April, 2007 1 Comments about nonlinear modal analysis of partial differential equations Roberto Suárez-Antola (1) Nonlinear modal analysis in a generalized sense In a generalized sense modal analysis of partial differential equations is a procedure that allows (Attle-Jackson, 1991; Zauderer, 2006): (a) The expansion of the solutions (fields) of partial differential equations in time and space variables as a series of given space functions weighted by time functions known as modal amplitudes. (b) The derivation of evolution equations for the mode amplitudes (a set of ordinary differential equations). As the space functions are known, by means of modal analysis a complex space- time field dynamics is reduced to the study of the evolution of a representative point in the space of mode amplitudes. If the original partial differential equation is nonlinear, the ordinary equations for mode amplitudes are nonlinear also: in this case we have nonlinear modal analysis. 1 (2) Nonlinear modal analysis in a restricted sense In a restricted sense, modal analysis is a procedure applied to the different fields that give the space-time evolution of a distributed parameters system with a bounded space domain D, expanding the fields in series of eigenfunctions of a certain suitably chosen linear operator  L . The domain U of this operator, which is a set of space dependent functions defined in D, is so defined to include the boundary conditions of the problem. These boundary conditions must be time independent, although they can be non-homogeneous. The abovementioned eigenfunctions (spatial modes) depend only of the space coordinates and, as already said, are defined in the bounded domain D. In each expansion corresponding to a given field, these spatial modes are weighted by unknown time dependent modal amplitudes. 1 An often used procedure to obtain the equations for mode a mplitudes is the application of Galerkin’s method (Gershenfeld, 1999). In general, in the evolution equations thus obtained, modal amplitudes appear coupled, even in the linear approximation.