Nonlinear Surface Waves in One Dimension
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In this chapter, we present some examples of nonlinear evolution equations in one space dimension. We re-discuss the traditional Korteweg–de Vries (KdV) equation for the shallow water long channel case, and its cnoidal waves and soliton solutions. Then we briefly present the MKdV equation and some nonlinear dispersion extension of it. In the last sections, we discuss some possible dynamical generalizations of the shallow water models on compact intervals, for any depth of the fluid. The resulting equation is an infinite-order differential one, and it reduces to a finite difference differential equation. We show that this generalized KdV equation approaches the KdV, MKdV, and Camassa–Holm limiting equations, both at the equation and at the solution level, in the appropriate physical conditions. In the last part we discuss the Boussinesq equations on a circle.