Philip Rosenau

The construction of special solutions to partial differential equations

Abstract Almost all the methods devised to date for constructing particular solutions to a partial differential equation can be viewed as manifestations of a single unifying method characterized by the appending of suitable “side conditions” to the equation, and solving the resulting overdetermined system of partial differential equations. These side conditions can also be regarded as specifying the invariance of the particular solutions under some generalized group of transformations.

Group-invariant solutions of differential equations

We introduce the concept of a weak symmetry group of a system of partial differential equations, that generalizes the “nonclassical” method introduced by Bluman and Cole for finding group-invariant...

On nonanalytic solitary waves formed by a nonlinear dispersion

We study the prototypical, genuinely nonlinear, K(m, n) equation, ut ± a(um)x + (un)xxx = 0, a = const, which exhibits a number of remarkable dispersive effects. In particular, the distinguished subclass wherein m = n + 2 is transformed into a new, purely dispersive equation free of convection. In addition to compactons, the K(m, n) can support both kinks and solitons with an infinite slope(s), periodic waves and dark solitons with cusp(s) all being manifestations of nonlinear dispersion in action. For n < 0 the enhanced dispersion at the tail may generate algebraically decaying patterns.

Compact and noncompact dispersive patterns

Abstract We discuss the pivotal role played by the nonlinear dispersion in shaping novel, compact and noncompact patterns. It is shown that if the normal velocity of a planar curve is U =−( k n ) s , n >1, where k is the curvature, then the solitary disturbances may propagate like compactons. We extend the KP and the Boussinesq equations to include nonlinear dispersion to the effect that the new equations support compact and semi-compact solitary structures in higher dimensions. We also discuss the relations between equations sharing the same scaling. We show how compacton supporting equations may be cast into a strong formulation wherein one avoids dealing with weak solutions.

Dynamics of nonlinear mass-spring chains near the continuum limit

Abstract We study the dynamics of mass-spring chains with arbitrary interparticle and substrate potentials and describe a systematic approach to derive the equations of motion near the continuum limit. Our method, which applies both to strictly one-dimensional problems and to one-dimensional chains free to move in three dimensions, correctly captures all terms to given order in discreteness and allows us to formulate well-behaved nonlinear partial differential equations for these systems. We re-examine the familiar cases of the Fermi-Pasta-Ulam problem and the Frenkel-Kontorova model and obtain new insights into these problems.

On a class of nonlinear dispersive-dissipative interactions

Abstract We study a prototype, dissipative-dispersive equation; ut+a(um)x+(un)xxx = μ(uk)xx, a, μ = consts., which represents a wide variety of interactions. At the critical value k = (m + n) 2 which separates dispersive- and dissipation-dominated phenomena, these effects are in a detailed balance and the patterns formed do not depend on the amplitude. In particular, when m = n + 2 = k + 1 the equation can be transformed into a form free of convection and dissipation, making it accessible to analysis. Both bounded and unbounded oscillations as well as solitary waves are found. A variety of exact solutions are presented, with a notable example being a solitary doublet. For n = 1 and a = (2μ 3) 2 the problem may be mapped into a linear equation, leading to rational, periodic or aperiodic solutions, among others.

Some symmetries of the nonlinear heat and wave equations

Abstract The Lie-group formalism is applied to deduce the classical symmetries of the nonlinear heat equation, the diffusion-convection equation and the nonlinear wave equations. Some nonclassical symmetries are also presented.

Thermal waves in an absorbing and convecting medium

The quasi-linear parabolic equation ∂tu = a∂xxuα + b∂xuβ − cuγ exhibits a wide variety of wave phenomena, some of which are studied in this work; and some solvable cases are presented. The motion of the wave front is characterized in terms of α, β and γ. Among the interesting phenomena we note the effect of fast absorption (b  0, 0 < γ < 1) that causes extinction within a finite time, may break the evolving pulse into several sub-pulses and causes the expanding front to reverse its direction. In the convecting case (c  0, b ≠ 0) propagation has many features in common with Burgers equation, α = 1; particularly, if 0 < a ≪ 1, a shock-like transit layer is formed.

On solitons, compactons, and Lagrange maps

Abstract Two local conservation laws of the K(m, n) equation, ut ± (um)x + (un)xxx = 0, are used to define two Lagrange-type transformations into mass and momentum space. These mappings help to identify new integrable cases (K(−1, −2), K(−2, −2), K( 3 2 , −1 2 ) ), transform conventional solitary waves into compactons - solitary waves on compactum - and relate certain soliton-carrying systems with compacton-carrying systems. Integrable equations are transformed into new integrable equations and interaction of N-solitons of the, say, m-KdV (m = 3, n = 1) is thus projected into an interaction in a compact domain from which N ordered stationary compactons emerge. The interaction of traveling compactons is the image of super-imposed equilibria of the corresponding soliton equation. For m = n + 2, the potential form of the K(m, n) equation may also be cast into a conserved form and thus transformed, yielding generalized Dym and Wadati equations and two new integrable cases. It is shown that r t + (1−r 2 ) 3 2 (r xx + r) x = 0 is integrable and supports compact kinds.

A quasi-continuous description of a nonlinear transmission line

We present a systematic approach to derive an equation describing the propagation of pulses along a transmission line made of a large number of nonlinear LC-circuits. This equation correctly captures the leading order effects due the discreteness of the line and describes a two-sided wave propagation, nonlinear wave-wave interactions, and wave-wall interactions. In contradistinction to previous works, our method is not limited to weakly nonlinear disturbances.