A. Atmani, A. Boukarou, D. Benterki, Kh. Zennir
The solution of a coupled system consisting of generalized Korteweg-de Vries-type equations is obtained for all time where the initial data are analytic on a band in the complex plane. We show that the width of this band decreases algebraically with time.
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We present estimates for the radius of analyticity of solutions to the Korteweg-de Vries equation, which improve earlier results due to Bona, Grujić and Kalisch.
Lower bound on the rate of decrease in time of the uniform radius of spatial analyticity of solutions to the quartic generalized KdV equation is derived, which improves an earlier result by Bona, Grujić and Kalisch.
The radius of spatial analyticity for solutions of the KdV equation is studied. It is shown that the analyticity radius does not decay faster than $t^{-1/4}$ as time $t$ goes to infinity. This improves the works [Selberg, da Silva, Lower bounds on the radius of spatial analyticity for the KdV equation, Annales Henri Poincaré, 2017, 18(3): 1009-1023] and [Tesfahun, Asymptotic lower bound for the radius of spatial analtyicity to solutions of KdV equation, arXiv preprint arXiv:1707.07810, 2017]. Our strategy mainly relies on a higher order almost conservation law in Gevrey spaces, which is inspir…
We show that the uniform radius of spatial analyticity $σ(t)$ of solution at time $t$ for the fifth order KdV-BBM equation cannot decay faster than $1/t$ for large $t>0$, given initial data that is analytic with fixed radius $σ_0$. This significantly improves a recent result by Carvajal and Panthee, where they established an exponential decay of $σ(t)$ for large $t$.
We show that the uniform radius of spatial analyticity $σ(t)$ of solutions at time $t$ to the fifth order KdV-BBM equation cannot decay faster than $1/ \sqrt{t}$ for large $t$, given initial data that is analytic with fixed radius $σ_0$. This improves a recent result by Belayneh, Tegegn and the third author, where they obtained a $1/t$ decay of $σ(t)$ for large time $t$.
On propagation of regularities and evolution of radius of analyticity in the solution of the fifth order KdV-BBM model
该研究针对描述单向水波的五阶KdV-BBM模型初值问题，证明了解的正则性传播特性、Gevrey类中的局部适定性，并给出解析半径演化的显式上下界公式。
We are concerned with the Cauchy problem for the KdV equation for nonsmooth locally integrable initial profiles q's which are, in a certain sense, essentially bounded from below and q(x)=O(e^{-cx^{ε}}),x\rightarrow+\infty, with some positive c and ε. Using the inverse scattering transform, we show that the KdV flow turns such initial data into a function which is (1) meromorphic (in the space variable) on the whole complex plane if ε>1/2, (2) meromorphic on a strip around the real line if ε=1/2, and (3) Gevrey regular if ε<1/2. Note that q's need not have any decay or pattern of behavior at -\…
Improved algebraic lower bound for the radius of spatial analyticity for the generalized KdV equation
本文研究gKdV方程解的空间解析半径，证明局部时间内半径保持不变，并在散焦偶数次情形给出改进的全局代数衰减下界。
On the radius of spatial analyticity for the Majda-Biello and Hirota-Satsuma systems
本文首次研究了Majda-Biello和Hirota-Satsuma系统中解的空间解析性持续性的保持问题。
It is shown that the uniform radius of spatial analyticity $σ(t)$ of solutions at time $t$ to the KdV equation cannot decay faster than $|t|^{-4/3}$ as $|t| \to \infty$ given initial data that is analytic with fixed radius $σ_0$. This improves a recent result of Selberg and Da Silva, where they proved a decay rate of $|t|^{-(4/3 + \varepsilon) }$ for arbitrarily small positive $\varepsilon$. The main ingredients in the proof are almost conservation law for the solution to the KdV equation in space of analytic functions and space-time dyadic bilinear $L^2$ estimates associated with the KdV equa…
Improved lower bound for the radius of analyticity for the modified KdV equation
本文针对散焦 mKdV 方程，在带 cosh 权的 Gevrey 空间中构造新的几乎守恒律，证明解全局存在，并将空间解析半径下界改进为 cT^{-1/2}。
Decay of the radius of spatial analyticity for the modified KdV equation and the nonlinear Schrödinger equation with third order dispersion
本文研究 mKdV 与带三阶色散的三次 NLS 的空间解析半径演化，利用解析型三线性估计证明局部适定性，并给出全局解解析半径的代数衰减下界。
Nonlinear smoothing implies improved lower bounds on the radius of spatial analyticity for nonlinear dispersive equations
本文通过将问题归结为具有特定额外导数分布的非线性光滑估计，为散焦广义KdV方程和奇次纯幂非线性薛定谔方程建立了改进的解析半径下界，得到对任意ε>0成立的 $σ(T)\gtrsim T^{-\frac{1}{2}-ε}$。
