Dynamics from a mathematical model of a two-state gas laser

• Published in: Physica A 2018-05, Vol.497, p.26-40
• Main Authors: Kleanthous, Antigoni, Hua, Tianshu, Manai, Alexandre, Yawar, Kamran, Van Gorder, Robert A.
• Format: Article
• Language: English
• Subjects: Nonlinear dynamics; Numerical simulations; Stability analysis; Two-state gas laser
• ISSN: 0378-4371; 1873-2119
• DOI: 10.1016/j.physa.2017.12.110

Motivated by recent work in the area, we consider the behavior of solutions to a nonlinear PDE model of a two-state gas laser. We first review the derivation of the two-state gas laser model, before deriving a non-dimensional model given in terms of coupled nonlinear partial differential equations. We then classify the steady states of this system, in order to determine the possible long-time asymptotic solutions to this model, as well as corresponding stability results, showing that the only uniform steady state (the zero motion state) is unstable, while a linear profile in space is stable. We then provide numerical simulations for the full unsteady model. We show for a wide variety of initial conditions that the solutions tend toward the stable linear steady state profiles. We also consider traveling wave solutions, and determine the unique wave speed (in terms of the other model parameters) which allows wave-like solutions to exist. Despite some similarities between the model and the inviscid Burger’s equation, the solutions we obtain are much more regular than the solutions to the inviscid Burger’s equation, with no evidence of shock formation or loss of regularity. •A non-dimensional PDE system for a two-state gas laser is derived.•Steady states of this PDE system are classified showing a linear profile in space is stable.•Numerical simulations are performed and resulting solutions tend to the linear steady state.•Solutions maintain regularity and do not show signs of turbulence.•Traveling wave solutions are also considered.