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】
: 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】
: 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】
: 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.
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