Class Of Nonlinear Dispersive Equations Research Articles

A high-order linearly implicit energy-preserving Partitioned Runge-Kutta scheme for a class of nonlinear dispersive equations

<abstract><p>In this paper, we design a novel class of arbitrarily high-order, linearly implicit and energy-preserving numerical schemes for solving the nonlinear dispersive equations. Based on the idea of the energy quadratization technique, the original system is firstly rewritten as an equivalent system with a quadratization energy. The prediction-correction strategy, together with the Partitioned Runge-Kutta method, is then employed to discretize the reformulated system in time. The resulting semi-discrete system is high-order, linearly implicit and can preserve the quadratic energy of the reformulated system exactly. Finally, we take the Camassa-Holm equation as a benchmark to show the efficiency and accuracy of the proposed schemes.</p></abstract>

On a class of derivative Nonlinear Schrödinger-type equations in two spatial dimensions

We present analytical results and numerical simulations for a class of nonlinear dispersive equations in two spatial dimensions. These equations are of (derivative) nonlinear Schrödinger type and have recently been obtained by Dumaset al.in the context of nonlinear optics. In contrast to the usual nonlinear Schrödinger equation, this new model incorporates the additional effects of self-steepening and partial off-axis variations of the group velocity of the laser pulse. We prove global-in-time existence of the corresponding solution for various choices of parameters. In addition, we present a series of careful numerical simulations concerning the (in-)stability of stationary states and the possibility of finite-time blow-up.

On the regularity of solutions to a class of nonlinear dispersive equations

We shall study special regularity properties of solutions to some nonlinear dispersive models. The goal is to show how regularity on the initial data is transferred to the solutions. This will depend on the spaces where regularity is measured.

The local Cauchy problem for a class of nonlinear dispersive equations

The local Cauchy problem for a class of nonlinear dispersive equations

Optimal distributed controls of a class of nonlinear dispersive equations with cubic nonlinearity

This paper is devoted to the optimal distributed control problem governed by a class of nonlinear dispersive equations with cubic nonlinearity, which contains the famous Novikov equation as special case. We first investigate the existence and uniqueness of the weak solution for the controlled system, and then find an optimal solution for the controlled system with the generalized cost functional. Moreover, by means of the method suggested by A.Ya. Dubovitskii and A.A. Milyutin, we establish the first order necessary optimality condition of optimal control for the controlled system in the fixed final horizon case.

Symmetry analysis for a class of nonlinear dispersive equations

A class of dispersive equations is studied within the framework of group analysis of differential equations. The enhanced Lie group classification is achieved. The complete list of equivalence transformations is presented. It is shown that certain equations from the class admit nonclassical reductions. Potential and potential nonclassical symmetries are also considered.

On some applications of transformation groups to a class of nonlinear dispersive equations

We consider a special class of nonlinear dispersive equations and search for classical and potential symmetries. We classify them in both cases. Reduced equations and wide classes of solutions of physical interest are obtained. Some equations of this special class are linearized via a symmetry approach, and an equivalence algebra is obtained.

Geometric finite difference schemes for the generalized hyperelastic-rod wave equation

Geometric integrators are presented for a class of nonlinear dispersive equations which includes the Camassa–Holm equation, the BBM equation and the hyperelastic-rod wave equation. One group of schemes is designed to preserve a global property of the equations: the conservation of energy; while the other one preserves a more local feature of the equations: the multi-symplecticity.

A note on the Cauchy problem of the generalized Davey–Stewartson equations

In this note, we establish some local and global existence results for the Cauchy problem of a class of nonlinear dispersive equations which generalize the nonlinear Schrödinger equations and the Davey–Stewartson equations. These results improve some previously obtained results by some other authors when they are restricted to certain special equations.

An energy-conserving Galerkin scheme for a class of nonlinear dispersive equations

A Galerkin scheme is presented for a class of conservative nonlinear dispersive equations, such as the Camassa–Holm equation and the regularized long wave equation. The scheme has two advantageous features: first, it is conservative in that it keeps the discrete analogue of the continuous energy conservation property in the original equations; second, it can be formulated only with cheap H 1 -elements even if the original equations include third derivative u xxx . Numerical experiments confirm the stability and effectiveness of the proposed scheme.

Smooth and non-smooth traveling wave solutions of a class of nonlinear dispersive equation

There is the widespread existence of wave phenomena in physics, mechanics. This clearly necessitates a study of traveling waves in depth and of the modeling and analysis involved. In this paper, we study a nonlinear dispersive K( n,− n,2 n) equation, which can be regarded as a generalized K( n, n) equation. Applying the bifurcation theory and the method of phase portraits analysis, we obtain the dynamical behavior and special exact solutions of the K( n,− n,2 n) equation. As a result, the conditions under which peakon and compacton solutions appear are also given and the analytic expressions of peakon solutions, compacton and periodic cusp wave solutions are obtained.

