New exact solutions for a generalization of the Korteweg-de Vries equation (KdV6)

The generalized tanh-coth method is used to construct periodic and soliton solutions for a new integrable system, which has been derived from an integrable sixth-order nonlinear wave equation (KdV6). The system is formed by two equations. One of the equations may be considered as a Korteweg-de Vries equation with a source and the second equation is a third-order linear differential equation.     

A non-homogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane

The Korteweg-de Vries equation was first derived by Boussinesq and Korteweg and de Vries as a model for long-crested small-amplitude long waves propagating on the surface of water. The same partial differential equation has since arisen as a model for unidirectional propagation of waves in a variety of physical systems. In mathematical studies, consideration has been given principally to pure initial-value problems where the wave profile is imagined to be determined everywhere at a given instant of time and the corresponding solution models the further wave motion. The practical, quantitative use of the Korteweg-de Vries equation and its relatives does not always involve the pure initial-value problem. Instead, initial-boundary-value problems often come to the fore. A natural example arises when modeling the effect in a channel of a wave maker mounted at one end, or in modeling near-shore zone motions generated by waves propagating from deep water. Indeed, the initial-boundary-value problem

studied here arises naturally as a model whenever waves determined at an entry point propagate into a patch of a medium for which disturbances are governed approximately by the Korteweg-de Vries equation. The present essay improves upon earlier work on (0.1) by making use of modern methods for the study of nonlinear dispersive wave equations. Speaking technically, local well-posedness is obtained for initial data in the class for \frac34$"> and boundary data in , whereas global well-posedness is shown to hold for when , and for when . In addition, it is shown that the correspondence that associates to initial data and boundary data the unique solution of (0.1) is analytic. This implies, for example, that solutions may be approximated arbitrarily well by solving a finite number of linear problems.

     

Attractor for nonlinear Schrödinger equation coupling with stochastic weakly damped, forced KdV equation

The nonlinear Schrödinger equation coupling with stochastic weakly damped, forced KdV equation with additive noise can be solved pathwise, and the unique solution generates a random dynamical system. Then we prove that the system possesses a global weak random attractor.     

Stability of periodic traveling waves for complex modified Korteweg-de Vries equation

We study the existence and stability of periodic traveling-wave solutions for complex modified Korteweg-de Vries equation. We also discuss the problem of uniform continuity of the data-solution mapping.     

Self-similar planar curves related to modified Korteweg-de Vries equation

We exhibit a time reversible geometric flow of planar curves which can develop singularities in finite time within the uniform topology. The example is based on the construction of selfsimilar solutions of modified Korteweg-de Vries equation of a given (small) mean.     

On the exponential decay of the critical generalized Korteweg-de Vries equation with localized damping

This paper is concerned with the asymptotic behavior of solutions of the critical generalized Korteweg-de Vries equation in a bounded interval with a localized damping term. Combining multiplier techniques and compactness arguments it is shown that the problem of exponential decay of the energy is reduced to prove the unique continuation property of weak solutions. A locally uniform stabilization result is derived.

     

Solitary waves for Korteweg-de Vries equation with small delay

This paper deals with the existence of solitary waves for Korteweg-de Vries equation with time delay. Based upon the inertial manifold theory and differential manifold geometric theory, the existence of solitary wave solution is proved when the delay is small enough. Up to now, studies on solitary wave for such delay differential equation are not available, so the results of this paper are new.     

A non-homogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain II

Studied here is an initial- and boundary-value problem for the Korteweg-de Vries equation

     

Computing exact solutions to a generalized Lax seventh-order forced KdV equation (KdV7)

In this paper, the Cole-Hopf transform is used to construct exact solutions to a generalization of both the seventh-order Lax KdV equation (Lax KdV7) and the seventh-order Sawada-Kotera-Ito KdV equation (Sawada-Kotera-Ito KdV7 ) with forcing term.     

Derivation of Korteweg-de Vries flow equations from the regularized long-wave (RLW) equation

We perform a multiple-time scales analysis and compatibility condition to the regularized long-wave (RLW) equation. We derive Korteweg-de Vries (KdV) flow equation in the bi-Hamiltonian form as an amplitude equation.     

Compact attractor for weakly damped driven Korteweg-de Vries equations on the real line

We investigate the long-time behaviour of solutions to the Korteweg-de Vries equation with a zero order dissipation and an additional forcing term, when the space variable varies over , and prove that it is described by a maximal compact attractor in H ²().     

