This chapter deals with the basic ideas and many major nonlinear model equations which arise in a wide variety of physical problems. Included are one-dimensional wave, Klein–Gordon (KG), sine–Gordon (SG), Burgers, Fisher, Korteweg–de Vries (KdV), Boussinesq, modified KdV, nonlinear Schrödinger (NLS), Benjamin–Ono (BO), Benjamin–Bona–Mahony (BBM), Ginzburg–Landau (GL), Burgers–Huxley (BH), KP, concentric KdV, Whitham, Davey–Stewartson, Toda lattice, Camassa–Holm (CH), and Degasperis–Procesi (DP) equations. This is followed by variational principles and the Euler–Lagrange equations. Also included are Plateau’s problem, Hamilton’s principle, Lagrange’s equations, Hamilton’s equations, the variational principle for nonlinear Klein–Gordon equations, and the variational principle for nonlinear water waves. Special attention is given to the Euler equation of motion, the continuity equation, the associated energy equation and energy flux, linear water wave problems and their solutions, nonlinear finite amplitude waves (the Stokes waves), gravity waves, gravity-capillary waves, and linear and nonlinear dispersion relations. Finally, the modern theory of nonlinear water waves is formulated.


Variational Principle Euler Equation Water Wave Lagrange Equation Nonlinear Partial Differential Equation 
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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Texas, Pan AmericanEdinburgUSA

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