Nonlinear Dispersive Partial Differential Equations Research Articles

Initial-boundary-value problems for linear and integrable nonlinear dispersive partial differential equations

It is suggested here that an interesting and important line of inquiry is the elaboration of methods of inverse scattering transform (IST) type in contexts where non-homogeneous boundary conditions intercede. The issue, which has practical relevance we indicate by example, appears ripe for development, thanks to recent new ideas interjected into the panoply of IST methodologies. A sketch of the principal steps envisaged in carrying out analysis of boundary-value problems using inverse scattering ideas is provided.

The Hydrodynamical Relevance of the Camassa–Holm and Degasperis–Procesi Equations

In recent years two nonlinear dispersive partial differential equations have attracted much attention due to their integrable structure. We prove that both equations arise in the modeling of the propagation of shallow water waves over a flat bed. The equations capture stronger nonlinear effects than the classical nonlinear dispersive Benjamin–Bona–Mahoney and Korteweg–de Vries equations. In particular, they accommodate wave breaking phenomena.

Degenerate dispersive equations arising in the study of magma dynamics

An outstanding problem in Earth science is understanding the method of transport of magma in the Earth's mantle. Two proposed methods for this transport are percolation through porous rock and flow up conduits. Under reasonable assumptions and simplifications, both means of transport can be described by a class of degenerate nonlinear dispersive partial differential equations of the form: where ϕ(z, 0) > 0 and ϕ(z, t) → 1 as z → ±∞.Although we treat arbitrary n and m, the exponents are physically expected to be between 2 and 5 and 0 and 1, respectively.In the case of percolation, the magma moves via the buoyant ascent of a less dense phase, treated as a fluid, through a denser, porous phase, treated as a matrix. In contrast to classical porous media problems where the matrix is fixed and the fluid is compressible, here the matrix is deformable, with a viscous constitutive relation, and the fluid is incompressible. Moreover, the matrix is modelled as a second, immiscible, compressible fluid to mimic the process of dilation of the pores. Flow via a conduit is modelled as a viscously deformable pipe of magma, fed from below.Analogue and numerical experiments suggest that these equations behave akin to KdV and BBM; initial conditions evolve into a collection of solitary waves and dispersive radiation. As ϕ → 0, the equations become degenerate. A general local well-posedness existence theory is given for a physical class of data (roughly H1) via fixed point methods. The strategy requires positive lower bounds on ϕ(z, t). The key to global existence is the persistence of these bounds for all time. Furthermore, we construct a Lyapunov energy functional, which is locally convex about the uniform porosity state, ϕ ≡ 1, and prove (global in time) nonlinear dynamic stability of the uniform state for any m and n. For data which are large perturbations of the uniform state, we prove global in time well-posedness for restricted ranges of m and n. This includes, for example, the case n = 4,m = 0, where an appropriate uniform in time lower global on ϕ can be proved using the conservation laws. We compare the dynamics with that of other problems and discuss open questions concerning a larger range of exponents, for which we conjecture global existence.

Marker method for the solution of nonlinear diffusion equations

The marker method for the solution of nonlinear diffusion equations is described. The method relies on the definition of a convective field associated with the underlying partial differential equation; the information about the approximate solution is associated with the response of an ensemble of markers to this convective field. Some key aspects of the method, such as the selection of the shape function and the initial loading, are discussed in some detail. Numerical experiments show that the method is accurate in determining the long time behavior of nonlinear diffusion equations. The marker method can be applied to an ensemble of nonlinear dispersive partial differential equations.

Modeling Solution of Nonlinear Dispersive Partial Differential Equations Using the Marker Method

A new method for the solution of nonlinear dispersive partial differential equations is described. The marker method relies on the definition of a convective field associated with the underlying partial differential equation; the information about the approximate solution is associated with the response of an ensemble of markers to this convective field. Some key aspects of the method, such as the selection of the shape function and the initial loading, are discussed in some detail.

