Nonlinear Schrödinger Equation - The Nonlinear Schrödinger Equation in Water Waves

The Nonlinear Schrödinger Equation in Water Waves

For water waves, the nonlinear Schrödinger equation describes the evolution of the envelope of modulated wave groups. In a paper in 1968, Vladimir E. Zakharov describes the Hamiltonian structure of water waves. In the same paper Zakharov shows, that for slowly modulated wave groups, the wave amplitude satisfies the nonlinear Schrödinger equation, approximately. The value of the nonlinearity parameter к depends on the relative water depth. For deep water, with the water depth large compared to the wave length of the water waves, к is negative and envelope solitons may occur.

For shallow water, with wavelengths longer than 4.6 times the water depth, the nonlinearity parameter к is positive and wave groups with envelope solitons do not exist. Note, that in shallow water surface-elevation solitons or waves of translation do exist, but they are not governed by the nonlinear Schrödinger equation.

The nonlinear Schrödinger equation is thought to be important for explaining the formation of rogue waves.

The complex field ψ, as appearing in the nonlinear Schrödinger equation, is related to the amplitude and phase of the water waves. Consider a slowly modulated carrier wave with water surface elevation η of the form:

where a(x₀, t₀) and θ(x₀, t₀) are the slowly modulated amplitude and phase. Further ω₀ and k₀ are the (constant) angular frequency and wavenumber of the carrier waves, which have to satisfy the dispersion relation ω₀ = Ω(k₀). Then

So its modulus |ψ| is the wave amplitude a, and its argument arg(ψ) is the phase θ.

The relation between the physical coordinates (x₀, t₀) and the (x, t) coordinates, as used in the nonlinear Schrödinger equation given above, is given by:

Thus (x, t) is a transformed coordinate system moving with the group velocity Ω'(k₀) of the carrier waves, The dispersion-relation curvature Ω"(k₀) is always negative for water waves under the action of gravity.

For waves on the water surface of deep water, the coefficients of importance for the nonlinear Schrödinger equation are:

so

where g is the Earth's acceleration gravity.