Analytical solutions for the generalized sine-Gordon equation with variable coefficients

Abstract The sine-Gordon equation in its standard form possesses soliton solution because of the delicate balance between dispersion and nonlinearity that is preserved in the equation. However, in practical situations, where the sine-Gordon equation is used to model either the movement of dislocations in crystals or the magnetic flux waves in Josephson junctions, because of impurities and non-uniformity, the delicate balance between dispersion and nonlinearity can not be maintained. This means that the model will not be the standard sine-Gordon equation, but, a generalized form of the sine-Gordon equation with variable coefficients that could depend on both time and space variables. In this paper, we study such a generalized sine-Gordon equation with the goal of obtaining soliton solutions. Employing the variable transformation used by Hirota [1, 2] in a judicious manner, we are able to obtain single soliton and multi-soliton solutions for the generalized sine-Gordon equation for a choice of variable coefficients.

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