Partial Differential Equations in Applied Mathematics (Jun 2022)
The generalized nonlinear Schrödinger-like equation of cosmogonical body forming: Justification and determination of its particular solutions
Abstract
This work justifies the generalized Schrödinger-like equation with logarithmic nonlinearity in the statistical theory of cosmogonical body formation. Within the framework of this theory, the models and evolution equations of the statistical mechanics have been proposed, while well-known problems of gravitational condensation of infinite distributed cosmic substances have been solved on the basis of the proposed statistical model of spheroidal bodies. Equation relative to the complete time-derivative of the common (hydrodynamic plus anti-diffusion) velocity is obtained. Taking into account equations obtained the generalized nonlinear time-dependent Schrödinger-like equation of cosmogonical body formation is derived. This paper investigates different particular forms of the generalized nonlinear time-dependent Schrödinger-like equation corresponding to the respective dynamical states of a forming spheroidal body. The cubic nonlinear time-dependent Schrödinger equation describing cosmogonical body formation in the state of nonlinear wave disturbances is derived. The soliton solution of the one-dimensional cubic nonlinear Schrödinger equation is obtained her e. This paper considers the determination of nonlinear wave solutions from the point of view of the inverse scattering problem method as well as the modified (G/G′)-expansion method.
