I don't understand the mean of solitons in pde. I study about it in wikipedia website. but I don't see it very well. Please help me if you have any reference/suggestion/etc.
Thanks in advance.
For the KdV equation : u_t+6uu_x+u_xxxx=0, look to a solution of type u=f(x-tv), then obtain an ODE for f, after solving the ODE , you obtain the soliton solution
We collide, lose some energy at expense of radiation - are solitaries, not solitons
The answer by Andriy O. Borisyuk is correct. Here is some addition to the answer. In a narrow sense, a soliton is a localized wave that propagates without a change of its shape and parameters, i.e. it behaves as a particle (so the name is similar to particles' names like "electron", "proton" etc). A soliton preserves its shape and parameters even after collision with another soliton. Usually, such localized waves are described by so-called integrable partial differential equations (PDEs). In a broad sense, a soliton is a localized wave that preserves its shape, more or less, when it propagates alone or collides with a similar object. Such waves can be described by any (integrable or non-integrable, conservative or dissipative) PDEs. Here are some examples of solitons: tsunami, pulses in optical fibers, density waves in Bose-Einstein condensates, contraction/expansion waves in lattices etc.
For a simple introduction, I would recommend a book by Alexandre T. Filippov, The Versatile Soliton (Modern Birkhäuser Classics). This is a popular science book. Also, there is an old, but still good book by P. L. Bhatnagar, Nonlinear Waves in One-Dimensional Dispersive Systems.
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