Induced generalized exact boundary synchronizations for a coupled system of wave equations

Taking a coupled system of wave equations with Dirichlet boundary controls as an example,by splitting and merging some synchronization groups of the state variables cor-responding to a given generalized synchronization matrix,this paper introduces two kinds of induced generalized exact boundary synchronizations to better determine its generalized exactly synchronizable states.

Localized waves in three-component coupled nonlinear Schrdinger equation

We study the generalized Darboux transformation to the three-component coupled nonlinear Schr ¨odinger equation.First-and second-order localized waves are obtained by this technique.In first-order localized wave,we get the interactional solutions between first-order rogue wave and one-dark,one-bright soliton respectively.Meanwhile,the interactional solutions between one-breather and first-order rogue wave are also given.In second-order localized wave,one-dark-one-bright soliton together with second-order rogue wave is presented in the first component,and two-bright soliton together with second-order rogue wave are gained respectively in the other two components.Besides,we observe second-order rogue wave together with one-breather in three components.Moreover,by increasing the absolute values of two free parameters,the nonlinear waves merge with each other distinctly.These results further reveal the interesting dynamic structures of localized waves in the three-component coupled system.

Exact Boundary Synchronization by Groups for a Kind of System of Wave Equations Coupled with Velocities

This paper deals with the exact boundary controllability and the exact boundary synchronization for a 1-D system of wave equations coupled with velocities.These problems can not be solved directly by the usual HUM method for wave equations,however,by transforming the system into a first order hyperbolic system,the HUM method for 1-D first order hyperbolic systems,established by Li-Lu(2022)and Lu-Li(2022),can be applied to get the corresponding results.

Exact Internal Controllability and Synchronization for a Coupled System of Wave Equations

In this paper the exact internal controllability for a coupled system of wave equations with arbitrarily given coupling matrix is established.Based on this result,the exact internal synchronization and the exact internal synchronization by p-groups are successfully considered.

BIFURCATIONS AND NEW EXACT TRAVELLING WAVE SOLUTIONS OF THE COUPLED NONLINEAR SCHRDINGER-KdV EQUATIONS

By using the method of dynamical system, the exact travelling wave solutions of the coupled nonlinear Schrdinger-KdV equations are studied. Based on this method, all phase portraits of the system in the parametric space are given. All possible bounded travelling wave solutions such as solitary wave solutions and periodic travelling wave solutions are obtained. With the aid of Maple software, the numerical simulations are conducted for solitary wave solutions and periodic travelling wave solutions to the coupled nonlinear Schrdinger-KdV equations. The results show that the presented findings improve the related previous conclusions.

A Note on the Indirect Controls for a Coupled System of Wave Equations

By a procedure of successive projections,the authors decompose a coupled system of wave equations into a sequence of sub-systems.Then,they can clarify the indirect controls and the total number of controls.Moreover,the authors give a uniqueness theorem of solution to the system of wave equations under Kalman’s rank condition.

Generalized Exact Boundary Synchronization for a Coupled System of Wave Equations with Dirichlet Boundary Controls

This paper deals with the generalized exact boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls in the framework of weak solutions.A necessary and sufficient condition for the generalized exact boundary synchronization is obtained,and some results for its generalized exactly synchronizable states are given.

Partial approximate boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls

In this paper,we propose the concept of partial approximate boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls,and make a deep discussion on it.We analyze the relationship between the partial approximate boundary synchronization and the partial exact boundary synchronization,and obtain sufficient conditions to realize the partial approximate boundary synchronization and necessary conditions of Kalman’s criterion.In addition,with the help of partial synchronization decomposition,a condition that the approximately synchronizable state does not depend on the sequence of boundary controls is also given.

Determination of generalized exact boundary synchronization matrix for a coupled system of wave equations

For a coupled system of wave equations with Dirichlet boundary controls,this paper deals with the possible choice of its generalized synchronization matrices so that the admissible generalized exact boundary synchronizations for this system are obtained.

On the Synchronizable System

In this paper,the synchronizable system is defined and studied for a coupled system of wave equations with the same wave speed or with different wave speeds.