Wave-Schrodinger systems
The Zakharov system is not the only wave-Schrodinger system studied. Another system of interest is the ``Yukawa-type system
i u_t + D u = -A u Box A = m2 A + |u|2
for d=3. A represents the meson field, while u is the nucleon field.
Global well posedness in the energy class (H1, H1 x L2) is in [Bch1984], [BlChd1978], [FuTs1978], [HaWl1987]. Modified wave operators were constructed for large energy data at infinity in [GiVl-p2].
With positive mass m=1, global well-posedness can be pushed to (Hs, Hm x Hm-1) whenever 1 ³ s,m > 7/10 and s+m > 3/2 [Pe-p2].
A generalized Zakharov system with a magnetic component was studied in [KeWg1998], with local existence of smooth solutions obtained.
Another such system is the Davey-Stewartson system [DavSte1974] in 2 spatial dimensions, a complex field u, and a real field phi:
i u_t + c_0 u_xx + u_yy = c_1 |u|^2 u + c_2 u phi_x phi_xx + c_3 phi_yy = partial_x ( |u|^2 )
The field phi depends elliptically on u when c_3 is positive and thus one usually only specifies the initial data for u, not phi. This equation is a modification of the cubic nonlinear Schrodinger equation and is completely integrable in the cases (c_0, c_1, c_2, c_3) = (-1,1,-2,1) (DS-I) and (1,-1,2,-1) (DS-II). When c_3 > 0 the situation becomes a nonlinear Schrodinger equation with non-local nonlinearity, and can be treated by Strichartz estimates [GhSau1990]; for c_3 < 0 the situation is quite different but one can still use derivative NLS techniques (i.e. gauge transforms and energy methods) to obtain some local existence results [LiPo1993]. Further results are in [HaSau1995].
The Davey-Stewartson system is a special case of the Zakharov-Schulman system
i u_t + L_1 u = phi u
L_2 phi = L_3( |u|^2 )
where L_1, L_2, L_3 are various constant coefficient differneital operators; these describe the interactions of small amplitude, high frequency waves with acoustic waves [ZkShl1980]. Using energy methods and gauge transformations, local existence for smooth data was established in [KnPoVe1995b]; see also [GhSau1992].
The Ishimori system [Im1984] has a complex field u and a real field phi in two dimensions, and has the form
iu_t + u_xx - a u_yy = 2 u (u_x^2 - u_y^2) / (1 + |u|^2) - i b(phi_x u_y - phi_y u_x) phi_xx + a' phi_yy = 8 Im(u_x u_y) / (1 + |u|^2)^2
The case (a,a') = (+1,-1) is studied in [Sy1992]. The case (a,a') = (-1,1) is studied in [HySau1995], [Ha-p], [KnPoVe2000]; in this case one has LWP for small data in the space H^4 intersect L^2( (x^2 + y^2)^4 dx dy) [KnPoVe2000].
