We consider solutions of the generalized Korteweg-de Vries equations (gKdV) which are non dispersive in some sense (in the spirit of [18]) and which remain close to multi-solitons. We show that these solutions are necessarily pure multi-solitons. For the Korteweg-de Vries equation (KdV) and the modified Korteweg-de Vries equation (mKdV) in particular, we obtain a characterization of multi-solitons and multi-breathers in terms of non-dispersion.
We consider the Korteweg-de Vries (KdV) equation, and prove that small localized data yields solutions which have dispersive decay on a quartic time-scale. This result is optimal, in view of the emergence of solitons at quartic time, as predicted by inverse scattering theory.
In this article, we prove that small localized data yield solutions to Higher order Korteweg-de Vries type equation with scattering-supercritical nonlinearity have linear dispersive decay in only a finite length of time. The proof is done by using space-time resonance method and analyzing the oscillatory integrals on the Fourier side.
In this paper we prove instability of the soliton for the focusing, mass-critical generalized KdV equation. We prove that the solution to the generalized KdV equation for any initial data with mass smaller than the mass of the soliton and close to the soliton in $L^{2}$ norm must eventually move away from the soliton.
We investigate the limit behavior of the solutions to the Kawahara equation $$ u_t +u_{3x} + \varepsilon u_{5x} + u u_x =0, $$ as $ 0<\varepsilon \to 0 $. In this equation, the terms $ u_{3x} $ and $ \varepsilon u_{5x} $ do compete together and do cancel each other at frequencies of order $ 1/\sqrt{\varepsilon} $. This prohibits the use of a standard dispersive approach for this problem. Nervertheless, by combining different dispersive approaches according to the range of spaces frequencies, we succeed in proving that the solutions to this equation converges in $ C([0,T];H^1(\R)) $ towards the…
We consider in this paper the well-posedness for the Cauchy problem associated to two-dimensional dispersive systems of Boussinesq type which model weakly nonlinear long wave surface waves. We emphasize the case of the {\it strongly dispersive} ones with focus on the "KdV-KdV" system which possesses the strongest dispersive properties and which is a vector two-dimensional extension of the classical KdV equation.
We consider KdV-type equations with $C^1$ nonhomogeneous nonlinearities and small dispersion $\varepsilon$. The first result consists in the conclusion that, in the leading term with respect to $\varepsilon$, the solitary waves in this model interact like KdV solitons. Next it turned out that there exists a very interesting scenario of instability in which the short-wave soliton remains stable whereas a small long-wave part, generated by perturbations of original equation, turns to be unstable, growing and destroying the leading term. At the same time, such perturbation can eliminate the colli…
In this paper we study the stability problem for KdV solitons on the left and right half-line. Unlike standard KdV, these are not exact solutions to the equations posed in the half-line. However, we are able to show that solitons placed initially far away from the origin are strongly stable for the problem posed on the right half-line, assuming homogeneous boundary conditions. For the problem posed on the left half-line, the positive infinitetime stability problem makes no sense for the case of KdV solitons, but in this setting we prove a result of stability for all negative times. The proof i…
In this work, we consider the stability of solitons for the KdV equation below the energy space, using spatially-exponentially-weighted norms. Using a combination of the $I$-method and spectral analysis following Pego and Weinstein, we are able to show that, in the exponentially weighted space, the perturbation of a soliton decays exponentially for arbitrarily long times. The finite time restriction is due to a lack of global control of the unweighted perturbation.
We revisit the phenomenon of instability of solitons in the generalized Korteweg-de Vries equation, $u_t + \partial_x(u_{xx} + u^p) = 0$. It is known that solitons are unstable for nonlinearities $p \geq 5$, with the critical power $p=5$ being the most challenging case to handle. The critical case was proved by Martel-Merle in [11], where the authors crucially relied on the pointwise decay estimates of the linear KdV flow. In this paper, we show simplified approaches to obtain the instability of solitons via truncation and monotonicity, which can be also useful for other KdV-type equations.
In this short note, we consider the global dynamics of the defocusing generalized KdV equations: u_t + u_{xxx} = (|u|^{p-1}u)_x. We use Tao's theorem that the energy moves faster than mass to prove a moment type dispersion estimate. As an application of the dispersion estimate, we show that there is no soliton-like solutions with decaying assumption.
In this note we prove scattering for perturbations of solitons in the scaling space appropriate for the quartic nonlinearity, namely $\dot{H}^{-\frac16}$. The article relies strongly on refined estimates for a KdV equation linearized at the soliton. In contrast to the work of Tao (2006), we are able to work purely in the scaling space without additional regularity assumptions, allowing us to prove some results on the existence of inverse wave operators.
