The term soliton comes from an abbreviation of “solitary wave”.
A soliton solution of a nonlinear wave equation is a solution whose large amplitude part is localized in space and is asymptotically stable in time. This asymptotic stability (more precisely non-damping and asymptotic preservation of shape, up to translation) is typically a feature of an infinite number of conservation laws, and many models of equations allowing soliton solutions are in fact integrable systems (with infinitely many degrees of reedom). Soliton solution often combine to multisoliton solutions in a nonlinear way, with a period of interaction when they “meet”, but after a passage of some time, the waves gradually uncouple and regain their original shape when outgoing to infinity. A typical example of a nonlinear wave equation exhibiting soliton solutions is the exactly solvable “nonlinear Schroedinger equation” appearing in optics.
M. Jimbo, T. Miwa, E. Date, Solitons: differential equations, symmetries and infinite dimensional algebras, Camb. tracts is math. 135 (transl. from Japanese by M. Reid?)
A. Newell, Solitons in mathematics and physics
Ludwig D. Faddeev, Leon Takhtajan, Hamiltonian methods in the theory of solitons, Springer, (transl. from Russian Гамильтонов подход в теории солитонов. — М.: Наука. 1986.)
L. D. Faddeev, V. E. Korepin, Quantum theory of solitons, 1987
David Tong, TASI Lectures on Solitons (web),