Multi-soliton solutions of the Korteweg-de Vries equation (KdV) are shown to be globally L2-stable, and asymptotically stable in the sense of Martel-Merle. The proof is surprisingly simple and combines the Gardner transform, which links the Gardner and KdV equations, together with the Martel-Merle-Tsai and Martel-Merle recent results on stability and asymptotic stability in the energy space, applied this time to the Gardner equation. As a by-product, the results of Maddocks-Sachs and Merle-Vega are improved in several directions.
Following Deift-Zhou's nonlinear steepest descent method, the long-time asymptotic behavior for the Cauchy problem of the 5th order modified Korteweg-de Vries equation is analyzed. Based on the inverse scattering transform, the 5th order MKdV is transformed to a 2 by 2 oscillatory Riemann-Hilbert problem, then by manipulating the Cauchy operator and reducing the degree of the phase function, the long-time asymptotics of the solution is given in terms of solutions of the parabolic cylinder equation.
In this paper, we determine spectral stability results of periodic waves for the critical Korteweg-de Vries and Gardner equations. For the first equation, we show that both positive and zero mean periodic traveling wave solutions possess a threshold value which may provides us a rupture in the spectral stability. Concerning the second equation, we establish the existence of periodic waves using a Galilean transformation on the periodic cnoidal solution for the modified Korteweg-de Vries equation and for both equations, the threshold values are the same. The main advantage presented in our pape…
We consider the modified Korteweg-de Vries equation, $ u_{xxx} + 6u^2 u_x + u_t \ =\ 0 $, and explore its dynamics in {\it spatial} direction. Higher $x$ derivatives bring about the {\it ghosts}. We argue that these ghosts are benign, i.e. the classical dynamics of this system does not involve a blow-up. This probably means that also the associated quantum problem is well defined.
We prove global existence and modified scattering for the solutions of the Cauchy problem to the fractional Korteweg-de Vries equation with cubic nonlinearity for small, smooth and localized initial data.
This paper is dedicated to finding the solutions of the equation of the loaded modified Korteweg-de Vries. By the way, it is shown to find the solutions via $(G'/G)$-expansion method that is one of the most effective ways of finding solutions.
For more than 20 years, the Korteweg-de Vries equation has been intensively explored from the mathematical point of view. Regarding control theory, when adding an internal force term in this equation, it is well known that the Korteweg-de Vries equation is exponentially stable in a bounded domain. In this work, we propose a weak forcing mechanism, with a lower cost than that already existing in the literature, to achieve the result of the global exponential stability to the Korteweg-de Vries equation.
We are concerned with the decay of long time solutions of the initial value problem associated with the Schrödinger-Korteweg-de Vries system. We use recent techniques in order to show that solutions of this system decay to zero in the energy space. The result is independent of the integrability of the equations involved and it does not require any size assumptions.
We consider a linear Korteweg-de Vries equation on a bounded domain with a left Dirichlet boundary control.The controllability to the trajectories of such a system was proved in the last decade by using Carleman estimates.Here, we go a step further by establishing the exact controllability in a space of analytic functions with the aid of the flatness approach.
Existence and a priori estimates for real-valued periodic solutions to the modified Korteweg-de Vries equation with initial data in $H^s$ are established for $s>0$. The short-time Fourier restriction norm method is employed to overcome the derivative loss. Further, non-existence of solutions below $L^2$ is proved conditional upon conjectured linear Strichartz estimates.
We prove a local well posedness result for the modified Korteweg-de Vries equation in a critical space designed so that is contains self-similar solutions. As a consequence, we can study the flow of this equation around self-similar solutions: in particular, we give an as-ymptotic description of small solutions as t $\rightarrow$ +$\infty$ and construct solutions with a prescribed blow up behavior as t $\rightarrow$ 0.
In this paper, we develop dispersive PDE techniques for the Fermi--Pasta--Ulam (FPU) system with infinitely many oscillators, and we show that general solutions to the infinite FPU system can be approximated by counter-propagating waves governed by the Korteweg--de Vries (KdV) equation as the lattice spacing approaches zero. Our result not only simplifies the hypotheses but also reduces the regularity requirement in the previous study [45].
In this paper, a viscous shock wave under space-periodic perturbation of generalized Korteweg-de Vries-Burgers equation is investigated. It is shown that if the initial periodic perturbation around the viscous shock wave is small, then the solution time asymptotically tends to a viscous shock wave with a shift partially determined by the periodic oscillations. Moreover the exponential time decay rate toward the viscous shock wave is also obtained for some certain perturbations.
We prove local well-posedness for the initial-boundary value problem (IBVP) associated to the Schrödinger-Korteweg de Vries system on right and left half-lines. The results are obtained in the low regularity setting by using two analytic families of boundary forcing operators, being one of these family developed by Holmer to study the IBVP associated to the Korteweg-de Vries equation (Communications in Partial Differential Equations, 31 (2006)) and the other family one was recently introduced by Cavalcante (Differential and Integral Equations (2017)) in the context of nonlinear Schrödinger wit…
