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A new method for solving boundary value problems has recently been introduced by the first author. Although this method was first developed for non-linear integrable PDEs (using the crucial notion of a Lax pair), it has also given rise to new analytical and numerical techniques for linear PDEs. Here we review the application of the new method to linear elliptic PDEs, using the modified Helmholtz equation as an illustrative example.
Almost forty years ago an ingenious new method was discovered for the solution of the initial value problem of the Korteweg–de Vries (KdV) equation [GGKM67]. This new method, which was later called the inverse scattering transform (IST) method was based on the mysterious fact that the KdV equation is equivalent to two linear eigenvalue equations called a Lax pair (in honor of Peter Lax, [Lax68] who first understood that the IST method was the consequence of this remarkable property). The KdV equation belongs to a large class of nonlinear equations which are called integrable. Although there exist several types of integrable equations, which include PDEs, ODEs, singular-integrodifferential equations, difference equations and cellular automata, the existence of an associated Lax pair provides a common feature of all these equations.
After several attempts to extend the inverse scattering transform method from initial value problems to boundary value problems, a unified method for solving boundary value problems for linear and integrable nonlinear PDEs was introduced by the first author in [Fok97] and reviewed in [Fok08].
P.C. Bressloff, Department of Mathematical Sciences, Loughborough University, Loughborough, Leics. LE11 3TU, U.K.,
A.S. Fokas, Department of Mathematics, Imperial College, University of London, London, SW7 2BZ, U.K.