Solving Second-Harmonic Generation Phase Mismatch - Exact Solution?

[andonrangelov]

Hi All,
I was searching in the net if the exact solution of the nonlinear system describing the second harmonic generation does exist, but I found nothing.
Does some one know about such solution and if yes can you give references? Thanks

 

[andonrangelov]

Well it seems that my question is very difficult …..

 

[Bill_K]

I suspect that you don't realize how complicated the answer is!

 

[AlephZero]

You haven't told us what nonlinear system you are asking about.

Post some equations, or a web link. Then you might get some answers.

 

[andonrangelov]

Ok, here is the nonlinear system that I want to solve

idA/dt=B.B.exp(i.s.t)

idB/dt=B*.A.exp(-i.s.t)

where B(t),A(t),s=constant but B and A are complex (B* means complex conjugation of B in the above equation)

 

[andonrangelov]

Nonlinear differential equation can, what are the solutions of this one?

Hi all,
I am searching for the solutions of the following nonlinear system:

idA/dt=B.B.exp(i.s.t)

idB/dt=B*.A.exp(-i.s.t)

where B(t),A(t),s=constant but B and A are complex (B* means complex conjugation of B in the above equation). Does some one know its solutions?
I know the solution in case when s=0 then B(t)=tanh(t); A(t)=i.sech(t), but I do not know the more general case. Can you help me?
Thanks

 

[hunt_mat]

What about is s is very small? I.e. when $0<s\ll 1$, you can get an analytical solution then, by expanding the exponential as $\exp (x)=1+x$.

Mat

 

[kosovtsov]

It seems to me that something is wrong here. Your B(t)=tanh(t); A(t)=i.sech(t) is not a solution to your system when s=0. Or I misunderstand your notations.

 

[hunt_mat]

I didn't check this, if s=0, then you can divide the equations to find:
$$\frac{dB}{dA}= \frac{A}{B}$$
Which integrates to $A=kB$, where k>0 is the integration constant, substuting this into the second equation shows that:
$$i\frac{dB}{dt}=kB^{2}$$
Which certainly doesn't give the solution that you have.