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Non dispersive solutions of the generalized KdV equations are typically multi-solitons
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Abstract
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.
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In this paper we consider the slightly -supercritical gKdV equations , with the nonlinearity and . In the previous paper \cite{L} we know that there exists an stable self-similar blow-up dynamics for slightly -supercritical gKdV equations. Such solution can be viewed as solutions with single blow-up point. In this paper we will prove the existence of solutions with multiple blow-up points, and give a description of the formation of the singularity near the blow-up time.
Considered herein are the stability and instability properties of
solitary-wave solutions of a general class of equations that arise as
mathematical models for the unidirectional propagation of weakly
nonlinear dispersive long waves. Special cases for which the present
analysis is decisive include equations of the Korteweg-de Vries and
Benjamin-Ono type. Necessary and sufficient conditions are formulated in
terms of the linearized dispersion relation and the nonlinearity for the
solitary waves to be stable.
Breather solutions of the modified Korteweg-de Vries equation are shown to be
globally stable in a natural H^2 topology. Our proof introduces a new Lyapunov
functional, at the H^2 level, which allows to describe the dynamics of small
perturbations, including oscillations induced by the periodicity of the
solution, as well as a direct control of the corresponding instability modes.
In particular, degenerate directions are controlled using low-regularity
conservation laws.
We study the problem of 2-soliton collision for the generalized Korteweg-de
Vries equations, completing some recent works of Y. Martel and F. Merle. We
classify the nonlinearities for which collisions are elastic or inelastic. Our
main result states that in the case of small solitons, with one soliton smaller
than the other one, the unique nonlinearities allowing a perfectly elastic
collision are precisely the integrable cases, namely the quadratic (KdV), cubic
(mKdV) and Gardner nonlinearities.
We consider the quintic generalized Korteweg–de Vries equation (gKdV)
which is a canonical mass critical problem, for initial data in H
1 close to the soliton. In earlier works on this problem, finite- or infinite-time blow up was proved for non-positive energy solutions, and the solitary wave was shown to be the universal blow-up profile, see [16], [26] and [20]. For well-localized initial data, finite-time blow up with an upper bound on blow-up rate was obtained in [18].
In this paper, we fully revisit the analysis close to the soliton for gKdV in light of the recent progress on the study of critical dispersive blow-up problems (see [31], [39], [32] and [33], for example). For a class of initial data close to the soliton, we prove that three scenarios only can occur: (i) the solution leaves any small neighborhood of the modulated family of solitons in the scale invariant L
2 norm; (ii) the solution is global and converges to a soliton as t → ∞; (iii) the solution blows up in finite time T with speed
Moreover, the regimes (i) and (iii) are stable. We also show that non-positive energy yields blow up in finite time, and obtain the characterization of the solitary wave at the zero-energy level as was done for the mass critical non-linear Schrödinger equation in [31].
We consider the quartic (nonintegrable) (gKdV) equation
and u(t) an outgoing 2-soliton of the equation, that is, a solution satisfying
where 0<c2<c1 and where Qcj(x−cjt) are explicit solitons of the equation. In [17], in the case , where ϵ0 is a small enough, not explicit constant, the solution u(t,x) is computed up to some order of ϵ, for all t and x. In particular, it is deduced that u(t) is not a multi-soliton as , proving the nonexistence of pure multi-soliton in this context. In the present paper, we prove the same result for an explicit
range of speeds: , by a different approach, which does not longer require a precise description of the solution. In fact, the nonexistence
result holds for outgoing N-solitons, for any N≥2, under the assumption: , which is a natural generalization of the condition for N=2.
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 (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].
The Korteweg-de Vries equation backslash[u_t+uu_x+u_xxx=0backslash] is a nonlinear partial differential equation arising in the study of a number of diferent physical systems, e.g., water waves, plasma physics, anharmonic lattices, and elastic rods. It describes the long time evolution of small-but-finite amplitude dispersive waves. From detailed studies of properties of the equation and its solutions, the concept of solitons was introduced and the method for exact solution of the initial-value problem using inverse scattering theory was developed. A survey of these and other results for the Korteweg-de Vries equation are given, including conservation laws an alternate method for exact solution, soliton solutions, asymptotic behavior of solutions, B-cklund transformation, and a nonlinear WKB method. The recent literature contains many extensions of these ideas to a number of other nonlinear evolution equations of physical interest and to other classes of equations. Some of these equations and results are indicated. The paper concludes with a list of open problems.
