Nonlinear Evolution Partial Differential Equations Research Articles

The Kadomtsev–Petviashvili equation as a source of integrable model equations

A new integrable and nonlinear partial differential equation (PDE) in 2+1 dimensions is obtained, by an asymptotically exact reduction method based on Fourier expansion and spatiotemporal rescaling, from the Kadomtsev–Petviashvili equation. The integrability property is explicitly demonstrated, by exhibiting the corresponding Lax pair, that is obtained by applying the reduction technique to the Lax pair of the Kadomtsev–Petviashvili equation. This model equation is likely to be of applicative relevance, because it may be considered a consistent approximation of a large class of nonlinear evolution PDEs.

An Adaptive Variational Finite Element Method for Computing Coupled Porous-Media Problems with Moving Fronts

SUMMARY An adaptive deforming-grid mixed-variable finite element method for solving two-dimensional porous-media transport problems governed by a system of coupled nonlinear partial differential evolution equations, is presented. The advantages of this methodology over standard fixed-grid formulations in improving solution accuracy and yielding better computational stability characteristics are discussed and demonstrated through solution of a representative set of problems. Accordingly, extremely accurate solutions were computed not only for the primary variables but also their fluxes, with a very small fixed number of deforming and moving finite elements, for a set of transient high Péclet number porous-media convection-diffusion (continuous moving front) problems and also for the front motion (discrete moving front) during resaturation by water of a porous rock.

The fully nonlinear development of Görtler vortices in growing boundary layers

The fully nonlinear development of small-wavelength Görtler vortices in a growing boundary layer is investigated by a combination of asymptotic and numerical methods. The starting point for the analysis is the weakly nonlinear theory of Hall ( J. Inst. Math. Applies 29, 173 (1982)) who discussed the initial development of large-wavenumber small-amplitude vortices in a neighbourhood of the location where they first become linearly unstable. That development is unusual in the context of nonlinear stability theory in that it is not described by the Stuart-Watson approach. In fact, the development is governed by a pair of coupled nonlinear partial differential evolution equations for the vortex flow and the mean flow correction. Here the further development of this interaction is considered for vortices so large that the mean flow correction driven by them is as large as the basic state. Surprisingly it is found that such a nonlinear interaction can still be described by asymptotic means. It is shown that the vortices spread out across the boundary layer and effectively drive the boundary layer. In fact, the system obtained by the equations for the fundamental component of the vortex generates a differential equation for the basic state. Thus the mean flow adjusts so as to make these large amplitude vortices locally neutral. Moreover in the region where the vortices exist the mean flow has a ‘square-root’ profile and the vortex velocity field can be written down in closed form. The upper and lower boundaries of the region of vortex activity are determined by a free-boundary problem involving the boundary-layer equations. In general it is found that this region ultimately includes almost all of the original boundary layer and much of the free stream. In this situation the mean flow has essentially no relation to the flow that exists in the absence of the vortices.

A modified modified Korteweg-de Vries equation

The solution of the Cauchy problem, an infinite set of conserved quantities, a variational formulation, the auto-Backlund transformations and the soliton phenomenology (featuring singly and doubly humped solitons, as well as breathers) are surveyed briefly for the integrable non-linear evolution partial differential equation (PDE) ut+(uxx-2 eta u3-23uux3/( eta +u2))x=0, eta =+or-1.

On the Dynamics of Nearly-Extinguished Non-Adiabatic Cellular Flames

Abstract We study the dynamics of a nearly-extinguished and weakly unstable non-adiabalic flame For simplicity sake, the analysis is conducted in the framework of a thermal-diffusional flame model Using high activation energy techniques and bifurcation methods, we derive a non-linear partial differential evolution equation for the changes of front shape and velocity By solving it approximately in a particular case, we show that the spontaneously growing front corrugations, due to diffusive instability, are sufficient to prevent flame quenching: a diffusively unstable flame front can still propagate with heat-loss intensities that would quench a planar one.

Nonlinear Degenerate Evolution Equations and Partial Differential Equations of Mixed Type

The Cauchy problem for the evolution equation $Mu'(t) + N(t,u(t)) = 0$ is studied, where M and $N(t, \cdot )$ are, respectively, possibly degenerate and nonlinear monotone operators from a vector space to its dual. Sufficient conditions for existence and for uniqueness of solutions are obtained by reducing the problem to an equivalent one in which M is the identity but each $N(t, \cdot )$ is multivalued and accretive in a Hilbert space. Applications include weak global solutions of boundary value problems with quasilinear partial differential equations of mixed Sobolev-parabolic-elliptic type, boundary conditions with mixed space-time derivatives, and those of the fourth or fifth type. Similar existence and uniqueness results are given for the semilinear and degenerate wave equation $Bu''(t) + F(t,u'(t)) + Au(t) = 0$, where each nonlinear $F(t, \cdot )$ is monotone and the nonnegative B and positive A are self-adjoint operators from a reflexive Banach space to its dual.