| 概要 | We study the stability and bifurcation of steady states
for a certain kind of damped driven nonlinear Schrodinger equation
with cubic nonlinearity and a detuning term in one space dimension, mathematically in a rigorous sense.
It is known by numerical simulations that the system shows lots of coexisting spatially localized structures
as a result of subcritical bifurcation.
Since the equation does not have a variational structure, unlike the conservative case,
we cannot apply a variational method for capturing the ground state.
Hence, we analyze the equation from a viewpoint of bifurcation theory. In the case of a finite interval,
we prove the fold bifurcation of nontrivial stationary solutions
around the codimension two bifurcation point of the trivial equilibrium
by exact computation of a fifth-order expansion on a center manifold reduction.
In addition, we analyze the steady-state mode interaction and prove the bifurcation of mixed-mode solutions,
which will be a germ of localized structures on a finite interval.
Finally, we study the corresponding problem on the entire real line by use of spatial dynamics.
We obtain a small dissipative soliton bifurcated adequately from the trivial equilibrium.
(This is a joint work with Prof. Isamu Ohnishi(Hiroshima University) and Prof. Yoshio Tsutsumi (Kyoto University).) |