Well-posedness of higher-order nonlinear Schrödinger equations in Sobolev spaces [formula omitted] and applications

In this paper we establish local well-posedness in the Sobolev space H s ( R n ) with s > s 0 for a general class of nonlinear dispersive equations of the type ∂ t u − i P ( D x ) u = F ( u ) , where P ( D x ) is an elliptic differential operator on R n with a real symbol, F ( u ) is a nonlinear function which behaves like | u | σ u for some constant σ > 0 , and s 0 is a critical index suggested by a standard scaling argument. By using such local result and conservation laws, we improve the known and obtain some new global well-posedness results for the fourth-order nonlinear Schrödinger equation i ∂ t u + a △ u + b △ 2 u = c | u | σ u .

On a model equation of traveling and stationary compactons

We introduce and study a new class of nonlinear dispersive equations: u t + ( u m ) x + [ Q ( u , u x , u x x ) ] x = 0 , where Q ( u , u x , u x x ) = q 0 ( u , u x ) u x x + q 1 ( u , u x ) u x 2 is the dispersive flux with typical q ′ s being monomials in u and u x (which amalgamates all KdV type equations with a monomial nonlinear dispersion) and show that it admits either traveling or stationary compactons. In the second case initial datum given on a compact support evolves into a sequence of stationary compactons, with the spatio-temporal evolution being confined to the initial support. We also discuss an N-dimensional extension u t + ( u m ) x + [ u a ( ∇ u ) 2 κ ∇ 2 u b ] x = 0 which induces N-dimensional compactons convected in x-direction. Two families of explicit solutions of N-dimensional compactons are also presented.

Existence and uniqueness of solutions with low regularity for a class of nonlinear dispersive equations

Working in Lagrangian coordinates, we prove the existence and uniqueness of solutions for a class of periodic nonlinear dispersive equations with continuously differentiable initial data. This lowers the regularity requirements available for the Cauchy problem by means of the semigroup approach for quasi-linear hyperbolic equations of evolution or by the viscosity method.

Local and global well-posedness for a class of nonlinear dispersive equations

We study the local and global well-posedness of the initial-value problem for the class of nonlinear dispersive PDEs of the form \[ u_t - M u_x + F(u)_x = 0, \;\;\; t\in \mathbb{R}, \] where $u=u(x,t)$, $x\in \mathbb{R} $ or $x\in \mathbb{T} $. Here $M$ is a linear operator, given in the Fourier space by the multiplication operator: $\hat{Mv} ( \xi ) = |\xi |^{2\beta } \hat{v} (\xi ),$ $ \beta \ge \frac12 $ and $F$ is a nonlinear, (sufficiently) smooth function. This equation is a generalization of the Korteweg--de Vries (KdV) equation ($\beta = 1 $), the Benjamin--Ono (BO) equation ($\beta = \frac12$) and the fifth-order KdV equation ($\beta = 2$). The nonlinearity can be very general, but a certain growth condition must be imposed in order to obtain global results. Roughly speaking, we impose that $(F'(r))_{+}$ grows at most like $\left| r\right| ^p$ as $r\to\infty $, for some $p < 4\beta $. Global existence of solutions is, therefore, intimately related to the balance between the strength of the nonlinearity and the dispersion relation. The semigroup methods developed by Goldstein--Oharu--Takahashi are being successfully applied here. Most of the results are presented in the periodic case (i.e., $x\in\mathbb{T} $), but they are also valid in the real-line case (when $x\in \mathbb{R} )$.

Effect of nonlinearity of the middle term of nonlinear dispersive equations on physical structures

In this work we study the effect of nonlinearity of the middle term of a class of nonlinear dispersive equations on the physical solutions. We show that the change of the exponent of the middle term plays a major role in exhibiting compactons, solitons and periodic solutions. Three model equations are adopted to carry out this study. The analysis reveals a change of the physical solutions as a result of the change of exponents.

Compactons in a class of nonlinear dispersive equations

In this work, we study the compactons structures in a class of nonlinear dispersive equations. The compactons, new form of solitary waves free of exponential tails and width independent of amplitude, are formally constructed. We further establish solitary patterns solutions for the defocusing branches of these models.

An analytic study of compactons structures in a class of nonlinear dispersive equations

In this work, we present an analytic study of the compactons structures in a class of nonlinear dispersive equations. The compactons: new form of solitary waves with compact support and width independent of amplitude, are formally constructed. We further establish solitary patterns solutions for the defocusing branches of these dispersive models.

Compact and noncompact structures in a class of nonlinearly dispersive equations

In this work, we study compact and noncompact dispersive structures formed by a class of nonlinear dispersive equations. We show that the focusing branches provide compactons solutions: solitons with compact support. We also show that the defocusing branches generate solitary patterns solutions. We test our work for a variety of nonlinear equations with positive and negative exponents.

Ill-Posedness Issues for the Benjamin--Ono and Related Equations

We establish that the Cauchy problem for the Benjamin--Ono equation and for a rather general class of nonlinear dispersive equations with dispersion slightly weaker than that of the Korteweg--de Vries equation cannot be solved by an iteration scheme based on the Duhamel formula. As a consequence, the flow map fails to be smooth.