Davey–Stewartson equation

From Wikipedia, the free encyclopedia

In mathematics, the Davey-Stewartson equation (DSE) was introduced in (Davey & Stewartson 1974) to describe the evolution of a three-dimensional wave-packet on water of finite depth. It is a system of partial differential equations for a complex field u and a real field φ:

iu_t + c₀u_xx + u_yy = c₁ | u | ²u + c₂uφ_x

φ_xx + c₃φ_yy = ( | u | ²)_x.

The DSE is an example of a soliton equation in 2+1 dimensions. The corresponding Lax representation for it is given in (Boiti, Martina & Pempinelli 1995).

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

iu_t + u_xx + 2k | u | ²u = 0.

Itself the DSE is the particular reductions of the Zakharov-Schulman equation. On the other hand, the equivalent counterpart of the DSE is the Ishimori equation.

[edit] See also

• Nonlinear systems
• Ishimori equation

[edit] References

• Boiti, M.; Martina, L. & Pempinelli, F. (1995), “Multidimensional localized solitons”, Chaos Solitons Fractals 5 (12): 2377--2417, MR1368226, DOI 10.1016/0960-0779(94)E0106-Y 
• Davey, A. & Stewartson, K (1974), “On three dimensional packets of surface waves”, Proc. R. Soc. A 338: 101-110, DOI 10.1098/rspa.1974.0076 
• Sattinger, David H.; Tracy, C. A. & Venakides, S., eds. (1991), Inverse Scattering and Applications, vol. 122, Contemporary Mathematics, Providence, RI: American Mathematical Society, MR1135850 

[edit] External links

• Davey-Stewartson_system at the dispersive equations wiki.