Linear superpositions in nonlinear wave equations (Q1381213)

In the note it is shown: (a) the existence of \textit{linear} generalized conditional symmetries (GCS) for a large class of nonlinear equations. In particular these equations are \[ L_1(\partial_x, \partial_t)u= L_2(\partial_x, \partial_t)(u_x\cdot u_t) \] and \[ L(\partial_x, \partial_t)u=f_1(u_t+ku_x)+f_2(u_t-ku_x), \] where \(L_1, L_2, L\) are arbitrary linear differential operators of \(\partial_x\) and \(\partial_t\), \(f_1\), \(f_2\) are arbitrary differentiable functions of arguments indicated and \(k\) is constant. (b) It is explicitly given the general form of all nonlinear evolution equations admitting a given linear GCS. A particular case of this construction is that the most general evolution equation of degree \(\leq 3\) admitting the linear GCS \(\sigma=u_{xxx}+u_x\) is \[ u_t=u_{xx}F_1(u_{xx}^2+u_x^2, u_{xx}+u) +u_xF_2(u_{xx}^2+u_x^2, u_{xx}+u)+ F_3(u_{xx}^2+u_x^2, u_{xx}+u) \] where \(F_1\), \(F_2\), \(F_3\) are arbitrary differentiable functions. Physically interesting examples are considered as well as general assertions.