Abstract
We consider the hierarchy of partial differential equations with arbitrary power-low nonlinearity which can be used for description of the propagation pulse in optical fiber. The Cauchy problem for these partial differential equations cannot be solved by the inverse scattering transform and we look for exact solutions of differential equations using the traveling wave reduction. It is proven that all equations of the hierarchy have exact solutions in the form of periodic and solitary waves that are determined by means of the elliptic functions. A more detailed study of the hierarchy is presented for the equations of the second, fourth and sixth order. The parameter values for existence of exact solutions for these equations are given. Exact solutions of differential equations are expressed in terms of the Weierstrass elliptic function. A formula for describing solitary waves is also given. Exact solutions in the form of periodic and solitary waves for differential equations are illustrated.
Published Version
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