A NEW PERTURBATIVE APPROACH TO NONLINEAR PARTIAL-DIFFERENTIAL EQUATIONS

This paper shows how to solve some nonlinear wave equations as perturbation expansions in powers of a parameter that expresses the degree of nonlinearity. For the case of the Burgers equation u(t) + uu(x) = u(xx), the general nonlinear equation u(t) + u-delta-u(x) = u(xx) is considered and expanded in powers of delta. The coefficients of the delta series to sixth order in powers of delta is determined and Pade summation is used to evaluate the perturbation series for large values of delta. The numerical results are accurate and the method is very general; it applies to other well-studied partial differential equations such as the Korteweg-de Vries equation, u(t) + uu(x) = u(xxx).