Athanassios S. Fokas

On the inverse scattering transform of multidimensional nonlinear equations related to first‐order systems in the plane

The inverse problem associated with a rather general system of n first‐order equations in the plane is linearized. When the system is hyperbolic, this is achieved by utilizing a Riemann–Hilbert problem; similarly, a ‘‘∂’’ (DBAR) problem is used when the system is elliptic. The above result can be employed to linearize the initial value problem associated with a variety of physically significant equations in 2+1, i.e., two spatial and one temporal dimensions. Concrete results are given for the n‐wave interaction in 2+1 and for various forms of the Davey–Stewartson equations. Lump solutions (solitons in 2+1) of the latter equation are given a definitive spectral characterization and are obtained through a linear system of algebraic equations.

On a unified approach to transformations and elementary solutions of Painlevé equations

An algorithmic method is developed for investigating the transformation properties of second‐order equations of Painleve type. This method, which utilizes the singularity structure of these equations, yields explicit transformations which relate solutions of the Painleve equations II–VI, with different parameters. These transformations easily generate rational and other elementary solutions of the equations. The relationship between Painleve equations and certain new equations quadratic in the second derivative of Painleve type is also discussed.

On the use of isospectral eigenvalue problems for obtaining hereditary symmetries for Hamiltonian systems

We present an algorithmic method for obtaining an hereditary symmetry (the generalized squared‐eigenfunction operator) from a given isospectral eigenvalue problem. This method is applied to the n×n eigenvalue problem considered by Ablowitz and Haberman and to the eigenvalue problem considered by Alonso. The relevant Hamiltonian formulations are also determined. Finally, an alternative method is presented in the case two evolution equations are related by a Miura type transformation and their Hamiltonian formulations are known.

Hodograph transformations of linearizable partial differential equations

In this paper an algorithmic method is developed for transforming quasilinear partial differential equations of the form

The inverse scattering transform for multidimensional (2+1) problems

u_t = g( u )u_{nx} + f( u, u_x , cdots ,u_{( n - 1)x } ),, u_{mx} equiv partial ^m u/partial x^m

ON THE INVERSE SCATTERING AND DIRECT LINEARIZING TRANSFORMS FOR THE KADOMTSEV-PETVIASHVILI EQUATION

, where

The direct linearization of a class of nonlinear evolution equations

dg/du notequiv 0

On the limit from the intermediate long wave equation to the Benjamin–Ono equation

, into semilinear equations (i.e., equations of the above form with

On the periodic intermediate long wave equation

g( u ) = 1 )

The direct linearizing transform and the Benjamin-Ono equation

. This crucially involves the use of hodograph transformations (i.e., transformations involving the interchange of dependent and independent variables). Furthermore, the most general quasilinear equation of the above form is found that can be mapped via a hodograph transformation to a semilinear form.This algorithm provides a method for establishing whether a given quasilinear equation is linearizable, i.e., is solvable in terms of either a linear partial differential equation or of a linear integral equation. In particular, this method is used to show how the Painleve tests may be applied to quasilinear equations. This appears to resolve the problem that solutions of lineariz...