- #1
Romik
- 14
- 0
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
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