Thomas J. Bridges has been researching water waves for over 25 years, with contributions in the areas of shallow water hydrodynamics, breaking waves, resonances, instabilities, Hamiltonian structures, solitary waves, and sloshing. He is currently Professor of Mathematics at the University of Surrey, has over 130 published papers, and is leader of the EPSRC-supported project on modelling water wave energy converters. Mark D. Groves has been researching the theory of water waves for over 20 years. He is currently Professor at the University of Saarlandes, Saarbrücken, Germany. His principal contributions have been in the areas of dimension breaking, Hamiltonian structures, spatial dynamics, and the applications of dynamical systems theory and nonlinear functional analysis to the problem of water waves. David P. Nicholls has studied water waves, electromagnetic and acoustic wave propagation, and high-order spectral methods for the past 20 years. He is currently a Professor of Mathematics, Statistics, and Computer Science at the University of Illinois, Chicago. He has over 60 published papers and has received continuous support from the NSF, DOE, and industry.

'In 2014, there was a special four-week programme on water waves at the Isaac Newton Institute for mathematical Sciences in Cambridge. A large group of experts gathered together to look at the state of the art and to develop a strategy for future directions of research … This is not an introductory text, in that it really requires some serious background from e.g. an undergraduate degree programme … However, it should provide an excellent overview for those keen to move into the field and would therefore be a good introduction for Ph.D. students, as well as being good for specialists wishing to widen their expertise … This would have been a stimulating and rewarding programme to have attended, and these papers give a very useful resource summarising the current state of this research area - one which is likely to prove interesting, demanding and probably elusive for a long time to come!' Frank Berkshire, Contemporary Physics

• Preface Thomas J. Bridges, Mark D. Groves and David P. Nicholls; 1. High-Order Perturbation of Surfaces (HOPS) Short Course – boundary value problems David P. Nicholls; 2. HOPS Short Course – traveling water waves Benjamin F. Akers; 3. High-Order Perturbation of Surfaces (HOPS) Short Course – analyticity theory David P. Nicholls; 4. HOPS Short Course – stability of travelling water waves Benjamin F. Akers; 5. A novel non-local formulation of water waves Athanassios S. Fokas and Konstantinos Kalimeris; 6. The dimension-breaking route to three-dimensional solitary gravity-capillary water waves Mark D. Groves; 7. Validity and non-validity of the nonlinear Schrödinger equation as a model for water waves Guido Schneider; 8. Vortex sheet formulations and initial value problems: analysis and computing David M. Ambrose; 9. Wellposedness and singularities of the water wave equations Sijue Wu; 10. Conformal mapping and complex topographies André Nachbin; 11. Variational water wave modelling: from continuum to experiment Onno Bokhove and Anna Kalogirou; 12. Symmetry, modulation and nonlinear waves Thomas J. Bridges.

Thomas J. Bridges has been researching water waves for over 25 years, with contributions in the areas of shallow water hydrodynamics, breaking waves, resonances, instabilities, Hamiltonian structures, solitary waves, and sloshing. He is currently Professor of Mathematics at the University of Surrey, has over 130 published papers, and is leader of the EPSRC-supported project on modelling water wave energy converters. Mark D. Groves has been researching the theory of water waves for over 20 years. He is currently Professor at the University of Saarlandes, Saarbrücken, Germany. His principal contributions have been in the areas of dimension breaking, Hamiltonian structures, spatial dynamics, and the applications of dynamical systems theory and nonlinear functional analysis to the problem of water waves. David P. Nicholls has studied water waves, electromagnetic and acoustic wave propagation, and high-order spectral methods for the past 20 years. He is currently a Professor of Mathematics, Statistics, and Computer Science at the University of Illinois, Chicago. He has over 60 published papers and has received continuous support from the NSF, DOE, and industry.

