Sine-Gordon is also a scalar field theory. If you mean $\phi^4$ then no - $\phi^4$ and sine-Gordon are different theories and they have different solitons. There is no unique soliton for all two dimensional field theories - each theory has its own spectrum of solitons (none, one or many) just like it has its own spectrum of particle excitations. $\endgroup$ – Michael Brown Apr 30 '13 at 6:34

The sine-Gordon equation is a nonlinear hyperbolic partial differential equation in 1 + 1 dimensions involving the d'Alembert operator and the sine of the unknown function. It was originally introduced by Edmond Bour (1862) in the course of study of surfaces of constant negative curvature as the Gauss–Codazzi equation for surfaces of curvature −1 in 3-space, and rediscovered by Frenkel and Kontorova (1939) in their study of crystal dislocations known as the Frenkel–Kontorova

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Abstract It was conjectured that the classical bosonic string in AdS times a sphere has a special action variable which corresponds to the length of the operator on the field theory side. We discuss the analogous action variable in the sine–Gordon model. We explain the relation between this action variable and the Backlund transformations and show that the corresponding hidden symmetry acts

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We discuss the analogous action variable in the sine-Gordon model. We explain the relation between this action variable and the Backlund transformations and show that the corresponding hidden symmetry acts on We propose an action for a sine-Gordon-like theory, which reproduces the classical equations of motion of the Green-Schwarz-Metsaev-Tseytlin superstring on AdS5 × S5. The action is relativistically invariant.

This chapter computes a numerical solution by the method of lines (MOL), including detailed discussion of the Matlab routines and the numerical and graphical output. The sine–Gordon model is one of the simplest exactly solvable models of interacting relativistic fields, and the bosonic string propagating on is one of the simplest nonlinear string worldsheet theories. On the level of classical equations of motion these two models are equivalent. Consider the classical string propagating on In both cases (1) and (2), the sine-Gordon equation admits a Lax representation ∂ L ∂ t = [ L, M], where L and M are linear operators and [ L, M] = L M − M L. This enables one to obtain solutions of the Cauchy problem by using the inverse scattering method.
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REAL TWO-ZONE SOLUTIONS OF THE SINE-GORDON EQUATION B. A. Dubrovin and S. M. Natanzon UDC 517.9 Introduction The goal of the present paper is to enumerate all smooth real solutions of the sine-Gordon equation Uxy = sinu, expressed in terms of theta-functions of two variables.

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