Borys Alvarez-Samaniego, Pascal Azerad
We study the existence of travelling-waves and local well-posedness in a subspace of $C_b^1(\mathbb{R})$ for a nonlinear evolution equation recently proposed by Andrew C. Fowler to study the dynamics of dunes.
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We investigate a fractional diffusion/anti-diffusion equation proposed by Andrew C. Fowler to describe the dynamics of sand dunes sheared by a fluid flow. In this paper, we prove the global-in-time well-posedness in the neighbourhood of travelling-waves solutions of the Fowler equation.
We are concerned with singular elliptic equations of the form $-Δu= p(x)(g(u)+ f(u)+|\nabla u|^a)$ in $\RR^N$ ($N\geq 3$), where $p$ is a positive weight and $0< a <1$. Under the hypothesis that $f$ is a nondecreasing function with sublinear growth and $g$ is decreasing and unbounded around the origin, we establish the existence of a ground state solution vanishing at infinity. Our arguments rely essentially on the maximum principle.
In this article, we are interested in a non-monotone system of logistic reaction-diffusion equations. This system of equations models an epidemics where two types of pathogens are competing, and a mutation can change one type into the other with a certain rate. We show the existence of minimal speed travelling waves, that are usually non monotonic. We then provide a description of the shape of those constructed travelling waves, and relate them to some Fisher-KPP fronts with non-minimal speed.
By studying the linearization of contour dynamics equation and using implicit function theorem, we prove the existence of co-rotating and travelling global solutions for the gSQG equation, which extends the result of Hmidi and Mateu \cite{HM} to $α\in[1,2)$. Moreover, we prove the $C^\infty$ regularity of vortices boundary, and show the convexity of each vortices component.
We prove existence and uniqueness of travelling waves for a reaction-diffusion system coupling a classical reaction-diffusion equation in a strip with a diffusion equation on a line. To do this we use a continuation method which leads to further insight into the system. In particular, the transition occurs through a singular perturbation which seems new in this context, connecting the system with a Wentzell type boundary value problem.
By applying implicit function theorem on contour dynamics, we prove the existence of co-rotating and travelling patch solutions for both Euler and the generalized surface quasi-geostrophic equation. The solutions obtained constitute a desingularization of points vortices when the size of patch support vanishes. In particular, solutions constructed in this paper consist of doubly connected components, which is essentially different from all known results.
We give a new approach on general systems of the form \[(G){[c]{c}% -Δ_{p}u=\operatorname{div}(|\nabla u| ^{p-2}\nabla u)=ε_{1}|x| ^{a}u^{s}v^δ, -Δ_{q}v=\operatorname{div}(|\nabla v|^{q-2}\nabla u)=ε_{2}|x|^{b}u^μv^{m},\] where $Q,p,q,δ,μ,s,m,$ $a,b$ are real parameters, $Q,p,q\neq1,$ and $ε_{1}=\pm1,$ $ε_{2}=\pm1.$ In the radial case we reduce the problem to a quadratic system of order 4, of Kolmogorov type. Then we obtain new local and global existence or nonexistence results. In the case $ε_{1}=ε_{2}=1,$ we also describe the behaviour of the ground states in two cases where the system is va…
We consider a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypical trait. To sustain the possibility of invasion in the case where an underlying principal eigenvalue is negative, we investigate the existence of travelling wave solutions. We identify a minimal speed $c^*>0$, and prove the existence of waves when $c\geq c^*$ and the non existence when $0\leq c
This paper presents recent results concerning the existence and qualitative properties of travelling wave solutions to the Gross-Pitaevskii equation posed on the whole space R^N. Unlike the defocusing nonlinear Schrödinger equations with null condition at infinity, the presence of non-zero conditions at infinity yields a rather rich and delicate dynamics. We focus on the case N = 2 and N = 3, and also briefly review some classical results on the one-dimensional case. The works we survey provide rigorous justifications to the impressive series of results which Jones, Putterman and Roberts estab…
We study existence, uniqueness and stability of radial solutions of the Lane-Emden-Fowler equation $-Δ_g u=|u|^{p-1}u$ in a class of Riemannian models $(M,g)$ of dimension $n\ge 3$ which includes the classical hyperbolic space $\mathbb H^n$ as well as manifolds with sectional curvatures unbounded below. Sign properties and asymptotic behavior of solutions are influenced by the critical Sobolev exponent while the so-called Joseph-Lundgren exponent is involved in the stability of solutions.
