of NLS(-) | |
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| A BT for NLS(+) | |
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| Transverse phenomena | |
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| Summary | |
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| Problems | |
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| References | |
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| Reaction-Diffusion Systems | |
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| Simple reaction-diffusion equations | |
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| The Zeldovich-Frank-Kamenetsky (Z-F) equation | |
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| The Burgers equation | |
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| The Hodgkin-Huxley (H-H) system | |
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| Space-clamped squid membrane dynamics | |
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| The H-H impulse | |
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| Simplified nerve models | |
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| The Markin-Chizmadzhev (M-C) model | |
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| The FitzHugh-Nagumo (F-N) model | |
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| Morris-Lecar (M-L) models | |
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| Stability analyses | |
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| The Z-F equation | |
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| The M-C model | |
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| The F-N model | |
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| The H-H and M-L systems | |
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| Decremental conduction | |
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| Nonuniform fibers | |
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| Tapered fibers | |
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| Leading-edge charge and impulse ignition | |
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| Dendritic logic | |
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| More space dimensions | |
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| Two-dimensional nonlinear diffusion | |
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| Nonlinear diffusion in three dimensions | |
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| Turing patterns | |
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| Hypercycles | |
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| Summary | |
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| Problems | |
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| References | |
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| Nonlinear Lattices | |
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| Spring-mass lattices | |
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| The Toda-lattice soliton | |
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| Lattice solitary waves | |
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| Existence of lattice solitary waves | |
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| Intrinsic localized modes and intrinsic gap modes | |
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| Lattices with nonlinear on-site potentials | |
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| The discrete sine-Gordon equation | |
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| Nonlinear Schrodinger lattices | |
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| The discrete self-trapping equation | |
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| Biological solitons | |
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| Alpha-helix solitons in protein | |
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| Self-trapping in globular proteins | |
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| Solitons in DNA | |
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| Nonconservative lattices | |
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| Quasiharmonic lattices | |
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| Myelinated nerves | |
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| Emergence of form by replication | |
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| Assemblies of neurons | |
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| Summary | |
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| Problems | |
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| References | |
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| Inverse Scattering Methods | |
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| Linear scattering revisited | |
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| Scattering solutions, bound states, and upper half plane poles | |
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| Why the upper half plane poles must be simple | |
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| The Gel'fand-Levitan equation again | |
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| Any questions? | |
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| Inverse scattering method for KdV | |
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| General description | |
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| Some examples | |
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| Reduction to Fourier analysis in the small amplitude limit | |
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| Two-component scattering theory | |
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| Linear theory | |
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| ISMs for two-component scattering | |
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| The sine-Gordon equation | |
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| The nonlinear Schrodinger equation | |
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| Conservation laws | |
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| Conservation laws for the KdV equation | |
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| Conserved densities for matrix scattering | |
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| Summary | |
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| Problems | |
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| References | |
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| Perturbation Theory | |
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| Perturbed matrices | |
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| A damped harmonic oscillator | |
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| Energy analysis | |
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| Multiple time scales | |
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| Energy analysis of soliton dynamics | |
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| Korteweg-de Vries solitons | |
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| Sine-Gordon solitons | |
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| Nonlinear Schrodinger solitons | |
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| More general soliton analyses | |
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| Multiple scale analysis of an SG kink | |
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| Variational analysis of an NLS soliton | |
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| Multisoliton perturbation theory | |
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| General theory | |
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| Kink-antikink collisions | |
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| Radiation from a fluxon | |
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| Neural perturbations | |
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| The FitzHugh-Nagumo system | |
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| Electrodynamic (ephaptic) coupling of nerves | |
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| Summary | |
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| Problems | |
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| References | |
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| Quantum Lattice Solitons | |
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| Quantum oscillators | |
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| A classical nonlinear oscillator | |
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| The birth of quantum theory | |
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| A quantum linear oscillator | |
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| The rotating wave approximation | |
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| The Born-Oppenheimer approximation | |
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| Dirac's notation | |
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| Pump-probe measurements | |
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| Self-trapping in the dihalomethanes | |
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| Classical analysis | |
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| Quantum analysis | |
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| Comparison with experiments | |
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| Boson lattices | |
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| The discrete self-trapping equation | |
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| A lattice nonlinear Schrodinger equation | |
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| Soliton wave packets | |
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| The Hartree approximation | |
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| More general quanta | |
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| The Ablowitz-Ladik equation | |
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| Salerno's equation | |
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| A fermionic polaron model | |
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| The Hubbard model | |
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| Energy transport in protein | |
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| Dynamic equations | |
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| Experimental observations | |
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| Recent comments | |
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| A quantum lattice sine-Gordon equation | |
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| Theoretical perspectives | |
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| Number state method | |
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| Quantum inverse scattering method | |
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| QISM analysis of the DST dimer | |
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| Comparison of the NSM and the QISM | |
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| Summary | |
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| Problems | |
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| References | |
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| Looking Ahead | |
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| References | |
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| Conservation Laws and Conservative Systems | |
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| References | |
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| Multisoliton Formulas | |
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| The KdV equation | |
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| The SG equation | |
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| The NLS equation | |
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| The Toda lattice | |
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| References | |