Article

Self-similar dynamics for the modified Korteweg-de Vries equation

Authors:
To read the full-text of this research, you can request a copy directly from the authors.

No full-text available

Request Full-text Paper PDF

To read the full-text of this research,
you can request a copy directly from the authors.

... Recently new interest on self-similar solutions to the mkdV equation appears due to the link with the behavior of vortex filaments in fluid dynamics. In a series of works [3], [4], [5] the local well-posedness for small subcritical perturbations of self-similar solutions to the mKdV equation is proven. In the critical and super critical case p ≥ 4, self-similar profiles were constructed with precise asymptotic in [1] by means of the associated ODE. ...
... Theorem 2. If p > 2α/(α−1), then the profile v obtained in Theorem 1 satisfies equation (4) in distributional sense and belongs to C ∞ (R) ∩ W 1,∞ (R). ...
... which gives (4). In particular ...
Preprint
We consider the Cauchy problem for the generalized fractional Korteweg-de Vries equation ut+Dαux+upux=0,1<α2,pN{0}, u_t+D^\alpha u_x + u^p u_x= 0, \quad 1<\alpha\le 2, \quad p\in {\mathbb N}\setminus\{0\}, with homogeneous initial data Φ\Phi. We show that, under smallness assumption on Φ\Phi, and for a wide range of (α,p)(\alpha, p), including p=3, we can construct a self-similar solution of this problem.
... In the refered articles, the analysis in performed for subcritical solutions. In [4], together with Luis Vega, we introduced a critical space where long-time asymptotics could be analyzed. In particular, we proved that any small critical object converges to the self-similar solution with the same zero Fourier mode. ...
... The corresponding solutions are critical in terms of time, space and frequency decay. In physical space, the profiles have decay like |x|´1 {4 as x Ñ´8, while their derivative grows as |x| 1{4 (implying a strong oscillatory behavior, see for example [4] for precise asymptotics). In frequency space, the solutions are merely bounded, with logarithmic oscillations at infinity and a jump discontinuity at ξ " 0 (induced by the parameter α): we refer to Proposition 1.2 below for more details. ...
... Indeed, the required L 2 bound for B ξũ corresponds (up to controlled terms) to an L 2 bound for the vector-field, and this can be obtained through a direct energy estimate. These two mechanisms are enough to bound solutions for times away from 0 (as done in [4]). If one introduces sufficiently smooth subcritical perturbations, the bounds on the remainder can actually be shown to be uniform up to t " 0, which lead to the blow-up stability result. ...
Preprint
Full-text available
We prove a first stability result of self-similar blow-up for the modified KdV equation on the line. More precisely, given a self-similar solution and a sufficiently small regular profile, there is a unique global solution which behaves at t tends to 0 as the sum of the self-similar solution and the smooth perturbation.
... Interaction between modes. In the preprints by Correia-Côte-Vega [9,2], we studied a dispersive model for the formation and evolution of vortex filaments making a corner. We studied the self-similar solutions to the modified Korteweg-de Vries equation (which corresponds to the curvature of the filaments), and we found a (critical) functional framework in which the flow can be studied. ...
Preprint
Full-text available
This is the final report of the ANR project 14-CE25-0009-01 entitled "Mathematical Analysis of Topological Singularities in some physical problems" (MAToS) that was developed by the authors between January 2015-December 2019. The central theme of this project lied in the area of nonlinear analysis (nonlinear partial differential equations and calculus of variations). We focused on the structure and dynamics of topological singularities arising in some variational physical models driven by the Landau-Lifshitz equation (in micromagnetics) and the Gross-Pitaevskii equation (in superconductivity, Bose-Einstein condensation, nonlinear optics). These included vortex singularities, traveling waves and domain walls in magnetic thin films. These structures are observed experimentally and in numerical simulations and play an important role in the dynamics of the corresponding physical systems. We made significant progress in the mathematical analysis of these structures (both at the stationary and dynamical level) that gives more insight into the physical phenomena.
Article
Full-text available
