Nonlinear Evolution Partial Differential Equations Research Articles

Lifespan estimates for solutions of some evolution inequalities

We consider nonlinear evolution partial differential equations and inequalities. For the solutions of the Cauchy problem for such equations and inequalities, we establish conditions for finite time blow-up and derive an estimate for the blow-up time.

Plus–minus algorithm — A method for derivation of the Bäcklund transformations

A new way of deriving Bäcklund transformations for nonlinear partial differential evolution equations is presented and applied to the following equations: Korteweg–de Vries, Gardner, Burgers, generalized KdV and the fifth order equations of the KdV hierarchies. The presented method is based on the assumption of the existence of particular forms of the Bäcklund transformations. This assumption is supported by the strong or semi-strong necessary condition concepts [Sokalski, K., Wietecha, T., Lisowski, Z., 2001. Acta Phys. Polon. B32, 17; Sokalski, K., Wietecha, T., Lisowski, Z., 2002. Int. J. Theor. Phys. Group Theory Nonlinear Opt., NOVA, 9, 331; Sokalski, K., Wietecha, T., Lisowski, Z., 2001. Acta Phys. Polon. B32, 2771; Sokalski, K., Wietecha, T., Sokalska, D. 2005. J. Nonlinear Math. Phys. 12, 31]. Its general form has been put within the framework of an algorithm and implemented in MAPLE.

On Periodic Wave Solutions to (1+1)-Dimensional Nonlinear Physical Models Using the Sine-Cosine Method

We investigate two interesting (1+1)-dimensional nonlinear partial differential evolution equations (NLPDEEs), namely the nonlinear dispersion equation with compact structures and the generalized Camassa–Holm (CH) equation describing the propagation of unidirectional shallow water waves on a flat bottom, and arising in the study of a certain non-Newtonian fluid. Using an interesting technique known as the sine-cosine method for investigating travelling wave solutions to NLPDEEs, we construct many new families of wave solutions to the previous NLPDEEs, amongst which the periodic waves, enriching the wide class of solutions to the above equations.

Constraint Preserving Schemes for Some Gauge Invariant Wave Equations

We study discretizations of the Maxwell–Klein–Gordon equation as an example of a constrained geometric nonlinear evolution partial differential equation. For the temporal gauge we propose a fully discrete scheme which preserves the nonlinear constraint thanks to a special application of Lagrange multipliers. We show that the method generalizes to Hamiltonian wave equations whose kinetic and potential energy are both invariant under a group of transformations, even though the Galerkin spaces are not invariant. We then extend the method to the Lorenz gauge. Numerical results illustrate the discussion.

On the Conversion of High-Frequency Soliton Solutions to a (1+1)-Dimensional Nonlinear Partial Differential Evolution Equation

From the dynamical equation of barotropic relaxing media beneath pressure perturbations, and using the reductive perturbative analysis, we investigate the soliton structure of a (1+1)-dimensional nonlinear partial differential evolution (NLPDE) equation δy(δ + uδy + (u2/2) δy)u + αuy + u = 0, describing high-frequency regime of perturbations. Thus, by means of Hirota's bilinearization method, three typical solutions depending strongly upon a characteristic dissipation parameter are unearthed.

Algebraic dynamics solutions and algebraic dynamics algorithm for nonlinear partial differential evolution equations of dynamical systems

Using functional derivative technique in quantum field theory, the algebraic dynamics approach for solution of ordinary differential evolution equations was generalized to treat partial differential evolution equations. The partial differential evolution equations were lifted to the corresponding functional partial differential equations in functional space by introducing the time translation operator. The functional partial differential evolution equations were solved by algebraic dynamics. The algebraic dynamics solutions are analytical in Taylor series in terms of both initial functions and time. Based on the exact analytical solutions, a new numerical algorithm—algebraic dynamics algorithm was proposed for partial differential evolution equations. The difficulty of and the way out for the algorithm were discussed. The application of the approach to and computer numerical experiments on the nonlinear Burgers equation and meteorological advection equation indicate that the algebraic dynamics approach and algebraic dynamics algorithm are effective to the solution of nonlinear partial differential evolution equations both analytically and numerically.

