On the linearized korteweg-devries equation

An extensive theory has developed about localized nondissipative waves called solitons. In 1895 Korteweg and deVries derived a third order nonlinear partial differential equation that demonstrated solitons exist. In this work, we introduce the concept of solitons and derive one solution of the nonlinear Korteweg-deVries equation. However, the difficulty of the nonlinear term leads us to consider a linearized Korteweg-deVries equation for an infinite space initial-boundary value problem and a finite space initial-boundary value problem. We use methods of Fourier transforms and separation of variables to solve these third order problems. For the infinite case our solution involves the special Airy function. Thus, we derive the Airy function from a second order differential equation and examine some of its properties. However, upon solving the finite initial-boundary value problem by separation of variables, we fail to obtain the usual orthogonality of the eigenfunctions since the eigenvalue problem is not self-adjoint. Therefore, we invoke a theorem which gives us a bi-orthogonality relationship between the eigenfunctions of the eigenvalue problem and the eigenfunctions of the adjoint problem provided the Green's function has simple poles at each eigenvalue of the eigenvalue problem.