In this paper, we are interested in the phenomenon of blow up in nite time (or formation of singularity in nite time) of solutions of the critical generalized KdV equation. Few results are known in the context of partial dierential equations with a Hamiltonian structure. For the semilinear wave equation, or more generally for hyperbolic systems, the nite speed of propagation allows one to build blowing up solutions by reducing the problem to an ordinary dierential equation. For the nonlinear Schrodinger equation, iut = uj ujp 1u; where u : R RN ! C; (1) the formation of singularity is related to the existence of a conformal invariance of the equation in the critical case, namely,
Consider an abstract Hamiltonian system which is invariant under a one-parameter unitary group of operators. By a “solitary wave” we mean a solution the time development of which is given exactly by the one-parameter group. We find sharp conditions for the stability and instability of solitary waves. Applications are given to bound states and traveling waves of nonlinear PDEs such Klein-Gordon and Schrödinger equations.
In this paper, we prove that there exist no blow-up solutions of
the critical generalized Korteweg‒de Vries (gKdV) equation with
minimal L2-mass, assuming an L2-decay on the right on the
initial data.
We consider the generalized Korteweg–de Vries (gKdV) equations
in the subcritical cases p = 2, 3 or 4. The first objective of this paper is to present a direct, simplified proof of the asymptotic stability of solitons in the energy space H¹, which was proved by the same authors in Martel and Merle (2001 Arch. Ration. Mech. Anal. 157 219–54).
Then, in the case of the KdV equation, we show the optimality of the result by constructing a solution which behaves asymptotically as a soliton located on a curve which is a logarithmic perturbation of a line. This example justifies the apparently weak control on the location of the soliton in the asymptotic stability result.
We consider the generalized Korteweg-de Vries equations u(t) + (u(xx) + u(P))(x) = 0, t,x is an element of R, in the subcritical and critical cases p = 2,3,4 or 5. Let R-j(t,x) = Qc(j)(x - c(j)t - x(j)), where j is an element of {1,..., N}, be N soliton solutions of this equation, with corresponding speeds 0 < c(1) < c(2) <...< c(N). In this paper, we construct a solution u(t) of the generalized Korteweg-de Vries equation such that N [GRAPHIC] This solution behaves asymptotically as t -> +infinity as the sum of N solitons without loss of mass by dispersion. This is an exceptional behavior, indeed, being given the parameters {cj}1 <= j <= N, {xj} 1 <= j <= N, we prove uniqueness of such a solution. In the integrable cases p = 2 and 3, such solutions are explicitly known and their properties were extensively studied in the literature (they are called N-soliton solutions). Therefore, the existence result is new only for the nonintegrable cases. The uniqueness result is new for all cases.
It was proved in papers by S. Martel and F. Merle that any H 1 solution of the critical or subcritical generalized KdV equations which is close to a soliton and satisfies a property of uniform localization of the L 2 norm (we call such solution an L 2 -compact solution) is exactly a soliton. This result was a key tool in those papers for studying the asymptotic behavior of global solutions close to the family of solitons and building a large class of blow up solutions in the critical case. In this article, we study properties of L 2 -compact solutions for the generalized KdV equations (without restriction of subcriticality) that are not necessarily close to solitons. We prove C ∞ regularity and uniform exponential decay properties for these solutions. For the KdV equation, as a consequence of this result, using the decomposition for large time of any smooth and decaying solution into the sum of N solitons we prove that any L 2 -compact solution is a soliton. This extends the above-mentioned result of Martel and Merle by removing the assumption of closeness to a soliton.
On presente une nouvelle demonstration de la stabilite orbitale des solitons d'etat fondamental de l'equation de Schrodinger non lineaire pour une grande classe de non-linearites. On demontre la stabilite de l'onde solitaire pour l'equation de Korteweg-de Vries generalisee
We study the solution u(x, t) of the Korteweg-de Vries equation ut − 6uux + uxxx = 0 with arbitrary initial conditions u(x, 0) = u0(x), u0(x) decaying sufficiently rapidly as |ξ|→∞. Using the method of the inverse scattering transformation we analyse the Gel'fand-Levitan equation in all coordinate systems moving to the right and give a complete and rigorous description of the emergence of solitons.