A marker method for the solution of the damped Burgers' equation

AbstractA new method for the solution of the damped Burgers' equation is described. The marker method relies on the definition of a convective field associated with the underlying partial differential equation; the information about the approximate solution is associated with the response of an ensemble of markers to this convective field. Some key aspects of the method, such as the selection of the shape function and the initial loading, are discussed in some details. The marker method is applicable to a general class of nonlinear dispersive partial differential equations. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006

Local discontinuous Galerkin methods for nonlinear dispersive equations

We develop local discontinuous Galerkin (DG) methods for solving nonlinear dispersive partial differential equations that have compactly supported traveling waves solutions, the so-called “compactons”. The schemes we present extend the previous works of Yan and Shu on approximating solutions for linear dispersive equations and for certain KdV-type equations. We present two classes of DG methods for approximating solutions of such PDEs. First, we generate nonlinearly stable numerical schemes with a stability condition that is induced from a conservation law of the PDE. An alternative approach is based on constructing linearly stable schemes, i.e., schemes that are linearly stable to small perturbations. The numerical simulations we present verify the desired properties of the methods including their expected order of accuracy. In particular, we demonstrate the potential advantages of using DG methods over pseudo-spectral methods in situations where discontinuous fronts and rapid oscillations co-exist in a solution.

TRAVELING WAVE SOLUTIONS FOR A CLASS OF NONLINEAR DISPERSIVE EQUATIONS

The method of the phase plane is emploied to investigate the solitary and periodic traveling waves for a class of nonlinear dispersive partial differential equations. By using the bifurcation theory of dynamical systems to do qualitative analysis, all possible phase portraits in the parametric space for the traveling wave systems are obtained. It can be shown that the existence of a singular straight line in the traveling wave system is the reason why smooth solitary wave solutions converge to solitary cusp wave solution when parameters are varied. The different parameter conditions for the existence of solitary and periodic wave solutions of different kinds are rigorously determined.

The effect of the order of nonlinear dispersive equation on the compact and noncompact solutions

The effect of the order of nonlinear dispersive partial differential equation on the compact and noncompact solutions is studied in this paper. Two variants of the KdV equation of odd orders are examined. Our analysis reveals that the physical nature of these nonlinear KdV-type of equations are different. Two distinct sets of general formulas for each type of variants are performed to present a fairly complete understanding of the compact and noncompact structures.

A study on nonlinear dispersive partial differential equations of compact and noncompact solutions

In this paper we study two genuinely nonlinear dispersive partial differential equations in one- and higher-dimensional spaces. We show that the focusing branch and the defocusing branch of these equations each leads to a different physical structure. In addition, we develop general solutions of compact and noncompact support which are not obtainable through classic techniques.

A construction of compact and noncompact solutions for nonlinear dispersive equations of even order

A construction of compact and noncompact solutions for nonlinear dispersive partial differential equations of even order is considered. Two model equations of even orders k⩾4 are adopted to carry out this study. The analysis reveals the presence of distinct roots of different indices that play a significant role in the solutions.

Smooth and non-smooth traveling waves in a nonlinearly dispersive equation

The method of the phase plane is employed to investigate the solitary and periodic traveling waves in a nonlinear dispersive integrable partial differential equation. It is shown that the existence of a singular straight line in the corresponding ordinary differential equation for traveling wave solutions is the reason that smooth solitary wave solutions converge to solitary cusp wave solutions when the parameters are varied. The different parameter conditions for the existence of different kinds of solitary and periodic wave solutions are rigorously determined.

Numerical Study of Barotropic and Baroclinic Behavior of a Nonlinear Two-Layer Model

The fluid motion in a strongly stratified lake can be described by a set of nonlinear dispersive partial differential equations derived from the equations of motion. These model equations are solved numerically by an implicit finite difference scheme. Variable topography and the influence of the free surface are taken into account as well as nonlinear and dispersive terms. The numerical solutions that are obtained are stable and in good agreement with experimental results. In the case of wind forcing, they show the typical transition from barotropic to baroclinic motion.

Modified dispersive equations

Modifications of solutions of various non-linear dispersive partial differential equations, due to the addition of finite terms to the equations, have been determined. General solutions for the first-order perturbations of solutions of the equations are obtained when the linearizations are taken in the neighborhood of non-constant states. Although the primary focus is on sine-Gordon and Korteweg-deVries equations, similar results are true for other equations.