Mode Excitation And Nonlinear Interaction In Continuous Systems

The partial differential equations satisfied by the dynamical evolution of continuous media and the corresponding boundary conditions constitute the definite solution problem.People often transform the partial differential equation in finite size system to eigenmode space to solve the infinite dimensional ordinary differential equation satisfied by mode coefficient.For certain boundary conditions(generally homogeneous and inhomogeneous of the first type,homogeneous and inhomogeneous of the second type,and a mixture of these boundary conditions),the symmetry of the system will also have an important impact on the properties of nonlinear dynamics.Onlinearity leads to couplings between modes,resulting in rich dynamic behavior and pattern structure of continuous media system.Symmetry will strengthen the coupling between some modes and weaken the coupling between others.There are numerous previous studies on mode excitation and evolution of one-dimensional nonlinear continuum systems or discrete lattice systems.In this paper,we focus on the problem of how the continuum field being excited and evolve under nonlinear effects and different symmetries.In this regard,we take the water surface wave system and the two-dimensional nonlinear Schrodinger equation as the research object and did the following two works:1.We consider the dynamical evolution of Faraday waves in a simple(integrable)rectangular tank and find a new class of Faraday waves(Alternately Localized Faraday Wave,ALFW).A full study of such systems will help us accumulate theoretical and experimental experiences and lay a foundation for the subsequent study of the dynamical behavior of water surface waves in non-integrable systems.In the experiment,the influence of the systematic error of the shaking table on the experiment is quantitatively analyzed,and the four main characteristics of ALFW wave are described in detail:(a)localization and "cantilever vibration”.This is the most obvious characteristic of the waveform that distinguishes it from other waveforms.In other words,the region with large amplitude not only distributes alternately in the rectangular direction of the water tank,but also presents a "cantilever" type oscillation in the narrow direction,where one end oscillates violently while the other end is flat and motionless.(b)two-mode DCT spectrum and phase locking.The special localization of ALFW wave is not made up of complex modes,but two simple modes--(12,0)mode and(8,1)mode.These two modes form a fixed phase difference through phase-locking,which generates a large interference enhancement and cancellation to form the ALFW waveform.(c)the "driven-stimulated" relationship between modes in the process of dynamic evolution.We will talk about this latter.(d)stability of ALFW wave with respect to parameters.ALFW wave can appear stably in a certain range of parameters,which is necessary for the experimental observation of this phenomenon.In theory,the linear and nonlinear dynamical mechanisms of ALFW wave' formation are given.The Mathieu equation for mode coefficients of water surface wave can be obtained by linearization analysis of water surface wave.The stability parameter space of the Mathieu equation with and without dissipation is discussed in detail.The weak nonlinear dynamic equation of low order is derived by small amplitude approximation of the nonlinear equation of water surface wave.On this basis,considering the requirements of physical images,a strongly nonlinear model is presented to describe the asymptotic dynamic behavior of ALFW waves.By systematically adjusting the undetermined parameters of the equations so that the boundary of the two types of parameter spaces obtained by numerical simulation and experiment agree with each other,the parameters of the equations in the phenomenological model are thus determined.In particular,the applicability of the phenomenological model to the description of ALFW waves is verified by comparison with the time series obtained respectively by experiments and obtained by numerical simulation with specific parameters.Finally,the sensitivity of the parameter space obtained by numerical simulation to the parameters obtained by fitting is analyzed in detail.On the basis of this work,we will further apply the experimental techniques and theoretical methods to the study of water surface wave mechanics of a non-integrable system(such as a tank with a stadium-shaped boundary).2.The nonlinear Schr?dinger equation in the stadium-shaped system is solved numerically,and the phenomenon of "exponential excitation" and "exponential recurrence" of modes driven by a single mode(primary mode,as the initial state)are found.Similarly,we give both linear and nonlinear dynamical mechanisms to explain these two phenomena.The evolution equation of the primary mode and its analytic solution can be obtained by linear analysis.We find that the instability caused by the coupling of the eigenmode with the primary mode under the nonlinear action comes from two types of mechanisms:(a)sine excitation caused by the symmetry and(b)exponential excitation caused by parametric instability(complex "Mathieu" equation).The analysis of the specific model(primary mode p=200)shows that the phenomenon of "exponential excitation" is precisely because the parameters of the exponential growth model fall into the unstable region of the parameter space of the complex "Mathieu" equation.Finally,we find that the "recurrence phenomenon" occurs between the primary mode and the instability mode,and the energy of the two modes increase or decrease exponentially through nonlinear coupling.By means of multi-scale perturbation,the asymptotic behavior of the evolution of two modes is obtained.The phase space structure is obtained by numerical simulation of the non-perturbation equation,which is symmetric about the two modes,forming an interlocking "comb structure".This indicates that the roles of "primary" and "instability" of the two modes are not absolute,but constantly changing in the evolutionary process.This mechanism of role transition allows the applicability of the complex "Mathieu" equation to shift from the "instability" mode to the previous "primary" mode whenever the energy is transferred from one mode to another,thus fully explaining the exponential excitation and decay in the recurrence process.Through these two works,we find that for the rectangular water surface wave system with high symmetry,the nonlinear interaction between modes can generate a "master-slave" mode pair,which facilitates the construction of strong nonlinear models to describe the dynamics of the system.For the stadium-shaped system with weak symmetry,the mechanism leading to mode instability depends on the residual symmetry,that is,symmetry makes the coupled modes lose their stability through sinusoidal excitation,while the non-coupled modes lose stability through exponential instability..