Periodic Dynamics Of A Nonlinear Partial Differential Equation

Nonlinear Schr?dinger equation with derivatives is widely used in the fields of fluid mechanics and optics.In this paper,we study the periodic dynamics of the derivative nonlinear Schr?dinger equation with a periodic coefficient,including the existence of large-amplitude periodic solutions and small-amplitude periodic solutions and the estimates of their rotation numbers.Considering the coherent structure of uniform propagation,we obtain the evolution equations of amplitude and phase,respectively.The amplitude equation is a second-order differential equation with singularity,and the phase functions depend on the variation of the amplitude functions.To obtain periodic solutions,we must prove that both the amplitude equation and the phase equation have periodic solutions.Firstly,we prove that the amplitude evolution equation has infinite large amplitude periodic solutions by using Poincaré-Birkhoff twist theorem.These large amplitude periodic solutions depend on integral constants.By using the continuity of the solutions with respect to the parameters,we can obtain infinite periodic solutions of the phase evolution equation.By estimating the solutions,we prove that the rotation numbers of these large-amplitude periodic solutions tend to infinity.Secondly,we study the small-amplitude periodic solutions by means of averaging theory.Thus,we not only have proved the existence of infinite small-amplitude periodic solutions,but also have given the exact expressions of the small-amplitude periodic solutions.In addition,we also obtain these small-amplitudes with nontrivial phases.The rotation numbers of the amplitude-periodic solution tend to a certain constant.Compared with the existing results,the innovation of this paper includes two aspects: on the one hand,we consider the derivative nonlinear Schr?dinger equation with variable coefficients,and the related integrable theory can not be directly applied;on the other hand,the large-amplitude and small-amplitude solutions obtained by us have nontrivial phase.Moreover,we characterize their rotation numbers.