Multisoliton solutions of the KdV equation satisfy nonlinear ordinary differential equations which are known as stationary equations for the KdV hierarchy, or sometimes as Lax-Novikov equations. An interesting feature of these equations, known since the 1970's, is that they can be explicitly integrated, by virtue of being finite-dimensional completely integrable Hamiltonian systems. Here we use the integration theory to investigate the question of whether the multisoliton solutions are the only nonsingular solutions of these ordinary differential equations which vanish at infinity. In particul…
We prove weighted estimates on the linear KdV group, which are scaling sharp. This kind of estimates are in the spirit of that used to prove small data scattering for the generalized KdV equations.
We study the long-time stability of soliton solutions to the Korteweg-deVries equation. We consider solutions $u$ to the KdV with initial data in $H^s$, $0 \leq s < 1$, that are initially close in $H^s$ norm to a soliton. We prove that the possible orbital instability of these ground states is at most polynomial in time. This is an analogue to the $H^s$ orbital instability result of \cite{CKSTT3}, and obtains the same maximal growth rate in $t$. Our argument is based on the {``}$I$-method{\rq\rq} used in \cite{CKSTT3} and other papers of Colliander, Keel, Staffilani, Takaoka and Tao, which pus…
We prove nonlinear modulational instability for both periodic and localized perturbations of periodic traveling waves for several dispersive PDEs, including the KDV type equations (e.g. the Whitham equation, the generalized KDV equation, the Benjamin-Ono equation), the nonlinear Schrödinger equation and the BBM equation. First, the semigroup estimates required for the nonlinear proof are obtained by using the Hamiltonian structures of the linearized PDEs; Second, for KDV type equations the loss of derivative in the nonlinear term is overcome in two complementary cases: (1) for smooth nonlinear…
The soliton resolution for the focusing modified Korteweg-de vries (mKdV) equation is established for initial conditions in some weighted Sobolev spaces. Our approach is based on the nonlinear steepest descent method and its reformulation through $\bar{\partial}$-derivatives. From the view of stationary points, we give precise asymptotic formulas along trajectory $x=\textrm{v}t$ for any fixed $ \textrm{v} $. To extend the asymptotics to solutions with initial data in low regularity spaces, we apply a global approximation via PDE techniques. As byproducts of our long-time asymptotics, we also o…
We prove the orbital stability of periodic traveling-wave solutions for systems of dispersive equations with coupled nonlinear terms. Our method is basically developed under two assumptions: one concerning the spectrum of the linearized operator around the traveling wave and another one concerning the existence of a conserved quantity with suitable properties. The method can be applied to several systems such as the Liu-Kubota-Ko system, the modified KdV system and a log-KdV type system.
In this paper, we reconsider the well-known result of Pego-Weinstein \cite{MR1289328} that soliton solutions to the Korteweg-deVries equation are asymptotically stable in exponentially weighted spaces. In this work, we recreate this result in the setting of modern well-posedness function spaces. We obtain asymptotic stability in the exponentially weighted space via an iteration argument. Our purpose here is to lay the groundwork to use the $I$-method to obtain asymptotic stability below $H^1$, which will be done in a second, forthcoming paper \cite{PR}. This will be possible because the expone…
In this article, we will prove $L^2(\mathbb{R})$-stability of $1$-solitons for the KdV equation by using exponential stability property of the semigroup generated by the linearized operator. The proof follows the lines of recent stability argument of Mizumachi [Asymptotic stability of lattice solitons in the energy space, Comm. Math. Phys., (2009)] and Mizumachi, Pego and Quintero [Asymptotic stability of solitary waves in the Benney-Luke model of water waves, Differential Integral Equations, (2013)] which show stability in the energy class by using strong linear stability of solitary waves in…
In this paper we obtain the following stability result for periodic multi-solitons of the KdV equation: We prove that under any given semilinear Hamiltonian perturbation of small size $\varepsilon > 0$, a large class of periodic multi-solitons of the KdV equation, including ones of large amplitude, are orbitally stable for a time interval of length at least $O(\varepsilon^{-2})$. To the best of our knowledge, this is the first stability result of such type for periodic multi-solitons of large size of an integrable PDE.
We consider the following variational problem: minimize the $(n+1)$st polynomial conserved quantity of KdV over $H^n(\mathbb{R})$ with the first $n$ conserved quantities constrained. Maddocks and Sachs used that $n$-solitons are local minimizers for this problem in order to prove that $n$-solitons are orbitally stable in $H^n(\mathbb{R})$. Given $n$ constraints that are attainable by an $n$-soliton, we show that there is a unique set of $n$ amplitude parameters so that the corresponding multisolitons satisfy the constraints. Moreover, we prove that these multisolitons are the unique global con…
We show how the inverse scattering transform can be used as a convenient tool to derive the long-time asymptotics of shock waves for the Korteweg-de Vries (KdV) equation in the soliton region. In particular, we improve the results previously obtained via the nonlinear steepest decent approach both with respect to the decay of the initial conditions as well as the region where they are valid.
