Mark J. Ablowitz

Solitons and the inverse scattering transform

Abstract : Under appropriate conditions, ocean waves may be modeled by certain nonlinear evolution equations that admit soliton solutions and can be solved exactly by the inverse scattering transform (IST). The theory of these special equations is developed in five lectures. As physical models, these equations typically govern the evolution of narrow-band packets of small amplitude waves on a long (post-linear) time scale. This is demonstrated in Lecture I, using the Korteweg-deVries equation as an example. Lectures II and III develop the theory of IST on the infinite interval. The close connection of aspects of this theory to Fourier analysis, to canonical transformations of Hamiltonian systems, and to the theory of analytic functions is established. Typical solutions, including solitons and radiation, are discussed as well. With periodic boundary conditions, the Korteweg-deVries equation exhibits recurrence, as discussed in Lecture IV. The fifth lecture emphasizes the deep connection between evolution equations solvable by IST and Painleve transcendents, with an application to the Lorenz model.

A connection between nonlinear evolution equations and ordinary differential equations of P‐type. II

It is known through the inverse scattering transform that certain nonlinear differential equations can be solved via linear integral equations. Here it is demonstrated ’’directly,’’ i.e., without the Jost‐function formalism that the solution of the linear integral equation actually solves the nonlinear differential equation. In particular, this extends the scope of inverse scattering methods to ordinary differential equations which are found to be of Painleve type. Some global properties of these nonlinear ODE’s are obtained rather easily by this approach.

Nonlinear differential–difference equations and Fourier analysis

The conceptual analogy between Fourier analysis and the exact solution to a class of nonlinear differential–difference equations is discussed in detail. We find that the dispersion relation of the associated linearized equation is prominent in developing a systematic procedure for isolating and solving the equation. As examples, a number of new equations are presented. The method of solution makes use of the techniques of inverse scattering. Soliton solutions and conserved quantities are worked out.

Nonlinear differential−difference equations

A method is presented which enables one to obtain and solve certain classes of nonlinear differential−difference equations. The introduction of a new discrete eigenvalue problem allows the exact solution of the self−dual network equations to be found by inverse scattering. The eigenvalue problem has as its singular limit the continuous eigenvalue equations of Zakharov and Shabat. Some interesting differences arise both in the scattering analysis and in the time dependence from previous work.

On the evolution of packets of water waves

We consider the evolution of packets of water waves that travel predominantly in one direction, but in which the wave amplitudes are modulated slowly in both horizontal directions. Two separate models are discussed, depending on whether or not the waves are long in comparison with the fluid depth. These models are two-dimensional generalizations of the Korteweg-de Vries equation (for long waves) and the cubic nonlinear Schrodinger equation (for short waves). In either case, we find that the two-dimensional evolution of the wave packets depends fundamentally on the dimensionless surface tension and fluid depth. In particular, for the long waves, one-dimensional (KdV) solitons become unstable with respect to even longer transverse perturbations when the surface-tension parameter becomes large enough, i.e. in very thin sheets of water. Two-dimensional long waves (‘lumps’) that decay algebraically in all horizontal directions and interact like solitons exist only when the one-dimensional solitons are found to be unstable. The most dramatic consequence of surface tension and depth, however, occurs for capillary-type waves in sufficiently deep water. Here a packet of waves that are everywhere small (but not infinitesimal) and modulated in both horizontal dimensions can ‘focus’ in a finite time, producing a region in which the wave amplitudes are finite. This nonlinear instability should be stronger and more apparent than the linear instabilities examined to date; it should be readily observable. Another feature of the evolution of short wave packets in two dimensions is that all one-dimensional solitons are unstable with respect to long transverse perturbations. Finally, we identify some exact similarity solutions to the evolution equations.

Explicit solutions of Fisher's equation for a special wave speed

The travelling waves for Fishers equation are shown to be of a simple nature for the special wave speeds\(c = \pm 5/\sqrt {(6)} \). In this case the equation is shown to be of Painleve type, i.e. solutions admit only poles as movable singularities. The general solution for this wave speed is found and a method is presented that can be applied to the solution of other nonlinear equations of biological and physical interest.

Two‐dimensional lumps in nonlinear dispersive systems

Two‐dimensional lump solutions which decay to a uniform state in all directions are obtained for the Kadomtsev–Petviashvili and a two‐dimensional nonlinear Schrodinger type equation. The amplitude of these solutions is rational in its independent variables. These solutions are constructed by taking a ’’long wave’’ limit of the corresponding N‐soliton solutions obtained by direct methods. The solutions describing multiple collisions of lumps are also presented.

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.

Solitons and rational solutions of nonlinear evolution equations

Rational solutions of certain nonlinear evolution equations are obtained by performing an appropriate limiting procedure on the soliton solutions obtained by direct methods. In this note specific attention is directed at the Korteweg–de Vries equation. However, the methods used are quite general and apply to most nonlinear evolution equations with the isospectral property, including certain multidimensional equations. In the latter case, nonsingular, algebraically decaying, soliton solutions can be constructed.

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.