Family Of Nonlinear Evolution Equations Research Articles

Integrable fractional modified Korteweg–deVries, sine-Gordon, and sinh-Gordon equations

The inverse scattering transform allows explicit construction of solutions to many physically significant nonlinear wave equations. Notably, this method can be extended to fractional nonlinear evolution equations characterized by anomalous dispersion using completeness of suitable eigenfunctions of the associated linear scattering problem. In anomalous diffusion, the mean squared displacement is proportional to t α , α > 0, while in anomalous dispersion, the speed of localized waves is proportional to A α , where A is the amplitude of the wave. Fractional extensions of the modified Korteweg–deVries (mKdV), sine-Gordon (sineG) and sinh-Gordon (sinhG) and associated hierarchies are obtained. Using symmetries present in the linear scattering problem, these equations can be connected with a scalar family of nonlinear evolution equations of which fractional mKdV (fmKdV), fractional sineG (fsineG), and fractional sinhG (fsinhG) are special cases. Completeness of solutions to the scalar problem is obtained and, from this, the nonlinear evolution equation is characterized in terms of a spectral expansion. In particular, fmKdV, fsineG, and fsinhG are explicitly written. One-soliton solutions are derived for fmKdV and fsineG using the inverse scattering transform and these solitons are shown to exhibit anomalous dispersion.

Energy and Large Time Estimates for Nonlinear Porous Medium Flow with Nonlocal Pressure in $$\mathbb {R}^N$$

We study the general family of nonlinear evolution equations of fractional diffusive type $$\partial _t u-\text{ div }\left( |u|^{m_1}\nabla (-\Delta )^{-s} [ |u|^{m_2-1}u]\right) =f$$ . Such nonlocal equations are related to the porous medium equations with a fractional Laplacian pressure. Our study concerns the case in which the flow takes place in the whole space. We consider $$m_1, m_2>0$$ , and $$s\in (0,1)$$ , and prove the existence of weak solutions. Moreover, when $$f\equiv 0$$ we obtain the $$L^p$$ - $$L^\infty $$ decay estimates of solutions, for $$p\geqq 1$$ . In addition, we also investigate the finite time extinction of solution.

Porous medium equation with nonlocal pressure in a bounded domain

We study a quite general family of nonlinear evolution equations of diffusive type with nonlocal effects. More precisely, we study porous medium equations with a fractional Laplacian pressure, and the problem is posed on a bounded space domain. We prove existence of weak solutions with suitable a priori bounds and regularity estimates.

New integrable models with <italic>N</italic>-peakons and algebro-geometric solutions of soliton equations

In the soliton theory, it is a basic and challenging problem to search for integrable dynamical models with peaked soliton solutions and obtain the explicit solutions to soliton equations. This paper can be mainly divided into two parts. First, the derivation of two integrable dynamical systems with N -peakon, which provides new models for the family of nonlinear evolution equations with N -peakon; on the other hand, based on the knowledge of hyperelliptic curve, the method of nite-order expansion of Lax pairs is improved to obtain the solution of KdV6 equations. Moreover, to avoid the limit of using the Riemann theorem, the meromorphic function and the Baker-Akhiezer vector are introduced by which quasi-periodic solutions to the mKdV type equations and mixed AKNS equations are constructed according to their asymptotic properties and algebro-geometric characters.

Tsallis’ maximum entropy ansatz leading to exact analytical time dependent wave packet solutions of a nonlinear Schrödinger equation

Tsallis maximum entropy distributions provide useful tools for the study of a wide range of scenarios in mathematics, physics, and other fields. Here we apply a Tsallis maximum entropy ansatz, the q-Gaussian, to obtain time dependent wave-packet solutions to a nonlinear Schrödinger equation recently advanced by Nobre, Rego-Monteiro and Tsallis (NRT) [F.D. Nobre, M.A. Rego-Monteiro, C. Tsallis, Phys. Rev. Lett. 106 (2011) 140601]. The NRT nonlinear equation admits plane wave-like solutions (q-plane waves) compatible with the celebrated de Broglie relations connecting wave number and frequency, respectively, with energy and momentum. The NRT equation, inspired in the q-generalized thermostatistical formalism, is characterized by a parameter q and in the limit q→1 reduces to the standard, linear Schrödinger equation. The q-Gaussian solutions to the NRT equation investigated here admit as a particular instance the previously known q-plane wave solutions. The present work thus extends the range of possible processes yielded by the NRT dynamics that admit an analytical, exact treatment. In the q→1 limit the q-Gaussian solutions correspond to the Gaussian wave packet solutions to the free particle linear Schrödinger equation. In the present work we also show that there are other families of nonlinear Schrödinger-like equations, besides the NRT one, exhibiting a dynamics compatible with the de Broglie relations. Remarkably, however, the existence of time dependent Gaussian-like wave packet solutions is a unique feature of the NRT equation not shared by the aforementioned, more general, families of nonlinear evolution equations.

