On a partially synchronizable system for a coupled system of wave equations in one dimension

In this paper, we study a partially synchronizable system for a coupled system of wave equations with different wave speeds in the framework of classical solutions in one dimensional. A partially synchronizable system is defined as a system with at least one partial synchronized solutions. In fact, we cannot consider partial synchronization as the case that the system has the same wave speeds, because the influence of different wave speeds cause only some of the function in a given space being in a partially synchronized state, rather than all functions. Therefore, we can only consider under what conditions the coupled system can have partially synchronized solutions. We will consider it in two ways. On the one hand, under the necessary conditions, we obtain an unclosed characteristic equation associated with the partially synchronizable state. We add conditions to the wave speed matrix and coupling matrix to make the equation closed. From this, the characteristic function can be obtained, and all partially synchronized solutions are obtained; then we obtain the conditions under which the initial value should be satisfied. On the other hand, we consider a system of three variables first, where there are only two synchronized variables. By subtracting them to obtain a new variable, the problem can be transformed into the problem wherein the system that satisfies the new variable should have only zero solutions. Then solving this problem can lead to obtaining the conditions required for a partially synchronized solution. After extending it to the case of $ N $ variables, similar conclusions can be obtained.