Nonl inear ideas of a "sol iton" variety have been a unifying influence on the na tura 1 sci ences for many decades. HO~/ever, thei r uni versa 1 a pprec i at i on in the physics community as a genuine paradigm is very much a current develop­ ment. All of us who have been associated with this recent wave of enthusiasm were impressed with the variety of applications, their inevitability once the mental contraint of linear normal modes is removed, and above all by the common mathematical structures underpinning applications with quite different (and often novel) physical manifestations. This has certainly been the situ­ ation in condensed matter, and when, during the Paris Lattice Dynamics Con­ ference (September 1977), one of us (T. S. ) first suggested a condensed matter soliton Meeting, the idea was strongly encouraged. It would provide an opportunity to exhibit the common mathematical problems, illuminate the new contexts, and thereby focus the "subject" of nonlinear physics at this embryonic stage of its evolution. The original conception was to achieve a balance of mathematicians and phy~cis~ such that each would benefit from the other's expertise and out­ look. In contrast to many soliton Meetings, hO~/ever, a deliberate attempt was made to emphasize physics contexts rather than mathematical details.

I. Introduction.- Solitons in Mathematics: Brief History.- Solitons in Physics.- II. Mathematical Aspects.- Numerical Studies of Solitons.- Poles of the Toda Lattice.- Perturbation Theory of the Double Sine-Gordon Equation.- Soliton Perturbations and Nonlinear Focussing.- Novel Class of Nonlinear Evolution Equations Solvable by the Spectral Transform Technique, Including the So-Called Cylindrical KdV Equation.- The Complex Modified Korteweg-de Vries Equation, a Non-Integrable Evolution Equation.- III. Statistical Mechanics and Solid-State Physics.- Soliton-Bound States in the Magnetic Gap.- Statistical Mechanics of Nonlinear Dispersive Systems.- Some Applications of Instantons in Statistical Mechanics.- The Theory of Structural Phase Transitions: Cluster Walls and Phonons.- Nonlinear Lattice Dynamics: Molecular Dynamics Studies.- Computer Simulation of Structural Phase Transitions.- Soliton-Like Features in a Two-Dimensional XY Model with Ouartic Anisotropy.- Behavior of a ?4-Kink in the Presence of an Inhomogeneous Perturbation.- Solitary Wave Solutions in a Diatomic Lattice.- Lattice Models of High Velocity Dislocation Motion.- Grain Boundaries as Solitary Waves.- The Relation of Solitons to Polaritons in Coupled Systems.- Solitons in CsNiF3: Their Experimental Evidence and Their Thermodynamics.- Structure and Stability of Domain Walls - Phase Transition.- Periodic Lattice Distortions and Charge Density Waves in One- and Two-Dimensional Systems.- Solitons in Incommensurate Systems.- Fluctuations and Freezing in a One-Dimensional Liquid: Hg3??AsF6.- Charge Density Wave Systems: The ?-Particle Model.- The Soliton Lattice: Application to the ? Phase.- The New Concept of Transitions by Breaking of Analyticity in a Crystallographic Model.- Textures in Superfluid3He.- Creation of Spin Waves in 3HeB.- The Interaction of Spin Waves in Liquid He3 in Several Dimensions.- Josephson Transmission Line Oscillators.- Dissipative Structures in Quasi-One-Dimensional Superconductors.- Solitary Phenomena in Finite Dissipative Discrete Systems.- Stability of Nonuniform States in Systems Exhibiting Continuous Bifurcation.- The Sine-Gordon Chain: Mass Diffusion.- Solitary-Wave Propagation as a Model for Poling in PVF2.- Theory of One-Dimensional Ionic and Solitary Wave Conduction in Potassium Hollandite.- IV. Summary.- Summary: Where Do Solitons Go From Here?.- Index of Contributors.