'In 2014, there was a special four-week programme on water waves at the Isaac Newton Institute for mathematical Sciences in Cambridge. A large group of experts gathered together to look at the state of the art and to develop a strategy for future directions of research … This is not an introductory text, in that it really requires some serious background from e.g. an undergraduate degree programme … However, it should provide an excellent overview for those keen to move into the field and would therefore be a good introduction for Ph.D. students, as well as being good for specialists wishing to widen their expertise … This would have been a stimulating and rewarding programme to have attended, and these papers give a very useful resource summarising the current state of this research area - one which is likely to prove interesting, demanding and probably elusive for a long time to come!' Frank Berkshire, Contemporary Physics

• Preface Thomas J. Bridges, Mark D. Groves and David P. Nicholls; 1. High-Order Perturbation of Surfaces (HOPS) Short Course – boundary value problems David P. Nicholls; 2. HOPS Short Course – traveling water waves Benjamin F. Akers; 3. High-Order Perturbation of Surfaces (HOPS) Short Course – analyticity theory David P. Nicholls; 4. HOPS Short Course – stability of travelling water waves Benjamin F. Akers; 5. A novel non-local formulation of water waves Athanassios S. Fokas and Konstantinos Kalimeris; 6. The dimension-breaking route to three-dimensional solitary gravity-capillary water waves Mark D. Groves; 7. Validity and non-validity of the nonlinear Schrödinger equation as a model for water waves Guido Schneider; 8. Vortex sheet formulations and initial value problems: analysis and computing David M. Ambrose; 9. Wellposedness and singularities of the water wave equations Sijue Wu; 10. Conformal mapping and complex topographies André Nachbin; 11. Variational water wave modelling: from continuum to experiment Onno Bokhove and Anna Kalogirou; 12. Symmetry, modulation and nonlinear waves Thomas J. Bridges.

Thomas J. Bridges has been researching water waves for over 25 years, with contributions in the areas of shallow water hydrodynamics, breaking waves, resonances, instabilities, Hamiltonian structures, solitary waves, and sloshing. He is currently Professor of Mathematics at the University of Surrey, has over 130 published papers, and is leader of the EPSRC-supported project on modelling water wave energy converters. Mark D. Groves has been researching the theory of water waves for over 20 years. He is currently Professor at the University of Saarlandes, Saarbrücken, Germany. His principal contributions have been in the areas of dimension breaking, Hamiltonian structures, spatial dynamics, and the applications of dynamical systems theory and nonlinear functional analysis to the problem of water waves. David P. Nicholls has studied water waves, electromagnetic and acoustic wave propagation, and high-order spectral methods for the past 20 years. He is currently a Professor of Mathematics, Statistics, and Computer Science at the University of Illinois, Chicago. He has over 60 published papers and has received continuous support from the NSF, DOE, and industry.

'In 2014, there was a special four-week programme on water waves at the Isaac Newton Institute for mathematical Sciences in Cambridge. A large group of experts gathered together to look at the state of the art and to develop a strategy for future directions of research … This is not an introductory text, in that it really requires some serious background from e.g. an undergraduate degree programme … However, it should provide an excellent overview for those keen to move into the field and would therefore be a good introduction for Ph.D. students, as well as being good for specialists wishing to widen their expertise … This would have been a stimulating and rewarding programme to have attended, and these papers give a very useful resource summarising the current state of this research area - one which is likely to prove interesting, demanding and probably elusive for a long time to come!' Frank Berkshire, Contemporary Physics

• Preface Thomas J. Bridges, Mark D. Groves and David P. Nicholls; 1. High-Order Perturbation of Surfaces (HOPS) Short Course – boundary value problems David P. Nicholls; 2. HOPS Short Course – traveling water waves Benjamin F. Akers; 3. High-Order Perturbation of Surfaces (HOPS) Short Course – analyticity theory David P. Nicholls; 4. HOPS Short Course – stability of travelling water waves Benjamin F. Akers; 5. A novel non-local formulation of water waves Athanassios S. Fokas and Konstantinos Kalimeris; 6. The dimension-breaking route to three-dimensional solitary gravity-capillary water waves Mark D. Groves; 7. Validity and non-validity of the nonlinear Schrödinger equation as a model for water waves Guido Schneider; 8. Vortex sheet formulations and initial value problems: analysis and computing David M. Ambrose; 9. Wellposedness and singularities of the water wave equations Sijue Wu; 10. Conformal mapping and complex topographies André Nachbin; 11. Variational water wave modelling: from continuum to experiment Onno Bokhove and Anna Kalogirou; 12. Symmetry, modulation and nonlinear waves Thomas J. Bridges.