We consider a bistable ($0\textless{}θ\textless{}1$ being the three constant steady states) delayed reaction diffusion equation, which serves as a model in population dynamics. The problem does not admit any comparison principle. This prevents the use of classical technics and, as a consequence, it is far from obvious to understand the behaviour of a possible travelling wave in $+\infty$. Combining refined {\it a priori} estimates and a Leray Schauder topological degree argument, we construct a travelling wave connecting 0 in $-\infty$ to \lq\lq something" which is strictly above the unstable …
We consider a one-dimensional reaction-diffusion equation of Fisher-Kolmogoroff-Petrovsky-Piscounoff type. We investigate the effect of the interaction between the nonlinear diffusion coefficient and the reaction term on the existence and nonexistence of travelling waves. Our diffusion coefficient is allowed to be degenerate or singular at both equilibrium points, 0 and 1, while the reaction term need not be differentiable. These facts influence the existence and qualitative properties of travelling waves in a substantial way.
We establish -among other things- existence and multiplicity of solutions for the Dirichlet problem $\sum_i\partial_{ii}u+\frac{|u|^{\crit-2}u}{|x|^s}=0$ on smooth bounded domains $Ω$ of $ \rn$ ($n\geq 3$) involving the critical Hardy-Sobolev exponent $\crit =\frac{2(n-s)}{n-2}$ where $0<s<2$, and in the case where zero (the point of singularity) is on the boundary $\partial Ω$. Just as in the Yamabe-type non-singular framework (i.e., when s=0), there is no nontrivial solution under global convexity assumption (e.g., when $Ω$ is star-shaped around 0). However, in contrast to the non-satisfacto…
Existence of undercompressive travelling waves of a non-local generalised Korteweg-de Vries-Burgers equation
针对含非局部扩散与凹凸通量的广义 KdV-Burgers 方程，本文利用延迟积分微分方程的动力系统方法证明了欠压缩行波存在，其极限对应违反 Lax 熵条件的非经典激波。
Travelling waves in nonlocal reaction-dispersion equations with diffuse L{é}vy measures
本文利用实分析工具，在广泛框架下统一证明了非局部反应-扩散方程行波前沿的存在性，填补文献空白，并以双稳非线性为中心推广至其他经典非线性。
Existence of a pulled or pushed travelling front invading a critical point for parabolic gradient systems
本文研究全空间多维非线性抛物梯度系统，证明任意非全局极小的临界点都存在推动或拉动型行进波前，且波速不低于线性传播速度，无需非退化或推动侵入等额外假设。
Travelling Waves in a Mathematical Model for Oncolytic Virotherapy
该研究通过反应扩散模型分析了溶瘤病毒治疗中行波解的存在性，确定了最小波速并探讨了肿瘤环境中的传播机制。
We use a geometric approach to prove the existence of smooth travelling wave solutions of a nonlinear diffusion-reaction equation with logistic kinetics and a convex nonlinear diffusivity function which changes sign twice in our domain of interest. We determine the minimum wave speed, c*, and investigate its relation to the spectral stability of the travelling wave solutions.
Travelling waves in a dispersion-saturating diffusion equation
本文在带扩散与色散正则化的双曲守恒律框架下研究广义Rosenau-KdV方程的单调行波，并在不同参数区域证明了行波的存在性及线性与非线性确定性结果。
This paper is concerned with the existence and the stability of travelling wave solutions to a bistable reaction-diffusion equation with a jump discontinuious point on nonlinear term. Sub-super solution method is used throughout this paper. As a matter of fact, this nonlinear discontinuity term is frequently emerging when study the travelling wave solution of reaction-diffusion-ODE systems. Hence, it is meaningful to consider this kind of problem.
The paper is devoted to a reaction-diffusion equation with delay arising in modelling the immune response. We prove the existence of travelling waves in the bistable case using the Leray-Schauder method. In difference with the previous works, we do not assume here quasi-monotonicity of the delayed reaction term.
