Davey–Stewartson equation explained

In fluid dynamics, the Davey–Stewartson equation (DSE) was introduced in a paper by A. Davey and Keith Stewartson to describe the evolution of a three-dimensional wave-packet on water of finite depth.

A and a real (mean-flow) field B :i (\partial A)/(\partial t) + c _{0} (\partial ^{2} A)/(\partial x ^{2}) + (\partial A)/(\partial y ^{2}) = c _{1} |A| ^{2} A + c _{2} A (\partial B)/(\partial x) , (\partial B)/(\partial x ^{2}) + c _{3} (\partial ^{2} B)/(\partial y ^{2}) = (\partial |A| ^{2})/(\partial x) .

The DSE is an example of a soliton equation in 2+1 dimensions. The corresponding Lax representation for it is given in .

In 1+1 dimensions the DSE reduces to the nonlinear Schrödinger equation

i (\partial A)/(\partial t) + (\partial ^{2} A)/(\partial x ^{2}) + 2 k |A| ^{2} A =0 .

Itself, the DSE is the particular reduction of the Zakharov–Schulman system. On the other hand, the equivalent counterpart of the DSE is the Ishimori equation.

The DSE is the result of a multiple-scale analysis of modulated nonlinear surface gravity waves, propagating over a horizontal sea bed.

See also

• Ginzburg–Landau equation
• Nonlinear systems
• Ishimori equation

External links

• Davey-Stewartson_system at the dispersive equations wiki.