We give the asymptotics of the Fourier transform of self-similar solutions for the modified Korteweg-de Vries equation. In the defocusing case, the self-similar profiles are solutions to the Painlevé II equation; although they were extensively studied in physical space, no result to our knowledge describe their behavior in Fourier space. These Fourier asymptotics are crucial in the study of stability properties of the self-similar solutions for the modified Korteweg-de Vries flow. Our result is obtained through a fixed point argument in a weighted W1,∞ space around a carefully chosen, two term ansatz, and we are able to relate the constants involved in the description in Fourier space with those of the description in physical space.
Article
Full-text available
We consider the Cauchy problem for the intermediate long-wave equation ut-∂xu2+1ϑux+VP∫R12ϑcothπy-x2ϑuyyt,ydy=0,utxu2+1ϑux+VPR12ϑcoth(π(yx)2ϑ)uyy(t,y)dy=0,\begin{aligned} u_{t}-\partial _{x}u^{2}+\frac{1}{\vartheta }u_{x}+VP\int _{\mathbb {R}}\frac{1}{2\vartheta }\coth \left( \frac{\pi \left( y-x\right) }{2\vartheta }\right) u_{yy}\left( t,y\right) \mathrm{d}y=0, \end{aligned}where ϑ>0ϑ>0\vartheta >0. Our purpose in this paper is to prove the large time asymptotic behavior of solutions under the nonzero mass condition ∫u0xdx≠0u0(x)dx0\int u_{0}\left( x\right) \mathrm{d}x\ne 0.
Article
Full-text available
We study the Cauchy problem for the modified KdV equation for data u_0 in the space ^H^r_s defined by the norm ||u_0||_{^H^r_s}:=||<\xi>^s u^_0||_{L^r'_\xi}. Local well-posedness of this problem is established in the parameter range 2>=r>1, s>=1/2-1/2r, so the case (s,r)=(0,1), which is critical in view of scaling considerations is almost reached. To show this result, we use an appropriate variant of the Fourier restriction norm method as well as bi- and trilinear estimates for solutions of the Airy equation.
Article
We prove a full asymptotic stability result for solitary wave solutions of the mKdV equation. We consider small perturbations of solitary waves with polynomial decay at infinity and prove that solutions of the Cauchy problem evolving from such data tend uniformly, on the real line, to another solitary wave as time goes to infinity. We describe precisely the asymptotics of the perturbation behind the solitary wave showing that it satisfies a nonlinearly modified scattering behavior. This latter part of our result relies on a precise study of the asymptotic behavior of small solutions of the mKdV equation.
Article
An asymptotic formula or asymptotic form for a function f(x) is the name usually given to an approximate formula f(x) ≈ g(x) in some domain of values of x, where g(x) is ‘simpler’ then f(x). For example, if f(x) is an integral, then g(x) must either be given in terms of the values of the integrand and its derivatives at a finite number of points, or in terms of some simpler integral. If f(x) is a solution of an ordinary differential equation, then g(x) must either be expressed in quadratures or be the solution of a ‘simpler’ differential equation. This list can be extended—there is an unwritten heirarchy of asymptotic formulae. Of course all these definitions are very blurred. ‘“What is asymptotics?” This question is about as difficult to answer as the question “What is mathematics?”’
Article
Integrable systems related to the Korteweg-de Vries (KdV) equation are shown to be associated with the dynamics of vortex patches in ideal two-dimensional fluids. This connection is based on a truncation of the exact contour dynamics analogous to the “localized induction approximation” which relates the nonlinear Schrödinger equation to the motion of a vortex filament. Single soliton solutions of the periodic modified KdV problem correspond to uniformly rotating shapes which approximate the Kirchoff ellipse and known generalizations. A simple geometrical interpretation of the dual Poisson bracket structure of the modified KdV hierarchies is given.
Article
In this paper we consider the long time behaviour of solutions to the modified Korteweg-de Vries equation on R. For sufficiently small, smooth, decaying data we prove global existence and derive modified asymptotics without relying on complete integrability. We also consider the asymptotic completeness problem. Our result uses the method of testing by wave packets, developed in the work of Ifrim and Tataru on the 1d cubic nonlinear Schr\"odinger and 2d water wave equations.
Article
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.
Article