The construction suggested by an inverse-scattering analysis establishes the existence of solutions u(x, t) of the Korteweg-de Vries equation subject to an initial condition u(x, 0)=U(x), where U has certain regularity and decay properties. It is assumed that UC3(), that U is piecewise of class C
4, and that U
(j) decays at an algebraic rate for j4. The faster the decay of U
(j) the smoother the solution will be for t0. If U and its first four derivatives decay faster than x–n for all n, then the solution will be infinitely differentiable for t0. For t>0, the decay rate of u(x, t) as x + increases with the decay rate of U; but the decay rate as x - depends on the regularity of U. A solution u
1 of the Korteweg-de Vries equation such that u
1(, 0)C() may fail to remain in class C
for all time if u
1(x, 0) does not decay fast enough as x.
We prove in this paper the instability of the solitons for the critical generalized Korteweg—de Vries equation. We obtain
this result by a qualitative study of the solution close to the soliton and obtain a large set of initial data giving the
instability. This study is a step towards the existence and description of blow up solutions.
This paper describes the interaction of two solitons with nearly equal speeds for the quartic (gKdV) equation. partial derivative(t)u + partial derivative(x)(partial derivative(2)(x)u + u(4)) = 0, t, x is an element of R. (0.1) We call soliton a solution of (0.1) of the form u(t, x) = Q(c)(x - ct - y(0)), where c > 0, y(0) is an element of R and Q ''(c) + Q(c)(4) = cQ(c). Since (0.1) is not an integrable model, the general question of the collision of two given solitons Q(c1)(x - c(1)t), Q(c2)(x - c(2)t) with c(1) not equal c(2) is an open problem. We focus on the special case where the two solitons have nearly equal speeds: let U(t) be the solution of (0.1) satisfying lim(t -> 8-infinity) parallel to U(t) - Q(c1)(-) (. - c(1)(-)t) - Q(c2-) (. - c(2)(-)t)parallel to(1)(H) = 0, for mu(0) = (c(2)(-) - c(1)(-))/(c(1)(-) + c(2)(-)) > 0 small. By constructing an approximate solution of ( 0.1), we prove that, for all time t is an element of R, U(t) = Q(c1)(t)(x - y(1)(t)) + Q(c2(t))(x - y(2)(t)) + w(t) where parallel to w(t)parallel to(H1) <= | 1n mu(0)|mu(2)(0), with y(1)(t)- y(2)(t) > 2|ln mu(0)|+C, for some C is an element of R. These estimates mean that the two solitons are preserved by the interaction and that for all time they are separated by a large distance, as in the case of the integrable KdV equation in this regime. However, unlike in the integrable case, we prove that the collision is not perfectly elastic, in the following sense, for someC > 0, lim(t ->+infinity) c(1)(t) > c(2)(-) (1 + mu(5)(0)/C), lim(t+infinity) c(2)(t) < c(1)(-) (1 - mu(5)(0)/C) and w(t) (->) over bar 0 in H(1) as t -> +infinity.
We prove in this paper a rigidity theorem on the flow of the critical generalized Korteweg–de Vries equation close to a soliton up to scaling and translation. To prove this result we introduce new tools to understand nonlinear phenomenon. This will give a result of asymptotic completeness.
Multi-soliton solutions, i.e. solutions behaving as the sum of N given solitons as , were constructed for the critical and subcritical (NLS) and (gKdV) equations in previous works (see [Merle, F.: Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity. Comm. Math. Phys. 129 (1990), no. 2, 223-240], [Martel, Y.: Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations. Amer. J. Math. 127 (2005), no. 5, 1103-1140] and [Martel, Y. and Merle, F.: Multi solitary waves for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), 849-864]). In this paper, we extend the construction of multi-soliton solutions to the supercritical case both for (gKdV) and (NLS) equations, using a topological argument to control the direction of instability.