This paper sheds new light on the stability properties of solitary wave solutions associated with models of Korteweg-de Vries and Benjamin\&Bona\&Mahoney type, when the dispersion is very lower. Via an approach of compactness, analyticity and asymptotic perturbation theory, we establish sufficient conditions for the existence of exponentially growing solutions to the linearized problem and so a criterium of linear instability of solitary waves is obtained for both models. Moreover, the nonlinear stability and linear instability of the ground states solutions for both models is obtained for som…
We prove the nonsqueezing property of the coupled Korteweg-de Vries (KdV) equation. Relying on Gromov's nonsqueezing theorem for finite dimensional Hamiltonian systems, the argument is to approximate the solutions to the original infinite dimensional Hamiltonian system by a frequency truncated finite dimensional system, and then the nonsqueezing property is transferred to the infinite dimensional system. This is the argument used by Bourgain for the 1D cubic NLS flow, and Colliander et. al. for the KdV flow. One of main ingredients of \cite{Colliander:2005vv} is to use the Miura transform to c…
We continue our study of the well-posedness theory of a one-parameter family of coupled KdV-type systems in the periodic setting. When the value of a coupling parameter α\in (0, 4) \setminus 1, we show that the Gibbs measure is invariant under the flow and the system is globally well-posed almost surely on the statistical ensemble, provided that certain Diophantine conditions are satisfied.
We prove the sharp global well-posedness results for the initial value problems (IVPs) associated to the modified Korteweg-de Vries (mKdV) equation and a system modeled by the coupled modified Korteweg-de Vries equations (mKdV-system). To obtain our results we use the second generation of the modified energy and almost conserved quantities, more precisely, the refined I-method introduced by Colliander, Keel, Staffilani, Takaoka and Tao.
In this article, we study an initial-boundary-value problem of a coupled KdV-KdV system on the half line $ \mathbb{R}^+ $ with non-homogeneous boundary conditions: \begin{equation*} \left\{ \begin{array}{l} u_t+v_x+u u_x+v_{xxx}=0, \quad v_t+u_x+(vu)_x+u_{xxx}=0, \quad u(x,0)=ϕ(x),\quad v(x,0)=ψ(x), \quad u(0,t)=h_1(t),\quad v(0,t)=h_2(t),\quad v_x(0,t)=h_3(t), \end{array} \right. \qquad x,\,t>0. \end{equation*} It is shown that the problem is locally unconditionally well-posed in $H^s(\mathbb{R}^+)\times H^s(\mathbb{R}^+)$ for $s> -\frac34 $ with initial data $(ϕ,ψ)$ in $H^s(\mathbb{R}^+)\tim…
We consider the existence of bound and ground states for a family of nonlinear elliptic systems in $\mathbb{R}^N$, which involves equations with critical power nonlinearities and Hardy-type singular potentials. The equations are coupled by what we call ``Schrödinger-Korteweg-de Vries'' non-symmetric terms, which arise in some phenomena of fluid mechanics. By means of variational methods, ground states are derived for several ranges of the positive coupling parameter $ν$. Moreover, by using min-max arguments, we seek bound states under some energy assumptions.
Asymptotic stability is with no doubts an essential property to be studied for any system. This analysis often becomes very difficult for coupled systems and even harder when different timescales appear. The singular perturbation method allows to decouple a full system into what are called the reduced order system and the boundary layer system, to get simpler stability conditions for the original system. In the infinite-dimensional setting, we do not have a general result making sure this strategy works. This papers is devoted to this analysis for some systems coupling the Korteweg-the Vries e…
We consider the local well-posedness problem of a one-parameter family of coupled KdV-type systems both in the periodic and non-periodic setting. In particular, we show that certain resonances occur, closely depending on the value of a coupling parameter αwhen α\ne 1. In the periodic setting, we use the Diophantine conditions to characterize the resonances, and establish sharp local well-posedness of the system in H^s(\mathbb{T}_λ), s \geq s^\ast, where s^\ast = s^\ast(α) \in ({1/2}, 1] is determined by the Diophantine characterization of certain constants derived from the coupling parameter α…
We prove the existence of bound and ground states for a system of coupled nonlinear Schrödinger-Korteweg-de Vries equations, depending on the size of the coupling coefficient.
