Exact Solution for the Vibrations of a Nonlinear Continuous Model String

An exact solution is given for the partial differential equation $$y_{\mathrm{tt}}{\text{=[1+εy}}_x{]}^{\alpha }{y}_{\mathrm{xx}},$$ which describes the standing vibrations of a finite, continuous, and nonlinear string. The nonlinearity studied, [1 + εy_x]^α, was motivated by the work of Fermi, Pasta, and Ulam (1955), where they reported on numerical studies of the ``equipartition of energy'' in nonlinear systems. To obtain the solution, the above equation is transformed into a linear equation by inverting the roles of the dependent (u = y<sub>x</sub> and v = y<sub>t</sub>) and independent (x and t) variables. Riemann's method of integration is applied to the problem and the solutions for t and x are written as integrals. The nature of the ``inverse Riemann plane,'' how it is related to the initial conditions, and how one unfolds it, are discussed in detail. A general procedure is described for reinverting the solution, so that y can be written as a function of x and t. It is illustrated to order ε for the above problem. It is demonstrated that y_xx becomes singular, that is, y_x develops a discontinuity after an elapsed time or order (1/ε). The methods described are applicable to any nonlinear string where the coefficient of y_xx is a function of y_x only. The effect of higher spatial derivatives on the formation of the singularity is discussed.