For the L ² subcritical and critical (gKdV) equations, Martel [11 Martel , Y. ( 2005 ). Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations . Amer. J. Math. 127 ( 5 ): 1103 – 1140 . [Google Scholar]] proved the existence and uniqueness of multi-solitons. Recall that for any N given solitons, we call multi-soliton a solution of (gKdV) which behaves as the sum of these N solitons asymptotically as t → + ∞. More recently, for the L ² supercritical case, C
te et al. [4 Côte , R. , Martel , Y. , Merle , F. Construction of multi-soliton solutions for the L ²-supercritical gKdV and NLS equations . To appear in Revista Matematica Iberoamericana . ] proved the existence of at least one multi-soliton. In the present paper, as suggested by a previous work concerning the one soliton case [3 Combet , V. ( 2010 ). Construction and characterization of solutions converging to solitons for supercritical gKdV equations . Diff. Int. Eqs. 23 : 513 – 568 . [Google Scholar]], we first construct an N-parameter family of multi-solitons for the supercritical (gKdV) equation, for N arbitrarily given solitons, and then prove that any multi-soliton belongs to this family. In other words, we obtain a complete classification of multi-solitons for (gKdV).
We show that the family of solitary waves (1-solitons) of the Korteweg-de Vries equation is asymptotically stable. Our methods also apply for the solitary waves of a class of generalized Korteweg-de Vries equations, In particular, we study the case wheref(u)=u
p+1/(p+1),p=1, 2, 3 (and 3<p<4, foru>0, withf∈C
4). The same asymptotic stability result for KdV is also proved for the casep=2 (the modified Korteweg-de Vries equation). We also prove asymptotic stability for the family of solitary waves for all but a finite number of values ofp between 3 and 4. (The solitary waves are known to undergo a transition from stability to instability as the parameterp increases beyond the critical valuep=4.) The solution is decomposed into a modulating solitary wave, with time-varying speedc(t) and phase γ(t) (bound state part), and an infinite dimensional perturbation (radiating part). The perturbation is shown to decay exponentially in time, in a local sense relative to a frame moving with the solitary wave. Asp→4−, the local decay or radiation rate decreases due to the presence of aresonance pole associated with the linearized evolution equation for solitary wave perturbations.
We consider the generalized Korteweg-de Vries equation \partial_t u + \partial_x (\partial_x^2 u + f(u))=0, \quad (t,x)\in [0,T)\times \mathbb{R}, (1) with general nonlinearity f. Under an explicit condition on f and , there exists a solution in the energy space of (1) of the type , called soliton. In this paper, under general assumptions on f and , we prove that the family of soliton solutions around is asymptotically stable in some local sense in , i.e. if u(t) is close to (for all ), then u(t) locally converges in the energy space to some as . Note in particular that we do not assume the stability of . This result is based on a rigidity property of equation (1) around in the energy space whose proof relies on the introduction of a dual problem. These results extend the main results in previous works devoted to the pure power case.
This paper concerns the problem of collision of two solitons for the quartic generalized Korteweg-de Vries equation. We introduce a new framework to describe the collision in the special case where one soliton is small with respect to the other. We prove that the two soliton survive the collision, we describe the collision phenomenon (computation of the first order of the resulting shifts on the solitons). Moreover, we prove that in this situation, there does not exist pure two-soliton solutions.
Soliton resolution for the modified KdV equation
- Gong Chen
- Jiaqi Liu
Gong Chen and Jiaqi Liu. Soliton resolution for the modified KdV equation, 2019.
III: Exotic regimes. Annali della Scuola Normale Superiore di Pisa
- Yvan Martel
- Frank Merle
- Pierre Raphaël
Yvan Martel, Frank Merle, and Pierre Raphaël. Blow up for the critical generalized Korteweg-de Vries equation.
III: Exotic regimes. Annali della Scuola Normale Superiore di Pisa, XIV:575-631, 2015.
Asymptotic Analysis of Solitons Problems
- C Peter
- Schuur
Peter C. Schuur. Asymptotic Analysis of Solitons Problems. Berlin: Springer-Verlag, 1986.
On the existence and uniqueness of multi-breathers of (mKdV)
- Alexander Semenov
Alexander Semenov. On the existence and uniqueness of multi-breathers of (mKdV). preprint.