c*****t 发帖数: 520 | 1 Perhaps it is a foolish question.
I think that some types of Schrodinger equations and reaction-diffusion
equations are similar.
Semilinear Schrodinger equation: iu_t=u_xx+f(u).
Semilinear reaction-diffusion equation: u_t=u_xx+f(u).
But the methods for them are quite different. For example, for reaction-
diffusion equations, people ofen study the long time behavior of the
solutions by constructing subsolutions and/or supersolutions and using
comparison principle. For Schrodinger equations, perhaps the well-posedness
is the main problem.
Of course, solutions of Schrodinger equations are complex. But I still do
not know why Schrodinger equations and reaction-diffusion equations are so
different.
Can someone explain it briefly to me? Thank you very much. | C********n 发帖数: 6682 | 2 yes they r similiar
and there are books about it
posedness
【在 c*****t 的大作中提到】![](/moin_static193/solenoid/img/up.png) : Perhaps it is a foolish question. : I think that some types of Schrodinger equations and reaction-diffusion : equations are similar. : Semilinear Schrodinger equation: iu_t=u_xx+f(u). : Semilinear reaction-diffusion equation: u_t=u_xx+f(u). : But the methods for them are quite different. For example, for reaction- : diffusion equations, people ofen study the long time behavior of the : solutions by constructing subsolutions and/or supersolutions and using : comparison principle. For Schrodinger equations, perhaps the well-posedness : is the main problem.
| Q******g 发帖数: 607 | 3 The factor i in front of time derivative makes the difference.
With i as in Shroedinger eqn, the solution is oscillatory, it
makes no sense to study the long time behavior.
posedness
【在 c*****t 的大作中提到】![](/moin_static193/solenoid/img/up.png) : Perhaps it is a foolish question. : I think that some types of Schrodinger equations and reaction-diffusion : equations are similar. : Semilinear Schrodinger equation: iu_t=u_xx+f(u). : Semilinear reaction-diffusion equation: u_t=u_xx+f(u). : But the methods for them are quite different. For example, for reaction- : diffusion equations, people ofen study the long time behavior of the : solutions by constructing subsolutions and/or supersolutions and using : comparison principle. For Schrodinger equations, perhaps the well-posedness : is the main problem.
| w*********o 发帖数: 434 | 4 exactly, appearance of i in front of time derivative matters. that means it
is wave equation instead of difusion equation. | C********n 发帖数: 6682 | 5 http://www.amazon.com/Schr%C3%B6dinger-Equations-Diffusion-Mono
posedness
【在 c*****t 的大作中提到】![](/moin_static193/solenoid/img/up.png) : Perhaps it is a foolish question. : I think that some types of Schrodinger equations and reaction-diffusion : equations are similar. : Semilinear Schrodinger equation: iu_t=u_xx+f(u). : Semilinear reaction-diffusion equation: u_t=u_xx+f(u). : But the methods for them are quite different. For example, for reaction- : diffusion equations, people ofen study the long time behavior of the : solutions by constructing subsolutions and/or supersolutions and using : comparison principle. For Schrodinger equations, perhaps the well-posedness : is the main problem.
|
|