PERIODIC, SOLITARY AND DISCONTINUOUS PERIODIC SOLUTION OF NONLINEAR WAVES IN THE ATMOSPHERE AND THEIR EXISTENCE (I)——ON THE PSEUDO-ENERGY, THE PSEUDO-ENERGY INFLUENCE FUNCTION AND THEIR USE

Abstract

In this paper various non-dispersion solutions of nonlinear waves in the atmosphere are discussed. We turn the nonlinear partial differential equations into the nonlinear ordinary differential equations after the phase angle function has been introduced. The nature around the equilibrium points and singular points of these ordinary differential equations is discussed and various analytic expressions of the nondispersion solutions are obtained. In part (Ⅰ), two problems are dealt with mainly. (ⅰ) The relation between pseudo-energy and the pseudo-energy influence function and nonlinear waves is discussed. Through the discussion of the pseudo-energy influence function, we can determine the existential condition of the periodic solution, the solitary wave solution, the discontinuous periodic solution and the discontinuous solitary wave solution. We also indicate that if there exists an external source, which occasions infinitely small changes in the pseudo-energy influence function, the nonlinear solitary wave can be produced. (ⅱ) The existence of the discontinuous periodic solution is discussed and the method of function approximation is used. In case the analytic solution is unable to be obtained, the approximate solution can be obtained usually by using the Taylor expansion, but this method can bring many troubles. In this paper we derive the approximate solution by using the function-fitting method for the pseudo-energy function. This method avoids the defects of Taylor expansion.