Local Evolution Equations Research Articles

On the polynomial Miura transformation

The polynomial Miura transformation ϕ= ω x + a( ω) ( a( ω) is an nth-power polynomial) connects continual classes of local evolution equations at n<3, whereas at n⩾3 it turns out to connect only the trivial equations ϕ t = ϕ x and ω t = ω x or some equivalent to them.

Gravitational instability in a primordial collapsing gas cloud

This paper presents an analysis of the linear evolution of short-wavelength perturbations in a background fluid flow which is undergoing gravitational collapse on large scales. Local evolution equations for perturbations to an arbitrary flow are derived in the linear regime and the short-wavelength limit. Local perturbation behavior in an inhomogeneous flow is found to be the same as that in a homogeneous anisotropic flow having the same local velocity field. Background flows in which the scale factors vary as power laws in time are considered to illustrate the relative effects of self-gravity, pressure and kinematics of the background flow on the density perturbation evolution. Perturbation analyses are then presented for more realistic background flows arising from the evolution into the nonlinear regime of initially small density perturbations in an isotropically expanding cosmological model. For low-pressure, inhomogeneous collapses, kinematic effects tend to dominate over self-gravity in driving perturbation growth as the collapse proceeds.

The Miura transformation and Lie-Bäcklund algebras of exactly solvable equations

All the one-dimensional one-component local evolution equations connected via the Miura transformation are found. Exactly solvable equations and their Lie-Bäcklund algebras are shown to generate interesting transformations of infinite classes of evolution equations.

Non-linear propagation of broadband noise signals

Measurements of aircraft flyover noise have strongly suggested that non-linear propagation effects can significantly modify the noise spectrum. The data show that over ranges of 500 to 1000 m the attenuation in the range 5 to 10 kHz may fall considerably short of that predicted on the basis of linear propagation, if the noise is sufficiently intense. Such distances are comparable to those used in aircraft noise certification measurements, and the frequencies involved have an important influence on the perceived noise level. Therefore a scheme is required for predicting the finite amplitude propagation of random signals over a wide range of source and atmospheric conditions. Taylor series solutions to a generalized Burgers' equation are generated for stationary noise signals. The expansions are in powers of the range variable, and any frequency dependence of attenuation and dispersion is allowed. The limitations of such an approach are discussed, and it is found that the number of terms needed increases rapidly with range. For propagation over a long range, when non-linear effects are important, it is recommended that one of the differential equations derived for the power spectrum be used, together with a closure hypothesis. Individual terms in the series may be treated separately as local spectrum evolution equations, valid at all points along the propagation path. With appropriate modelling of the non-linear terms, such an equation can form a basis for aircraft noise prediction schemes.

On a new hierarchy of nonlinear evolution equations containing the Pohlmeyer–Lund–Regge equation

A hierarchy of local nonlinear evolution equations associated with a new spectral problem is derived. It is shown that each equation is Hamiltonian and that their fluxes commute and a local infinite set of conserved densities is given. An interesting reduction is considered. In this case a hierarchy of local nonlinear evolution equations is generated by a recursion operator and its explicit inverse. Also this hierarchy satisfies a canonical geometrical scheme. It contains as a special case the Pohlmeyer–Lund–Regge equation.

A recursive generation of local higher-order sine–Gordon equations and their Bäcklund transformation

A new hierarchy of local nonlinear evolution equations is generated by a recursion operator and its explicit inverse. It is shown that this hierarchy satisfies a canonical geometrical scheme and that it contains as special cases the sine–Gordon and Liouville equations in laboratory coordinates. A generalization of the well-known Bäcklund transformation and nonlinear superposition formula for the sine–Gordon equation is also obtained.