The Inverse Scattering Transform-Fourier Analysis for Nonlinear Problems

A systematic method is developed which allows one to identify certain important classes of evolution equations which can be solved by the method of inverse scattering. The form of each evolution equation is characterized by the dispersion relation of its associated linearized version and an integro-differential operator. A comprehensive presentation of the inverse scattering method is given and general features of the solution are discussed. The relationship of the scattering theory and Backlund transformations is brought out. In view of the role of the dispersion relation, the comparatively simple asymptotic states, and the similarity of the method itself to Fourier transforms, this theory can be considered a natural extension of Fourier analysis to nonlinear problems.     

Pole Dynamics for Elliptic Solutions of the Korteweg-deVries Equation

The real, nonsingular elliptic solutions of the Korteweg-de Vries equation are studied through the time dynamics of their poles in the complex plane. The dynamics of these poles is governed by a dynamical system with a constraint. This constraint is solvable for any finite number of poles located in the fundamental domain of the elliptic function, often in many different ways. Special consideration is given to those elliptic solutions that have a real nonsingular soliton limit. This revised version was published online in July 2006 with corrections to the Cover Date.     

Three-Phase Solutions of the Kadomtsev–Petviashvili Equation

The Kadomtsev–Petviashvili (KP) equation is known to admit explicit periodic and quasiperiodic solutions with N independent phases, for any integer N , based on a Riemann theta-function of N variables. For N =1 and 2, these solutions have been used successfully in physical applications. This article addresses mathematical problems that arise in the computation of theta-functions of three variables and with the corresponding solutions of the KP equation. We identify a set of parameters and their corresponding ranges, such that every real-valued, smooth KP solution associated with a Riemann theta-function of three variables corresponds to exactly one choice of these parameters in the proper range. Our results are embodied in a program that computes these solutions efficiently and that is available to the reader. We also discuss some properties of three-phase solutions.     

The modulation instability revisited

The modulational instability (or “Benjamin-Feir instability”) has been a fundamental principle of nonlinear wave propagation in systems without dissipation ever since it was discovered in the 1960s. It is often identified as a mechanism by which energy spreads from one dominant Fourier mode to neighboring modes. In recent work, we have explored how damping affects this instability, both mathematically and experimentally. Mathematically, the modulational instability changes fundamentally in the presence of damping: for waves of small or moderate amplitude, damping (of the right kind) stabilizes the instability. Experimentally, we observe wavetrains of small or moderate amplitude that are stable within the lengths of our wavetanks, and we find that the damped theory predicts the evolution of these wavetrains much more accurately than earlier theories. For waves of larger amplitude, neither the standard (undamped) theory nor the damped theory is accurate, because frequency downshifting affects the evolution in ways that are still poorly understood.     

Long Internal Waves in Fluids of Great Depth

An equation is derived that governs the evolution in two spatial dimensions of long internal waves in fluids of great depth. The equation is a natural generalization of Benjamin's (1967) one-dimensional equation, and relates to it in the same way that the equation of Kadomtsev and Petviashvili relates to the Kortewegde-Vries equation. The stability of one-dimensional solitons with respect to long transverse disturbances is studied in the context of this equation. Solitons are found to be unstable with respect to such perturbations in any system in which the phase speed is a minimum (rather than a maximum) for the longest waves. Internal waves do not have this property, and are not unstable with respect to such perturbations.     

Explosive Instability due to four-wave mixing

It is known that an explosive instability can occur when nonlinear waves propagate in certain media that admit 3-wave mixing. The purpose of this Letter is to show that explosive instabilities can occur even in media that admit no 3-wave mixing. Instead, the instability is caused by 4-wave mixing: four resonantly interacting wave trains gain energy from a background, and all blowup in a finite time. Unlike singularities associated with self-focussing, these singularities can occur with no spatial structure-the waves blowup everywhere in space simultaneously. We have not yet investigated the effect of spatial structure on a 4-wave explosive instability.     

Asymptotic Solutions of the Korteweg-deVries Equation

The long-time asymptotic solution of the Korteweg-deVries equation, corresponding to initial data which decay rapidly as |x|→∞ and produce no solitons, is found to be considerably more complicated than previously reported. In general, the asymptotic solution consists of exponential decay, similarity, rapid oscillations and a “collisionless shock” layer. The wave amplitude in this layer decays as [(lnt)/t]^2/3. Only for very special initial conditions is the shock layer absent from the solution.     

Asymptotics beyond All Orders in a Model of Crystal Growth

In its simplest form, the geometric model of crystal growth is a third-order, nonlinear, ordinary differential equation for θ(s, ε): A needle crystal is a solution that satisfies boundary conditions The geometric model admits a needle-crystal solution for ε = 0; for small ε, it admits an asymptotic expansion that is valid to all orders for such a solution. Even so, we prove that the geometric model in this form admits no needle crystal for any small, nonzero ε, a fact that lies beyond all orders of the asymptotic expansion. A more complicated version of the geometric model is where α represents crystalline anisitropy. We show that for 0 < α < 1, the geometric model admits needle crystals for a discrete set of values of α. The number of such values of α increases like ε^?1 as ε → 0.