Exact Solutions to the Langmuir Lattice Equation

Our primary goal is to develop a large class of exact solutions to the integrable nonlinear partial differential-difference equation, known as the Langmuir lattice equation (also known as the Volterra lattice equation). This equation has important applications in electromagnetic wave propagation in plasmas (ionized gases) and in one dimensional crystals, where the nearest neighbors interact with forces that depend on the difference of their displacements. Using a recently developed general approach, we construct a formula for exact solutions for certain wave solutions to the Langmuir lattice equation. Our formula uses three matrices as input and contains so-called matrix exponentials. Such solutions represent solitary waves (solitons) with particle-like behavior. We investigate the relationship between the parameters appearing in our solution formula and the physical properties of waves represented by our solutions. In particular, we analyze the relationship between the velocities of the individual solitons in our solutions and the eigenvalues of one of the input matrices. When our solutions are positive everywhere and at all times, with the help of our formula we construct a solution formula for another nonlinear partial differential-difference equation known as the Kac-van Moerbeke lattice equation. Such exact solutions are important not only physically but also mathematically, as they may be used as test functions to check the accuracy of numerical methods to solve nonlinear partial differential-difference equations.