In this paper, the two-component Fornberg-Whitham system is studied. We firstly investigate the well-posedness in classical Sobolev Space and show a blow-up scenario by local-in-time a priori estimates, then we present some sufficient conditions on the initial data to lead to wave breaking. Furthermore, we establish analytically the existence of periodic travelling waves.
In this paper, we study the existence and stability of travelling wave solutions of a kinetic reaction-transport equation. The model describes particles moving according to a velocity-jump process, and proliferating thanks to a reaction term of monostable type. The boundedness of the velocity set appears to be a necessary and sufficient condition for the existence of positive travelling waves. The minimal speed of propagation of waves is obtained from an explicit dispersion relation. We construct the waves using a technique of sub- and supersolutions and prove their \eb{weak} stability in a we…
Addendum to "Travelling waves for a non-local Korteweg-de Vries-Burgers equation" [J. Differential Equations 257 (2014), no. 3, 720--758]
本文补证了非局部 KdV-Burgers 方程行波存在唯一性证明所依赖的关键假设：线性化方程的有界解只能是指数衰减函数。
In this paper, we study the traveling wave solutions to the density-suppressed motility model describing the ``self-trapping'' mechanism that induces spatio-temporal pattern formations observed in the experiment. We establish the existence of traveling wavefronts with a minimal wave speed and discuss the selection of wave profiles supplemented with numerical simulations illustrating the wave patterns which are well consistent with experimental observations.
The purpose of this paper is to provide a rigorous mathematical proof of the existence of travelling wave solutions to the Gross-Pitaevskii equation in dimensions two and three. Our arguments, based on minimization under constraints, yield a full branch of solutions, and extend earlier results, where only a part of the branch was built. In dimension three, we also show that there are no travelling wave solutions of small energy.
We are interested in the existence of depolarization waves in the human brain. These waves propagate in the grey matter and are absorbed in the white matter. We consider a two-dimensional model $u_t=Δu + f(u) \1_{|y|\leq R} - αu \1_{|y|>R}$, with $f$ a bistable nonlinearity taking effect only on the domain $\Rm\times [-R,R]$, which represents the grey matter layer. We study the existence, the stability and the energy of non-trivial asymptotic profiles of possible travelling fronts. For this purpose, we present dynamical systems technics and graphic criteria based on Sturm-Liouville theory and …
We consider the mass critical fractional (NLS). We show the existence of travelling waves for all mass below the ground state mass, and give a complete description of the associated profiles in the small mass limit. We therefore recover a situation similar to the one discovered in [Gerard P.; Lenzmann E.; Pocovnicu O.; Raphaël, P., A two soliton with transient turbulent regime for the one dimensional cubic half wave, submitted] for the critical case s = 1, but with a completely different asymptotic profile when the mass vanishes.
Existence and Limiting Profiles of Normalized Travelling Wave Solutions for the Pseudo-Relativistic Schrödinger Equation with Logarithmic Nonlinearity
研究具有对数非线性的伪相对论薛定谔方程归一化解，用变分方法对其极小值存在性分类，给出质量临界情形阈值，分析爆破行为及不同情形下极小值的极限轮廓。
For the nonlinear Klein-Gordon equation in R 1+d , we prove the existence of multi-solitary waves made of any number N of decoupled bound states. This extends the work of C{ô}te and Mu{ñ}oz (Forum Math. Sigma 2 (2014)) which was restricted to ground states, as were most previous similar results for other nonlinear dispersive and wave models.
We study the dynamics of a domain wall under the influence of applied magnetic fields in a one-dimensional ferromagnetic nanowire, governed by the Landau--Lifshitz--Gilbert equation. Existence of travelling-wave solutions close to two known static solutions is proven using implicit-function-theorem-type arguments.
