A class of coupled nonlinear Schrödinger equations: Painlevé property, exact solutions, and application to atmospheric gravity waves

Abstract

The Painlevé integrability and exact solutions to a coupled nonlinear Schrödin-ger (CNLS) equation applied in atmospheric dynamics are discussed. Some parametric restrictions of the CNLS equation are given to pass the Painlevé test. Twenty periodic cnoidal wave solutions are obtained by applying the rational expansions of fundamental Jacobi elliptic functions. The exact solutions to the CNLS equation are used to explain the generation and propagation of atmospheric gravity waves.

Exact solutions and residual symmetries of the Ablowitz–Kaup–Newell–Segur system

“…The Painlevé analysis is considered as one of the most powerful and systematic methods to analyze the integrability of nonlinear systems. [17][18][19][20][21][22] Several years ago, Painlevé and his contemporaries identified a class of second-order nonlinear ordinary differential equations, in which the movable singularities exhibited by the general solution are only poles in the complex time plane. [17] Such systems are called the Painlevé type, possessing the Painlevé property.…”

Exact solutions of atmospheric (3+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq equations with viscosity

“…They can break and produce organized cloud and have a great effect on severe hazardous weather and climate, such as rainstorm of typhoon, deep convection and orographic precipitation. [1][2][3][4] The governing equations for gravity waves consist of the nonlinear horizontal momentum equations, mass continuity equations, and the nonlinear thermodynamic equations. For perfect atmosphere, the governing equations of gravity waves can be considered as the (3+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq (INHB) equations as follows 5…”

Residual symmetries of the modified Korteweg-de Vries equation and its localization

“…Finding exact solutions of NLEEs has become one of the most exciting and extremely active areas of research investigation. Many effective methods for obtaining exact solutions of the mKdV equation and other NLEEs have been presented, such as the classical and non-classical Lie group approaches [2,3], truncated Painlevé expansion method [4,5], Bäcklund transformation [6], variational method, Multi-linear variable separa-tion approach, function expansion method [7], the mapping method and so on [8][9][10][11][12][13]. The Painlevé analysis and the truncated Painlevé analysis is considered as one of the most powerful and systematic methods to analyze the integrability of nonlinear systems.…”