In this article we present a new and general approach to analyzing the asymptotics of oscillatory Riemann-Hilbert problems. Such problems arise, in particular, when evaluating the long-time behavior of nonlinear wave equations solvable by the inverse scattering method. We will restrict ourselves here exclusively to the modified Korteweg-de Vries (MKdV) equation
Article
We consider the large time asymptotic behavior of solutions to the Cauchy problem for the modified Korteweg–de Vries equation ut+a(t)(u3)x+13uxxx=0,(t,x)R×Ru_t + a\left( t \right)\left( {u^3 } \right)_x + \frac{1}{3}u_{xxx} = 0,\left( {t,x} \right) \in R \times R, with initial data u(0,x)=u0(x),xRu\left( {0,x} \right) = u_0 \left( x \right),x \in R. We assume that the coefficient a(t)C1(R)a\left( t \right) \in C^1 \left( R \right) is real, bounded and slowly varying function, such that a(t)Ct76\left| {a'\left( t \right)} \right| \leqslant C\left\langle t \right\rangle ^{ - \frac{7}{6}}, where t=(1+t2)12\left\langle t \right\rangle = \left( {1 + t^2 } \right)^{\frac{1}{2}}. We suppose that the initial data are real-valued and small enough, belonging to the weighted Sobolev space H1,1={ϕL2;1+x21x2ϕ<}H^{1,1} = \left\{ {\phi \in L^2 ;\left\| {\sqrt {1 + x^2 } \sqrt {1 - \partial _x^2 } \phi } \right\| < \infty } \right\}. In comparison with the previous paper (Internat. Res. Notices 8 (1999), 395–418), here we exclude the condition that the integral of the initial data u 0 is zero. We prove the time decay estimates t23t3u(t)ux(t)Cε\sqrt[3]{{t^2 }}\sqrt[3]{{\left\langle t \right\rangle }}\left\| {u\left( t \right)u_x \left( t \right)} \right\|_\infty \leqslant C\varepsilon and t1313βu(t)βCε\left\langle t \right\rangle ^{\frac{1}{3} - \frac{1}{{3\beta }}} \left\| {u\left( t \right)} \right\|_\beta \leqslant C\varepsilon for all tRt \in R, where 4<β4 < \beta \leqslant \infty. We also find the asymptotics for large time of the solution in the neighborhood of the self-similar solution.
Article
The initial value problem associated with the second Painlev Transcendent is linearized via a matrix, discontinuous, homogeneous Riemann-Hilbert (RH) problem defined on a complicated contour (six rays intersecting at the origin). This problem is mapped through a series of transformations to three different simple Riemann-Hilbert problems, each of which can be solved via a system of two Fredholm integral equations. The connection of these results with the inverse scattering transform in one and two dimensions is also pointed out.
Article
The differential equation considered is yxy=yyαy'' - xy = y|y|^\alpha . For general positive α this equation arises in plasma physics, in work of De Boer & Ludford. For α=2, it yields similarity solutions to the well-known Korteweg-de Vries equation. Solutions are sought which satisfy the boundary conditions (1) y(∞)=0 (2) y()=0y{\text{(}}\infty {\text{)}} = {\text{0}} (1) y{\text{(}}x{\text{) \~( - }}\tfrac{{\text{1}}}{{\text{2}}}x{\text{)}}^{{{\text{1}} \mathord{\left/ {\vphantom {{\text{1}} \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} {\text{ as }}x \to - \infty (2) It is shown that there is a unique such solution, and that it is, in a certain sense, the boundary between solutions which exist on the whole real line and solutions which, while tending to zero at plus infinity, blow up at a finite x. More precisely, any solution satisfying (1) is asymptotic at plus infinity to some multiple kA i(x) of Airy's function. We show that there is a unique k*(α) such that when k=k*(α) the condition (2) is also satisfied. If 0<k<k *, the solution exists for all x and tends to zero as x→-∞, while if k>k * then the solution blows up at a finite x. For the special case α=2 the differential equation is classical, having been studied by Painlevé around the turn of the century. In this case, using an integral equation derived by inverse scattering techniques by Ablowitz & Segur, we are able to show that k*=1, confirming previous numerical estimates.
Article
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.
Article
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.
  • V Banica
  • L Vega
V. Banica and L. Vega. Evolution of polygonal lines by the binormal flow. arXiv:1807.06948, 2018.
Soliton resolution for the modified kdv equation
  • Gong Chen
  • Jiaqi Lui
Gong Chen and Jiaqi Lui. Soliton resolution for the modified kdv equation. Preprint, arXiv:1907.07115, 2019.
Painlevé transcendents
  • Athanassios S Fokas
  • Alexander R Its
  • Andrei A Kapaev
  • Victor Yu
  • Novokshenov
Athanassios S. Fokas, Alexander R. Its, Andrei A. Kapaev, and Victor Yu. Novokshenov. Painlevé transcendents, volume 128 of Math. Surveys Monogr. American Mathematical Society, Providence, RI, 2006. The Riemann-Hilbert approach.