Soil water redistribution and extraction flow models: Conservation laws

We determine all the nontrivial conservation laws for soil water redistribution and extraction flow equations which are modelled by a class of (2+1) nonlinear evolution partial differential equations with three arbitrary elements. It is shown that for arbitrary elements in the model equation there exist trivial conservation laws. We point out that nontrivial conservation laws exist for certain classes of equations which admit point symmetries.

On High-Frequency Soliton Solutions to a (2+1)-Dimensional Nonlinear Partial Differential Evolution Equation

A (2+1)-dimensional nonlinear partial differential evolution (NLPDE) equation is presented as a model equation for relaxing high-rate processes in active barothropic media. With the aid of symbolic computation and Hirota's method, some typical solitary wave solutions to this (2+1)-dimensional NLPDE equation are unearthed. As a result, depending on the dissipative parameter, single and multivalued solutions are depicted.

A Nonlinear Parabolic Model in Processing of Medical Image

The image's restoration is an essential step in medical imaging. Several Filters are developped to remove noise, the most interesting are filters who permits to denoise the image preserving semantically important structures. One class of recent adaptive denoising methods is the nonlinear Partial Differential Equations who knows currently a significant success. This work deals with mathematical study for a proposed nonlinear evolution partial differential equation for image processing. The existence and the uniqueness of the solution are established. Using a finite differences method we experiment the validity of the proposed model and we illustrate the efficiently of the method using some medical images. The Signal to Noise Ration (SNR) number is used to estimate the quality of the restored images.

Sur un modèle non-linéaire pour le débruitage de l'image

This work deals with a mathematical study for a proposed non-linear evolution partial differential equations model for image processing. The existence and the uniqueness of the solution are established, the model is numerically tested. The Signal to Noise Ratio (SNR) number is used to estimate the quality of the restored images. To cite this article: R. Aboulaich et al., C. R. Acad. Sci. Paris, Ser. I 345 (2007).

Nonlinear Fourier transforms, integrability and nonlocality in multidimensions

We present a nonlinear Fourier transform pair in any number of dimensions. This pair cannot be expressed in closed form but is defined through the solution of certain linear integral equations. The derivation of this result is based on the formulation of a nonlocal d-bar problem. The above nonlinear Fourier transform pair can be used for the solution of the Cauchy problem of a large class of nonlinear evolution equations in any number of spatial dimensions. Unfortunately, these nonlinear equations are highly nonlocal. Nevertheless, for the case of two dimensions they do reduce to the Davey–Stewartson equation which is a prototypical example of an integrable nonlinear evolution partial differential equation.

On a certain class of a nonlinear evolution partial differential equations

An existence result and a priori bound for the solution of a second-order nonlinear parabolic equation are established. Also a generalized tanh-function method is used for constructing exact travelling wave solutions for the nonlinear diffusion equation of Fisher type originated from the considered partial differential equation. And new multiple soliton solutions are obtained.

On infinite order and fully nonlinear partial differential evolution equations

In this paper we study the existence, uniqueness and propagation of regularity to infinite order partial differential evolution equations. Our approach is essentially functional and brings interesting results even when we restrict ourselves to finite order equations.

Integrable Nonlinear Evolution Partial Differential Equations in4+2and3+1Dimensions

The derivation and solution of integrable nonlinear evolution partial differential equations in three spatial dimensions has been the holy grail in the field of integrability since the late 1970s. The celebrated Korteweg-de Vries and nonlinear Schrödinger equations, as well as the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations, are prototypical examples of integrable evolution equations in one and two spatial dimensions, respectively. Do there exist integrable analogs of these equations in three spatial dimensions? In what follows, I present a positive answer to this question. In particular, I first present integrable generalizations of the KP and DS equations, which are formulated in four spatial dimensions and which have the novelty that they involve complex time. I then impose the requirement of real time, which implies a reduction to three spatial dimensions. I also present a method of solution.

Conservation laws for equations related to soil water equations

We obtain all nontrivial conservation laws for a class of (2+1) nonlinear evolution partial differential equations which are related to the soil water equations. It is also pointed out that nontrivial conservation laws exist for certain classes of equations which admit point symmetries. Moreover, we associate symmetries with conservation laws for special classes of these equations.