Application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations

The application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations is considered. Some classes of solitary wave solutions for the families of nonlinear evolution equations of fifth, sixth and seventh order are obtained. The efficiency of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations is demonstrated.

Traveling wave solutions of the generalized nonlinear evolution equations

Solitary wave solutions for a family of nonlinear evolution equations with an arbitrary parameter in the exponents are constructed. Some of the obtained solutions seem to be new.

Gauge transformations and symmetries of integrable systems

We analyse several integrable systems in zero-curvature form within the framework of invariant gauge theory. In the Drinfeld–Sokolov gauge, we derive a two-parameter family of nonlinear evolution equations which as special cases include the Korteweg–de Vries (KdV) and Harry Dym (HD) equations. We find residual gauge transformations which lead to infinitesimal symmetries of this family of equations. For KdV and HD equations we find an infinite hierarchy of such symmetry transformations, and we investigate their relation with local conservation laws, constants of the motion and the bi-Hamiltonian structure of the equations. Applying successive gauge transformations of Miura type we obtain a sequence of gauge-equivalent integrable systems, among them the modified KdV and Calogero KdV equations.

Non-uniform dependence on initial data for a family of non-linear evolution equations

We show that solutions to the periodic Cauchy problem for a family of non-linear evolution equations, which contains the Camassa-Holm equation, do not depend uniformly continuously on initial data in the Sobolev space $H^s(\mathbb{T})$, when $s=1$ or $s\geq 2$.

A class on non-local linear operators for vorticity waves

A steady longitudinal current in the nearshore can, in some conditions, support oscillations known as vorticity waves or shear waves. In this article, we consider a family of nonlinear evolution equations derived by Shrira and Voronovitch to describe the dynamics of vorticity waves near the coastal line and make the study of the dispersion and smoothing properties of the associated nonlocal free problems. More precisely, after establishing long and short time uniform estimates for a certain class of oscillatory integrals, we derive “L p −L q ” and Strichartz-type estimates for the solutions of the linearized equations.

Stability for evolving graphs by nonlocal weighted curvature

A general stability and convergence theorem is established for generalized solutions of a family of nonlinear evolution equations with non-local diffusion in one space dimension. As the first application we justify the motion by crystalline energy as a limit of regularized problems. As the sec-ond application we show the convergence of crystalline algorithm for general curvature flow equations. Our general results are also important to explain that geometric evolution of crystals depends continuously on temperature even if facets appear.

Gel'fand-Dikii method and nonlinear equations associated to Schr�dinger operators with energy-dependent potentials

The Gel'fand-Dikii method is applied to Schrodinger operators with energy-dependent potentials. New families of nonlinear evolution equations with an infinite number of constants of motion are found.

Spiky Soliton in Circular Polarized Alfvén Wave

A new type of nonlinear evolution equation for the Alfven waves, propagating parallel to the magnetic field, is now registered to the completely integrable family of nonlinear evolution equations. In spite of the extensive studies of Kaup and Newell, and of Kawata and Inoue, these analysis have been dealing with solutions for restricted boundary conditions. The present paper presents full account of stationary solitary wave solutions for the plane wave boundary condition. The obtained results exhibit peculiar structure of spiky modulation of amplitude and phase, which arises from the derivative nonlinear coupling term. A nonlinear equation for complex amplitude associated with the carrier wave is shown to be a mixed type of nonlinear Schrodinger equation, having an ordinary cubic nonlinear Schrodinger equation, having an ordinary cubic nonlinear term and the derivative of cubic nonlinear term.