Algebraic and Geometric Properties of Matrix Solutions of Nonlinear Wave Equations

We improve the construction of exact matrix solutions for nonlinear wave equations by using unitary anti-Hermitian and anticommuting matrices. We prove the theorem that constructs the matrix functions un satisfying the nonlinear wave equation for a set of special potentials. In this case, the graph of complex solution u1 has a soliton-like form with a finite number of coils. Exponential representation of matrix solutions un is associated with continuous rotations that can be used for describing intrinsic rotations and state changes of elementary particles. We also prove the theorem on the decomposition of continuous rotation (described by solution u2) onto three simultaneous rotations about coordinate vectors. Each of the three constructed matrix solutions u3 is also decomposed into the triplet of elementary matrix solutions.