It is a matter of experience that nonlinear waves in dispersive media, propagating primarily in one direction, may appear periodic in small space and time scales, but their characteristics --- amplitude, phase, wave number, etc. --- slowly vary in large space and time scales. In the 1970's, Whitham developed an asymptotic (WKB) method to study the effects of small "modulations" on nonlinear periodic wave trains. Since then, there has been a great deal of work aiming at rigorously justifying the predictions from Whitham's formal theory. We discuss recent advances in the mathematical understandi…
In this paper, the local well-posedness of periodic fifth order dispersive equation with nonlinear term $P_1(u)\p_xu + P_2(u)\p_x u\p_xu $. Here $P_1(u)$ and $P_2(u)$ are polynomials of $u$. We also get some new Strichartz estimates.
We revisit the asymptotic analysis of the KdV shock problem in the soliton region. Our approach is based on the analysis of the associated Riemann-Hilbert problem and we extend the domain of validity of the asymptotic formulas while at the same time requiring less decay and smoothness for the initial data.
We complete the known results on the local Cauchy problem in Sobolev spaces for the KdV-Burgers equation by proving that this equation is well-posed in $ H^{-1}(\R) $ with a solution-map that is analytic from $H^{-1}(\R) $ to $C([0,T];H^{-1}(\R))$ whereas it is ill-posed in $ H^s(\R) $, as soon as $ s<-1 $, in the sense that the flow-map $u_0\mapsto u(t) $ cannot be continuous from $ H^s(\R) $ to even ${\cal D}'(\R) $ at any fixed $ t>0 $ small enough. As far as we know, this is the first result of this type for a dispersive-dissipative equation. The framework we develop here should be very us…
We consider the problem of the soliton propagation, in a slowly varying medium, for a generalized Korteweg - de Vries equations (gKdV). We study the effects of inhomogeneities on the dynamics of a standard soliton. We prove that slowly varying media induce on the soliton solution large dispersive effects at large time. Moreover, unlike gKdV equations, we prove that there is no pure-soliton solution in this regime.
We prove in this paper the stability and asymptotic stability in H^1 of a decoupled sum of N solitons for the subcritical generalized KdV equations $u_t+(u_{xx}+u^p)_x=0$ (1<p<5). The proof of the stability result is based on energy arguments and monotonicity of local L^2 norm. Note that the result is new even for p=2 (the KdV equation). The asymptotic stability result then follows directly from a rigidity theorem in [15].
In this article we prove local well-posedness of quasilinear dispersive systems of PDE generalizing KdV. These results adapt the ideas of Kenig- Ponce-Vega from the Quasi-Linear Schrödinger equations to the third order dispersive problems. The main ingredient of the proof is a local smoothing estimate for a general linear problem that allows us to proceed via the artificial viscosity method.
A noncommutative KdV-type equation is introduced extending the Baecklund chart in [S. Carillo, M. Lo Schiavo, and C. Schiebold, SIGMA 12 (2016)]. This equation, called meta-mKdV here, is linked by Cole-Hopf transformations to the two noncommutative versions of the mKdV equations listed in [P.J. Olver and V.V. Sokolov Commun. Math. Phys. 193 (1998), Theorem 3.6]. For this meta-mKdV, and its mirror counterpart, recursion operators, hierarchies and an explicit solution class are derived.
