As many people define solitary waves they are localized pulses that propagate without changing the shape. As far as I know the same pulses exist in ordinary wave equation ! why should we look for solitons in nonlinear wave equation?
$\begingroup$
$\endgroup$
1
-
2$\begingroup$ The point of view is opposite... it's not just that one goes looking for solutions that propagate without changing shape and then end up looking at nonlinear equations... one of the important reasons one looks for such solutions is that they might help you solve and understand nonlinear phenomena. And non-linear dynamics is like non-elephant biology i.e. pretty much ubiquitous and very important. $\endgroup$– SivaCommented Sep 22, 2013 at 9:26
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
In a linear wave equation, there is nothing to pull a pulse or envelope of running waves apart. But there is nothing to hold it together, either. A minor disturbance such as a small obstacle or some dispersion, will change the waveshape, or break it up, such as losing some of its energy to outward spherical waves from the obstacle. Two or more pulses in a linear wave equation will pass through each other and continue without change.
For a soliton, nonlinear forces push and pull actively maintaining the shape. The details of this depends on the equation. Minor disturbances don't have lasting effects. Also, two solitons colliding likely won't just silently pass through each other, but whatever happens, the result can include new solitons heading off in new directions.
If what I say about solitons sounds wishy-washy, it's because there are a wide variety of nonlinear equations, and for scalars, vectors, spinors, etc. Finding out what can happen is part of the fun of studying soliton equations.
