Abstract
We consider the mathematical model with arbitrary power of nonlinearity which is described by the generalized Schrödinger equation. The Cauchy problem for this equation is not solved by the inverse scattering transform and we use the traveling wave reduction for the nonlinear partial differential equation. Using the traveling wave solutions we find the first integrals for the system of equations corresponding to real and imaginary parts of the profile pulse for a complex function. The system of equations is reduced by means of transformations to the first-order nonlinear ordinary differential equations with solutions expressing via the Weierstrass and Jacobi elliptic functions. The influence of the nonlinearity degree on the structure of periodic and solitary waves is studied. It is demonstrated that the degree of nonlinearity allows to control the amplitude and pulse length of periodic and solitary waves.
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