The term soliton has recently been coined to describe a pulselike nonlinear wave (solitary wave) which emerges from a collision with a similar pulse having unchanged shape and speed. To date at least seven distinct wave systems, representing a wide range of applications in applied science, have been found to exhibit such solutions. This review paper covers the current status of soliton research, paying particular attention to the very important "inverse method" whereby the initial value problem for a nonlinear wave system can be solved exactly through a succession of linear calculations.

The soliton: A new concept in applied science

This article reviews acoustic microfiuidics: the use of acoustic fields, principally ultrasonics, for application in microfiuidics. Although acoustics is a classical field, its promising, and indeed perplexing, capabilities in powerfully manipulating both fluids and particles within those fluids on the microscale to nanoscale has revived interest in it. The bewildering state of the literature and ample jargon from decades of research is reorganized and presented in the context of models derived from first principles. This hopefully will make the area accessible for researchers with experience in materials science, fluid mechanics, or dynamics. The abundance of interesting phenomena arising from nonlinear interactions in ultrasound that easily appear at these small scales is considered, especially in surface acoustic wave devices that are simple to fabricate with planar lithography techniques common in microfluidics, along with the many applications in microfluidics and nanofluidics that appear through the literature.

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Microscale acoustofluidics: Microfluidics driven via acoustics and ultrasonics

The Korteweg–de Vries equation \[ u_t + uu_x + u_{xxx} = 0\] is a nonlinear partial differential equation arising in the study of a number of different physical systems, e.g., water waves, plasma physics, anharmonic lattices, and elastic rods. It describes the long time evolution of small-but-finite amplitude dispersive waves. From detailed studies of properties of the equation and its solutions, the concept of solitons was introduced and the method for exact solution of the initial-value problem using inverse scattering theory was developed. A survey of these and other results for the Korteweg–deVries equation are given, including conservation laws, an alternate method for exact solution, soliton solutions, asymptotic behavior of solutions, Backlund transformation, and a nonlinear WKB method. The recent literature contains many extensions of these ideas to a number of other nonlinear evolution equations of physical interest and to other classes of equations. Some of these equations and results are indica...

The Korteweg–deVries Equation: A Survey of Results

We study a type of ‘eigenvalue’ problem for systems of linear ordinary differential equations with asymptotically constant coefficients by using the analytic function D (λ) introduced by J. W. Evans (1975) in his study of the stability of nerve impulses. We develop a general theory of D (λ) that clarifies the role of the essential spectrum in applications. New formulae for derivatives of D (λ) are used to study linear exponential instabilities of solitary waves for generalizations of: (1) the Korteweg-de Vries equation (KdV); (2) the Benjamin-Bona-Mahoney equation (BBM); and (3) the regularized Boussinesq equation. A pair of real eigenvalues exists, indicating a non-oscillatory instability, when the ‘momentum’ of the wave (a time-invariant functional associated with the hamiltonian structure of the equation) is a decreasing function of wave speed. Also we explain the mechanism of the transition to instability. Unexpectedly, these transitions are unlike typical transitions to instability in finite-dimensional hamiltonian systems. Instead they can be understood in terms of the motion of poles of the resolvent formula extended to a multi-sheeted Riemann surface. Finally, for a generalization of the KdV-Burgers equation (a model for bores), we show that a conjectured transition to instability does not involve real eigenvalues emerging from the origin, suggesting an oscillatory type of instability.

Eigenvalues, and Instabilities of Solitary Waves

We show that the family of solitary waves (1-solitons) of the Korteweg-de Vries equation

$$\partial _t u + u\partial _x u + \partial _x^3 u = 0 ,$$

is asymptotically stable Our methods also apply for the solitary waves of a class of generalized Korteweg-de Vries equations,

$$\partial _t u + \partial _x f(u) + \partial _x^3 u = 0 $$

In particular, we study the case wheref(u)=up+1/(p+1),p=1, 2, 3 (and 3 0, withf∈C4) The same asymptotic stability result for KdV is also proved for the casep=2 (the modified Korteweg-de Vries equation) We also prove asymptotic stability for the family of solitary waves for all but a finite number of values ofp between 3 and 4 (The solitary waves are known to undergo a transition from stability to instability as the parameterp increases beyond the critical valuep=4) The solution is decomposed into a modulating solitary wave, with time-varying speedc(t) and phase γ(t) (bound state part), and an infinite dimensional perturbation (radiating part) The perturbation is shown to decay exponentially in time, in a local sense relative to a frame moving with the solitary wave Asp→4−, the local decay or radiation rate decreases due to the presence of aresonance pole associated with the linearized evolution equation for solitary wave perturbations

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Asymptotic stability of solitary waves

This article is concerned with wave propagation processes where some balance occurs in the competition between a nonlinear effect and a higher order derivative effect which might be of a dispersive or a dissipative nature. The Burgers equation and the Korteweg–de Vries equation, which are prototype scalar nonlinear dissipative and dispersive equations, are shown to be fundamental to this study, even when quite general systems of equations are involved. The role of the solitary wave solution is shown to be central to the study which is applied to gravity waves, plasma waves and to waves in lattices. Both steady state solutions and initial value problems are reviewed together with questions of stability, existence and uniqueness.

A. Jeffrey

Papers

Weak Nonlinear Dispersive Waves: A Discussion Centered Around the Korteweg–De Vries Equation