The Korteweg-de Vries equation in a cylindrical pipe

A mathematical model of nonlinear wave propagation in a pipeline is constructed. The Korteweg-de Vries equation is derived by determining asymptotic solutions and changing variables. A particular solution to the model equations is found that has the fluid velocity function in the form of a solitary wave. Thus, the class of nonlinear fluid dynamics problems described by the KdV equation is expanded.     

Odd periodic waves and stability results for the defocusing mass-critical Korteweg-de Vries equation

In this paper, we present results of existence and stability of odd periodic traveling wave solutions for the defocusing mass-critical Korteweg-de Vries equation. The existence of periodic wave trains is obtained by solving a constrained minimization problem. Concerning the stability, we use the Floquet theory to determine the behavior of the first three eigenvalues of the linearized operator around the wave, as well as the positiveness of the associated Hessian matrix.     

On the Propagation of Regularity and Decay of Solutions to the k-Generalized Korteweg-de Vries Equation

We study special regularity and decay properties of solutions to the IVP associated to the k-generalized KdV equations. In particular, for datum u ₀ ∈ H ^3/4+ (?) whose restriction belongs to H ^l ((b, ∞)) for some l ∈ ?⁺ and b ∈ ? we prove that the restriction of the corresponding solution u(·, t) belongs to H ^l ((β, ∞)) for any β ∈ ? and any t ∈ (0, T). Thus, this type of regularity propagates with infinite speed to its left as time evolves.     

Exact controllability of the nonlinear third-order dispersion equation

Exact controllability of a nonlinear dispersion system has been studied. This work extends the work of Russell and Zhang [D.L. Russell, B.Y. Zhang, Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain, SIAM J. Control Optim. 31 (1993) 659-676], in which the authors considered a linear dispersion system. We obtain controllability results using two standard types of nonlinearities, namely, Lipschitzian and monotone. We also obtain the exact controllability of the same system through the approach of Integral Contractors which is a weaker condition than Lipschitz condition.     

On the numerical solution of Korteweg-de Vries equation by the iterative splitting method

In this paper, we apply the method of iterative operator splitting on the Korteweg-de Vries (KdV) equation. The method is based on first, splitting the complex problem into simpler sub-problems. Then each sub-equation is combined with iterative schemes and solved with suitable integrators. Von Neumann analysis is performed to achieve stability criteria for the proposed method applied to the KdV equation. The numerical results obtained by iterative splitting method for various initial conditions are compared with the exact solutions. It is seen that they are in a good agreement with each other.     

Control and Stabilization of the Korteweg-de Vries Equation on a Periodic Domain

In [38 Russell , D.L. , Zhang , B.-Y. ( 1996 ). Exact controllability and stabilizability of the Korteweg-de Vries equation . Trans. Amer. Math. Soc. 348 : 3643 – 3672 . [Google Scholar]], Russell and Zhang showed that the Korteweg-de Vries equation posed on a periodic domain  with an internal control is locally exactly controllable and locally exponentially stabilizable when the control acts on an arbitrary nonempty subdomain of . In this paper, we show that the system is in fact globally exactly controllable and globally exponentially stabilizable. The global exponential stabilizability is established with the aid of certain properties of propagation of compactness and regularity in Bourgain spaces for the solutions of the associated linear system. With Slemrod's feedback law, the resulting closed-loop system is shown to be locally exponentially stable with an arbitrarily large decay rate. A time-varying feedback law is further designed to ensure a global exponential stability with an arbitrarily large decay rate.     

Three types of exact solutions to Wick-type generalized stochastic Korteweg-de Vries equation

A Wick-type generalized stochastic Korteweg-de Vries equation is researched. By means of Hermite transformation, white noise theory and Riccati equation mapping method, three types of exact solutions to the generalized stochastic Korteweg-de Vries equation, which include the functional solutions of hyperbolic-exponential type, trigonometric-exponential type and exponential type, are derived.     

The breather wave solutions, M-lump solutions and semi-rational solutions to a (2+1)-dimensional generalized Korteweg-de Vries equation

Under investigation in this work is a (2+1)-dimensional generalized Korteweg-de Vries equation, which can be used to describe many nonlinear phenomena in plasma physics. By using the properties of Bell"s polynomial, we obtain the bilinear formalism of this equation. The expression of $N$-soliton solution is established in terms of the Hirota"s bilinear method. Based on the resulting $N$-soliton solutions, we succinctly show its breather wave solutions. Furthermore, with the aid of the corresponding soliton solutions, the $M$-lump solutions are well presented by taking a long wave limit. Two types of hybrid solutions are also represented in detail. Finally, some graphic analysis are provided in order to better understand the propagation characteristics of the obtained solutions.