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I'm studying soliton (solitary waves). They are many theory which explain the phenomenon, like sine-Gordon model. But sine-Gordon model has limitations when it applies to 4 dimension because it is valid for only two dimensions only.

So which topics/theory will cover to understands solitons fully?

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Solitons are magical in the sense that having an exact solution to a nonlinear field theory is rare and there is no real systematic way to go about solving it. I seem to recall that José and Saletan's Classical Dynamics textbook has a good discussion of solitons, with the sine Gordon and Kortweig-de Vries equations as the examples of choice.

There is also talk of "solitons" for the nonlinear Maxwell-Vlasov equations in plasma physics, but these are not solitons in the Hamiltonian field theory sense.

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Can you give the the download link of that book? – Laser-boy Apr 7 '13 at 19:49

The KDV theory is based on approximations to the governing equations, and hence it is only weakly-nonlinear. Fully nonlinear internal solitary waves may be obtained through the Dubreil-Jacotin-Long (DJL) equation [Long, 1953], for which there are special nonlinear cases amenable to analytical methods. The DJL equation is the formal equivalent to the Euler equations which yield solutions for steady NLIWs of arbitrary amplitude in continuously stratified, incompressible, inviscid fluids. Its solution may be used as an initial condition in a numerical model and involves solving an elliptic eigenvalue problem for the streamfunction yielding an infinite number of internal wave modes where the lowest mode is also the fastest. A number of studies have sought numerical solutions of the DJL equation, deep mode-2 solitary waves were first explored by Benjamin [1966], Davis and Acrivos [1967], Tung et al. [1982]. One limitation is that since this theory is steady it is not evolving in time.

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Solitons are just nonlinear waves. They appear in almost any nonlinear system, similar to usual (linear) waves that characterize excitations in different systems (deformation waves, acoustic waves, electromagnetic waves). A distinguishing feature of a soliton is that it is localized in space. Usually, a soliton has a bell-shaped form (sometimes, this type is called "dynamical soliton"), or a shock-wave or kink form (called as "topological soliton"). Another feature of the soliton is that it behaves like a particle, when interacting with another soliton or some obstacle (potential). In early days (in 60-80-s of 20th century), a term "soliton" was referred to excitations in integrable systems (such as the sine-Gordon model, the KdV equation, the nonlinear Schroedinger equation etc) only. But today, this term is applied to almost any particle-like excitation in different nonlinear systems.

For beginner, I would recommend a book by M. Remoissenet, "Waves called solitons", which is a good introduction to the topic. Also, there is relatively old, but still good and short book by P.Bhatnagar, "Nonlinear Waves in One-Dimensional Dispersive Systems".

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