Inspired by the recent successful completion of the study of the well-posedness theory for the Cauchy problem of the Korteweg-de Vries (KdV) equation \[ u_t +uu_x +u_{xxx}=0, \quad \left. u \right |_{t=0}=u_{0} \] in the space $H^{s} (\mathbb{R})$ (or $H^{s} (\mathbb{T})$), we study the well-posedness of the Cauchy problem for a class of coupled KdV-KdV (cKdV) systems \[\left\{\begin{array}{rcl} u_t+a_{1}u_{xxx} &=& c_{11}uu_x+c_{12}vv_x+d_{11}u_{x}v+d_{12}uv_{x},\\ v_t+a_{2}v_{xxx}&=& c_{21}uu_x+c_{22}vv_x +d_{21}u_{x}v+d_{22}uv_{x},\\ \left. (u,v)\right |_{t=0} &=& (u_{0},v_{0}) \end{array}\…
Insensitizing control problem for the Hirota-Satsuma system of KdV-KdV type
本文研究Hirota-Satsuma耦合KdV系统的不灵敏控制问题，将其转化为扩展系统的零控制问题，通过Carleman估计分析线性化模型的零能控性，再用逆映射定理证明非线性扩展系统的局部零能控性，从而得到不灵敏控制的存在性。
We study that a solution of the initial value problem associated for the coupled system of equations of Korteweg - de Vries type which appears as a model to describe the strong interaction of weakly nonlinear long waves, has analyticity in time and smoothing effect up to real analyticity if the initial data only has a single point singularity at $x=0.$
Studied in this paper is the well-posedness of the Cauchy problem for the coupled KdV-KdV systems \[ u_t+a_1u_{xxx} = c_{11}uu_x+c_{12}vv_x+d_{11}u_{x}v+d_{12}uv_{x}, \quad u(x,0)= u_0(x) \] \[ v_t+a_2v_{xxx}= c_{21}uu_x+c_{22}vv_x +d_{21}u_{x}v+d_{22}uv_{x}, \quad v(x,0)=v_0(x)\] posed on the torus $\mathbb{T}$ in the spaces \[ {\cal H}^s_1:=H^s_0 (\mathbb{T})\times H^s_0 (\mathbb{T}), \quad {\cal H}^s_2:=H^s_0 (\mathbb{T})\times H^s(\mathbb{T}), \quad {\cal H}^s_3:=H^s (\mathbb{T})\times H^s_0 (\mathbb{T}), \quad {\cal H}^s_4:=H^s (\mathbb{T})\times H^s (\mathbb{T}).\] For $k=1,2,3,4$, it is…
The integrability of a coupled KdV-mKdV system is tested by means of singularity analysis. The true Lax pair associated with this system is obtained by the use of prolongation technique.
On the space-time analyticity of the Keller-Segel-Navier-Stokes system
本文研究周期条带上耦合Keller-Segel-Navier-Stokes系统，通过能量方法证明了解直到边界的同步时空解析性。
We prove that one system of coupled KdV equations, claimed by Hirota, Hu, and Tang to pass the Painleve test for integrability, actually fails the test at the highest resonance of the generic branch and therefore must be non-integrable.
It is shown that one system of coupled KdV equations, found in J. Nonlin. Math. Phys., 1999, Vol.6, Nr.3, 255--262 [arXiv:solv-int/9901005] to possess the Painlevé property, is integrable but not new.
The Painlevé property of coupled, non-autonomous Korteweg-de Vries (KdV) type of systems is studied. The conditions under which the systems pass the Painlevé test for integrability are obtained. For some of the integrable cases, exact solutions are given.
