Wave-Schrodinger systems: Difference between revisions

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The [[Zakharov system]] is not the only wave-Schrodinger system studiedAnother system of interest is the ``Yukawa-type'' system
A '''wave-Schrodinger system''' is any coupled system of a [[wave equations|nonlinear wave equation]] and a [[Schrodinger equation|nonlinear Schrodinger equation]]The main examples are:


i u_t + D u = -A u
* The [[Zakharov equation]] ([[Zakharov equation on R|on R]], [[Zakharov equation on T|on T]], [[Zakharov equation on R^2|on R^2]], or [[Zakharov equation on R^3|on R^3]])
Box A = m2 A + |u|2
* The [[Klein-Gordon-Zakharov equation]]
 
* The [[Yukawa-type system]]
for d=3.  A represents the meson field, while u is the nucleon field.
* The [[Davey-Stewartson system]]
 
* The [[Zakharov-Schulman system]]
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].
* The [[Ishimori system]]
 
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].


[[Category:Equations]]
[[Category:Equations]]

Revision as of 06:07, 27 July 2006