Intro; Preface; Acknowledgements; Contents; 1 Introduction and Motivation of Models; 1.1 Dynamics of Hertzian Beads; 1.2 The Linear Discrete Wave Equation; 1.3 The Nonlinear States: Dispersive Shock Waves, Traveling Solitary Waves, and Discrete Breathers; References; 2 Dispersive Shock Waves; 2.1 Overview; 2.2 Theoretical Analysis of Shock Waves; 2.3 Prototypical Numerical Computations; 2.4 Outlook; References; 3 Traveling Waves; 3.1 Overview; 3.2 Traveling Waves With Precompression: The KdV and Toda Limits; 3.3 Traveling Waves Without Precompression I: The Continuum Limit

3.4 Traveling Waves Without Precompression II: Asymptotic and Numerically Exact ResultsReferences; 4 Discrete (Dark) Breathers; 4.1 Discrete Breathers: Overview and Theoretical Analysis; 4.2 Connection with Experimental Results; References; 5 Heterogeneous Media; 5.1 Overview; 5.2 Traveling Solitary Waves in Heterogeneous Chains; 5.3 From Isolated Defect Modes to Bright Breathers in Dimer Chains; References; 6 Media with Onsite Forces: The Newton's Cradle and Beyond; 6.1 Breathers in the Newton's Cradle; 6.2 Solitary Waves and Breathers in Locally Resonant Granular Chains; References

7 Higher Dimensional Lattices7.1 Overview; 7.2 Conical Diffraction in the Hexagonal Packing; 7.3 Other Work in Higher Dimensional Lattices and Future Directions; References; Appendix; A.1 Numerical Computation of Traveling Solitary Waves and Their Stability; A.2 Numerical Computation of Breathers and Their Stability; A.3 Derivation of the Nonlinear Schrödinger Equation; References