We review the theory of modulation equations or Whitham equations for the travelling wave solution of KdV. We then apply the Whitham modulation equations to describe the long-time asymptotics and small dispersion asymptotics of the KdV solution
We consider the approximate control of solitons in generalized Korteweg-de Vries equations. By introducing a suitable internal bilinear control on the equation, we prove that any soliton is locally null controllable, and moreover, any soliton can be accelerated to any particular positive velocity, after a suitable large amount of time. Precise estimates on the error terms and the rate of decay in the approximate null controllability result are also given. Our method introduces a new insight on the control of nonlinear objects, from the point of view of interaction and collision problems for no…
The KdV equation models the propagation of long waves in dispersive media, while the NLS equation models the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves. A system that couples the two equations to model the interaction of long and short waves seems attractive and such a system has been studied over the last decades. We evaluate the validity of this system, discussing two main problems. First, only the system coupling the linear Schrödinger equation with KdV has been derived in the literature. Second, the time variables appearing in the equations are of a diff…
Solitary waves are localized gravity waves that preserve their consistency and henceforth their visibility through properties of nonlinear hydrodynamics. Solitary waves have finite amplitude and spread with constant speed and constant shape. In this paper, we have used Lie group of transformation method to solve (3 + 1)-dimensional KdV type equation. We have obtained the infinitesimal generators, commutator table of Lie algebra for the KdV type equation. We have achieved a number of exact solutions of KdV type equation in the explicit form through similarity reduction. All the reported results…
We analytically study the long time and large space asymptotics of a new broad class of solutions of the KdV equation introduced by Dyachenko, Zakharov, and Zakharov. These solutions are characterized by a Riemann--Hilbert problem which we show arises as the limit $N\to \infty$ of a gas of $N$-solitons. We show that this gas of solitons in the limit $N \to \infty$ is slowly approaching a cnoidal wave solution for $x \to - \infty$ (up to terms of order $\mathcal{O} (1/x)$), while approaching zero exponentially fast for $x\to+\infty$. We establish an asymptotic description of the gas of solitons…
The local and global control results for a general higher-order KdV-type operator posed on the unit circle are presented. Using spectral analysis, we are able to prove local results, that is, the equation is locally controllable and exponentially stable. To extend the local results to the global one we captured the smoothing properties of the Bourgain spaces, the so-called propagation of singularities, which are proved with a new perspective. These propagation, together with the Strichartz estimates, are the key to extending the local control properties to the global one, precisely, higher-ord…
We study the dynamics of solitons as solutions to the perturbed KdV (pKdV) equation $\partial_t u = -\partial_x (\partial_x^2 u + 3u^2-bu)$, where $b(x,t) = b_0(hx,ht)$, $h\ll 1$ is a slowly varying, but not small, potential. We option an explicit description of the trajectory of the soliton parameters of scale and position on the dynamically relevant time scale $δh^{-1}\log h^{-1}$, together with an estimate on the error of size $h^{1/2}$. In addition to the Lyapunov analysis commonly applied to these problems, we use a local virial estimate due to Martel-Merle (2005). The results are support…
It is universally accepted that the cubic, nonlinear Schrodinger equation (NLS) models the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves, while the Kortewegde Vries equation (KdV) models the propagation of long waves in dispersive media. A system that couples the two equations seems attractive to model the interaction of long and short waves and such a system has been studied over the last few decades. However, questions about the validity of the system in the study of water waves were raised in our previous work where we presented our analysis using the fifth…
In this paper we express tau functions for the Korteweg de Vries (KdV) equation, as Laplace transforms of iterated Skorohod integrals. Our main tool is the notion of Fredholm determinant of an integral operator. Our result extends the paper of Ikeda and Taniguchi who obtained a stochastic representation of tau functions for the $N$-soliton solutions of KdV as the Laplace transform of a quadratic functional of $N$ independent Ornstein-Uhlenbeck processes. Our general result goes beyond the $N$-soliton case and enables us to consider a non soliton solution of KdV associated to a Gaussian process…
We study the long-time dynamics of complex-valued modified Korteweg-de Vries (mKdV) solitons, which are recognized because they blow-up in finite time. We establish stability properties at the H^1 level of regularity, uniformly away from each blow-up point. These new properties are used to prove that mKdV breathers are H^1 stable, improving our previous result, where we only proved H^2 stability. The main new ingredient of the proof is the use of a Bäcklund transformation which links the behavior of breathers, complex-valued solitons and small real-valued solutions of the mKdV equation. We als…
We prove that solitons (or solitary waves) of the Zakharov-Kuznetsov (ZK) equation, a physically relevant high dimensional generalization of the Korteweg-de Vries (KdV) equation appearing in Plasma Physics, and having mixed KdV and nonlinear Schrödinger (NLS) dynamics, are strongly asymptotically stable in the energy space in the physical region. We also prove that the sum of well-arranged solitons is stable in the same space. Orbital stability of ZK solitons is well-known since the work of de Bouard. Our proofs follow the ideas by Martel and Martel and Merle, applied for generalized KdV equat…
We present a rigorous theory of a unified and simple inverse scattering transform (IST) for both focusing and defocusing real nonlocal (reverse-space-time) modified Korteweg-de Vries (mKdV) equations with non-zero boundary conditions (NZBCs) at infinity. The IST problems for the nonlocal equations with NZBCs are more complicated then ones for the local equations with NZBCs. The suitable uniformization variable is introduced in order to make the direct and inverse problems be established on a complex plane instead of a two-sheeted Riemann surface. The direct scattering problem establishes the a…
