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

u′′=au^{3}+bu^{2}+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

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# Initial condition effect in Nonlinear PDE of a wave

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