Initial condition effect in Nonlinear PDE of a wave

[Romik]

Hey there,

I have modeled a propagating wave in a 1D dispersive media, in which square and cubic nonlinear terms are present.

u′′=au³+bu²+cu

the propagating pulse starts to steepen with time which is the effect of nonlinearity, but there is an effect which I can't understand.

so here is the effect,

case a)
u(x,0)=sech(x-ct) and v(x,0)=0
from d'alambert solution, I have 2 equal propagating wave in (+) and (-) directions carrying the effect of nonlinearity. which make sense.

case b)
u(x,0)=sech(x-ct) and v(x,0)=g(x,t)
where the initial velocity,g, is the first time derivative of initial displacement, sech(x-ct).
In this case I expect to have pulse propagating in one direction (+), but I get two waves, one with big amplitude in main direction (+), and another one with small amplitude in opposite direction (-).

I can't understand how does the small amplitude wave develop and propagate. any explanation?


I would appreciate any hint or comment.
Thanks

 

[Greg Bernhardt]

I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?

 

[Strum]

I am not even sure I understand the question. In the first equation is the differentiation with respect to $\xi = x-ct$? And what is v(x,t) defined as? I am not used to this solitonic language so it is hard to follow what is going on.