Dispersive blow-up for a coupled Schrödinger-fifth order KdV system
本文针对耦合薛定谔-五阶KdV系统的初值问题，通过证明Bourgain空间中的局部适定性及解非线性部分的正则性，构造出可引发色散爆破现象的初值，建立了该系统的色散爆破结果。
Evaluation of analytical solutions of non-linear partial differential equations (both classical and fractional) is a rising subject in Applied Mathematics because its applications in Physical biological and social sciences. In this paper we have used generalized Tanh method to find the exact solution of KP-Burger equation and coupled KdV equation. The fractional Sub-equation method has been used to find the solution of fractional KP-Burger equation and fractional coupled KdV equations. The exact solution obtained by fractional sub-equation method reduces to classical solution when order of fra…
Convergence of a numerical scheme for a coupled Schrödinger--KdV system
通过截断技巧和能量估计，证明了有界域上耦合Schrödinger--KdV系统有限差分半离散格式在强范数下的收敛性，并给出数值算例。
In this paper, we study the persistence of spatial analyticity for the solutions to the Klein-Gordon-Schrödinger system, which describes a physical system of a nucleon field interacting with a neutral meson field, with analytic initial data. Unlike the case of a single nonlinear dispersive equation, not much is known about nonlinear dispersive systems as it is harder to show the spatial analyticity of coupled equations simultaneously. The only results known so far are rather recent ones for the Dirac-Klein-Gordon system which governs the physical system when the nucleon is described by Dirac s…
It is shown that the system of two coupled Korteweg-de Vries equations passes the Painlevé test for integrability in nine distinct cases of its coefficients. The integrability of eight cases is verified by direct construction of Lax pairs, whereas for one case it remains unknown.
This paper seeks to derive the modified KdV (mKdV) equation using a novel approach from systems generated from abstract Lagrangians that possess a two-parameter symmetry group. The method to do uses a modified modulation approach, which results in the mKdV emerging with coefficients related to the conservation laws possessed by the original Lagrangian system. Alongside this, an adaptation of the method of Kuramoto is developed, providing a simpler mechanism to determine the coefficients of the nonlinear term. The theory is illustrated using two examples of physical interest, one in stratified …
This paper addresses the problem of global well-posedness of a coupled system of Korteweg-de Vries equations, derived by Majda and Biello in the context of nonlinear resonant interaction of Rossby waves, in a periodic setting in homogeneous Sobolev spaces $\dot H^s$, for $s\geq 0$. Our approach is based on a successive time-averaging method developed by Babin, Ilyin and Titi [1].
We prove existence and stability results for a two-parameter family of solitary-wave solutions to a system in which an equation of nonlinear Schrödinger type is coupled to an equation of Korteweg-de Vries type. Such systems model interactions between short and long dispersive waves. The results extend earlier results of Angulo, Albert and Angulo, and Chen. Our proof involves the characterization of solitary-wave solutions as minimizers of an energy functional subject to two constraints. To establish the precompactness of minimizing sequences via concentrated compactness, we establish the sub-a…
Through Borel summation methods, we analyze the Boussinesq equations for coupled fluid velocity and temperature fields. We prove that an equivalent system of integral equations in the Borel variable p dual to 1/t has a unique solution in a class of exponentially bounded functions, implying the existence of a classical solution to the Boussinesq equations in a complex t-region that includes a real positive time axis segment. For analytic initial data and forcing, it is shown that the solution is Borel summable, implying that that formal series in powers of t is Gevrey-1 asymptotic, and within t…
Analyticity of the data-to-solution map for a stationary Navier-Stokes fluid-structure interaction problem
该研究针对定常纳维-斯托克斯与欧拉-伯努利梁耦合的流固问题，通过复化方法和全纯隐函数定理证明数据到解映射的实解析性，给出小数据存在性与局部唯一性结果，相关结论可用于参数偏微分方程的数据驱动降阶建模。
Local null-controllability of a system coupling Kuramoto-Sivashinsky-KdV and elliptic equations
本文研究了Kuramoto-Sivashinsky-KdV与椭圆方程耦合系统的零能控性，利用Carleman方法证明线性化系统的全局零能控性，并结合源项方法与Banach不动点定理得出非线性系统的小时间局部零能控性结论。
Bifurcations of solitary waves in a coupled system of long and short waves
研究长波与短波耦合系统中孤立波的分岔行为，通过局部分岔分析揭示孤立波与线性薛定谔方程基态及激发态的关联，证明其为能量约束最小解。
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…
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.
Using the theory developed by Kenig, Ponce, and Vega, we prove that the Hirota-Satsuma system is locally well-posed in Sobolev spaces $H^s(\mathbb{R}) \times H^{s}(\mathbb{R})$ for $3/4<s\le1$. We introduce some Bourgain-type spaces $X_{s,b}^a$ for $a\not =0$, $s,b \in \mathbb{R}$ to obtain local well-posedness for the Gear-Grimshaw system in $H^s(\mathbb{R})\times H^s(\mathbb{R})$ for $s>-3/4$, by establishing new mixed-bilinear estimates involving the two Bourgain-type spaces $X_{s,b}^{-α_-}$ and $X_{s,b}^{-α_+}$ adapted to $\partial_t+α_-\partial_x^3$ and $\partial_t+α_+\partial_x^3$ respec…
Boundary Effects on the Controllability of Coupled KdV Systems
本文研究有界区间上耦合KdV系统的精确边界可控性，通过线性化系统的对偶方法、Paley-Wiener谱分析及压缩映射定理，在最多四个控制输入下建立了完整系统的局部可控性。
Rapid Exponential Stabilization of a Boussinesq System of KdV--KdV Type
针对KdV型Boussinesq耦合系统，论文采用基于Gramian的反馈控制与谱分析，证明仅用一个标量输入即可实现衰减率任意大的快速指数镇定。
A (2+1)-dimensional perturbed KdV equation, recently introduced by W.X. Ma and B. Fuchssteiner, is proven to pass the Painlevé test for integrability well, and its 4$\times $4 Lax pair with two spectral parameters is found. The results show that the Painlevé classification of coupled KdV equations by A. Karasu should be revised.