A Nonlinear Galerkin Scheme Involving Vector and Tensor Spherical Harmonics for Solving the Incompressible Navier--Stokes Equation on the Sphere

This work is concerned with a nonlinear Galerkin method for solving the incompressible Navier--Stokes equation on the sphere. It extends the work of [A. Debussche, T. Dubois, and R. Temam, Theoret. Comput. Fluid Dyn., 7 (1995), pp. 279--315; M. Marion and R. Temam, SIAM J. Numer. Anal., 26 (1989), pp. 1139--1157; J. Shen and R. Temam, Proceedings of the International Conference on Nonlinear Evolution Partial Differential Equations, AMS, Providence, RI, 1997, pp. 363--376] from one-dimensional or toroidal domains to the spherical geometry. In the first part, the method based on type 3 vector spherical harmonics is introduced and convergence is indicated. Further it is shown that the occurring coupling terms involving three vector spherical harmonics can be expressed algebraically in terms of Wigner-$3j$ coefficients. To improve the numerical efficiency and economy we introduce an FFT-based pseudospectral algorithm for computing the Fourier coefficients of the nonlinear advection term. The resulting method scales with $O(N^3)$ if N denotes the maximal spherical harmonic degree. The latter is demonstrated in an extensive numerical example.

New Integrable Equations of Nonlinear Schrödinger Type

New integrable matrix nonlinear evolution partial differential equations in (1 + 1)‐dimensions are derived, via a treatment which starts from an appropriate matrix generalization of the Zakharov–Shabat spectral problem. Via appropriate parametrizations, multi‐vector versions of these equations are also exhibited. Generally these equations feature solitons that do not move with constant velocities: they rather behave as boomerons or as trappons, namely, up to a Galileian transformation, they typically boomerang back to where they came from, or they are trapped to oscillate around some fixed position determined by their initial data. In this paper, meant to be the first of a series, we focus on the derivation and exhibition of new coupled evolution equations of nonlinear Schrödinger type and on the behavior of their single‐soliton solutions.

Nonlinear anisotropic diffusion filtering for multiscale edge enhancement

Nonlinear anisotropic diffusion filtering is a procedure based on nonlinearevolution partial differential equations which seeks to improve imagesqualitatively by removing noise while preserving details and even enhancingedges. However, well known implementations are sensitive to parameterswhich are necessarily tuned to sharpen a narrow range of edge slopes;otherwise, edges are either blurred or staircased. In this work, nonlinearanisotropic diffusion filters have been developed which sharpen edges over awide range of slope scales and which reduce noise conservatively withdissipation purely along feature boundaries. Specifically, the range ofsharpened edge slopes is widened as backward diffusion normal to level setsis balanced with forward diffusion tangent to level sets. Also, noise isreduced by selectively altering the balance toward diminishing normalbackward diffusion and particularly toward total variation filtering. Thetheoretical motivation for the proposed filters is presented together withcomputational results comparing them with other nonlinear anisotropicdiffusion filters on both synthetic images and magnetic resonance images.

Nonlinear Schrödinger-type equations from multiscale reduction of PDEs. I. Systematic derivation

In this article we begin a systematic investigation via multiscale expansions of nonlinear evolution PDEs (partial differential equations). In this first article we restrict consideration to a single, autonomous, but otherwise generic, PDE in 1+1 variables (space+time), of first order in time, whose linear part is dispersive, and to solutions dominated by a single plane wave satisfying the linear part of the PDE. The expansion parameter is an, assumedly small, coefficient multiplying this plane wave. The main (indeed, asymptotically exact) effect of the (weak) nonlinearity is then to cause a modulation of the amplitude of the plane wave and of its harmonics, which is generally described, in (appropriately defined) coarse-grained time and space variables, by evolution equations of nonlinear Schrödinger type. A systematic analysis of such equations is presented, corresponding to various assumptions on the “resonances” occurring for the first few harmonics.

Reduction of dimension of approximate intertial manifolds by symmetry

In this paper, we study approximate inertial manifolds for nonlinear evolution partial differential equations which possess symmetry. The relationship between symmetry and dimensions of approximate inertial manifolds is established. We demonstrate that symmetry can reduce the dimensions of an approximate inertial manifold. Applications for concrete evolution equations are given.