Normalized ground state solutions of Schrödinger-KdV system in $\mathbb{R}^3$
针对 $\mathbb{R}^3$ 中具有质量约束且呈 $L^2$ 超临界增长的耦合 Schrödinger-KdV 系统，提出新的约束极小化方法，证明局部极小解及归一化基态解的存在性。
A Nonhomogeneous Boundary-Value Problem For The Nonlinear KdV Equation on Star Graphs
本文提出了一种统一框架，用于研究星形图上非线性KdV方程的非齐次边界值问题，建立了线性和非线性问题在$H^s$水平上的尖锐全局适定性理论。
Regularity Analysis for Two Coupled Second Order Evolution Equations
本文研究具有间接阻尼的两个耦合二阶发展方程组的半群正则性，通过特征值渐近分析划分参数空间，并结合虚轴预解式估计，给出了解析性与Gevrey类正则性的完整精确刻画。
We explain how the invariant subspace method can be extended to a scalar and coupled system of time-space fractional partial differential equations. The effectiveness and applicability of the method have been illustrated through time-space (i) fractional diffusion-convection equation, (ii) fractional reaction-diffusion equation, (iii) fractional diffusion equation with source term, (iv) two-coupled system of the fractional diffusion equation, (v) two-coupled system of fractional stationary transonic plane-parallel gas flow equation and (vi) three-coupled system of fractional Hirota-Satsuma KdV…
Nonlinear stability of vector multi-solitons in coupled NLS and modified KdV equations
该研究证明耦合NLS和修正KdV方程的多孤子（含呼吸子）在特定Sobolev空间中非线性稳定，并给出Lyapunov泛函负特征值个数，方法基于可积性，同时推广至经典方程及李代数框架。
Diophantine conditions in well-posedness theory for a coupled modulated Korteweg-de Vries system
研究圆环上耦合调制KdV系统适定性，耦合参数为1时已证全局适定，对于不同耦合用丢番图条件刻画共振，证明在一定条件下对任意\(s\in \mathbb{R}\)该系统在\(H^s(\mathbb{T})\times H^s(\mathbb{T})\)中全局适定，与未调制情况不同。
The Cauchy problem for a coupled system of the Schroedinger and the KdV equation is shown to be globally well-posed for data with infinite energy. The proof uses refined bilinear Strichartz estimates and the I-method introduced by Colliander, Keel, Staffilani, Takaoka, and Tao.
A-priori estimates for generalized Korteweg-de Vries equations in $H^{-1}(\mathbb{R})$
本文研究了广义Korteweg-de Vries方程在$H^{-1}(\mathbb{R})$中的先验估计，提出了一种结合bootstrap论证和局部平滑范数的方法，适用于浅水渠道中长波模型的非积分扰动分析。
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…
We study the relevance of various scalar equations, such as inviscid Burgers', Korteweg-de Vries (KdV), extended KdV, and higher order equations (of Camassa-Holm type), as asymptotic models for the propagation of internal waves in a two-fluid system. These scalar evolution equations may be justified with two approaches. The first method consists in approximating the flow with two decoupled, counterpropagating waves, each one satisfying such an equation. One also recovers homologous equations when focusing on a given direction of propagation, and seeking unidirectional approximate solutions. Th…
A Gevrey class semigroup, exponential decay and Lack of analyticity for a system formed by a Kirchhoff-Love plate equation and the equation of a membrane-like electric network with indirect fractional damping
本文研究 Kirchhoff-Love 板与膜状电网耦合系统的半群性质，证明其在 $θ\in[0,1[$ 非解析、$θ=1$ 解析，指数衰减，且 $θ\in]0,1[$ 时为 Gevrey 类 $s>1/θ$。
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