Asymptotic stability of travelling waves for general nonlinear Schrödinger equations with non-zero condition at infinity
本文研究非线性薛定谔方程行波在接近声速时的渐近稳定性，证明初始接近行波的解最终收敛于速度相近的行波，方法借鉴了Béthuel等人和Martel等人之前的工作。
We study the stability/instability of the subsonic travelling waves of the Nonlinear Schrödinger Equation in dimension one. Our aim is to propose several methods for showing instability (use of the Grillakis-Shatah-Strauss theory, proof of existence of an unstable eigenvalue via an Evans function) or stability. For the later, we show how to construct in a systematic way a Liapounov functional for which the travelling wave is a local minimizer. These approaches allow to give a complete stability/instability analysis in the energy space including the critical case of the kink solution. We also t…
Minimizing travelling waves for the one-dimensional nonlinear Schrödinger equation with non-zero condition at infinity
针对无穷远处非零的一维非线性 Schrödinger 方程，论文用固定动量约束下的能量极小化和集中紧性方法构造行波解，并证明极小化子的稳定性。
We prove the existence of travelling-wave solutions for a system of coupled nonlinear Schrödinger equations arising in nonlinear optics. Such a system describes second-harmonic generation in optical materials with $χ^{(2)}$ nonlinearity. To prove the existence of travelling waves, we employ the method of concentration compactness to prove the relative compactness of minimizing sequences of the associated variational problem.
A uniqueness result for the two vortex travelling wave in the Nonlinear Schrodinger equation
本文证明了2维非线性薛定谔方程在大动量极限下双涡旋行波极小值子的唯一性，从而确立了该行波的轨道稳定性。
We study an imbibition problem for porous media. When a wetted layer is above a dry medium, gravity leads to the propagation of the water downwards into the medium. In experiments, the occurrence of fingers was observed, a phenomenon that can be described with models that include hysteresis. In the present paper, we describe a single finger in a moving frame and set up a free boundary problem to describe the shape and the motion of one finger that propagates with a constant speed. We show the existence of solutions to the travelling wave problem and investigate the system numerically.
The existence of travelling waves for a coupled system of hyperbolic/ parabolic equations is established in the case of a finite number of velocities in the kinetic equation. This finds application in collective motion of chemotactic bacteria. The analysis builds on the previous work by the first author (arXiv:1607.00429) in the case of a continuum of velocities. Here, the proof is specific to the discrete setting, based on the decomposition of the population density in special Case's modes. Some counter-intuitive results are discussed numerically, including the coexistence of several travelli…
Considered herein are a number of variants of the Boussinesq type systems modeling surface water waves. Such equations were derived by different authors to describe the two-way propagation of long gravity waves. A question of existence of special solutions, the so called solitary waves, is of a particular interest. There are a number of studies relying on a variational approach and a concentration-compactness argument. These proofs are technically very demanding and may vary significantly from one system to another. Our approach is based on the implicit function theorem, which makes the treatm…
We investigate the connection between the existence of an explicit travelling wave solution and the travelling wave with minimal speed in a scalar monostable reaction-diffusion equation.
We provide a mathematical analysis of a thermo-diffusive combustion model of lean spray flames, for which we prove the existence of travelling waves. In the high activation energy singular limit we show the existence of two distinct combustion regimes with a sharp transition -- the diffusion limited regime and the vaporisation controlled regime. The latter is specific to spray flames with slow enough vaporisation. We give a complete characterisation of these regimes, including explicit velocities, profiles, and upper estimate of the size of the internal combustion layer. Our model is on the on…
The paper deals with a problem of interaction between hydrodynamics and mechanics of nonlinear elastic bodies. The existence question for two-dimensional symmetric steady waves travelling on the surface of a deep ocean beneath a heavy elastic membrane is analyzed as a problem in bifurcation theory. The behaviour of the two-dimensional cross-section of the membrane is modelled as a thin (unshearable), heavy, hyperelastic Cosserat rod, following Antman's elasticity theory, and the fluid beneath is supposed to be in steady 2D irrotational motion under gravity. Assuming that gravity and the densit…
We consider the Euler equations in ${\mathbb R}^3$ expressed in vorticity form. A classical question that goes back to Helmholtz is to describe the evolution of solutions with a high concentration around a curve. The work of Da Rios in 1906 states that such a curve must evolve by the so-called binormal curvature flow. Existence of true solutions concentrated near a given curve that evolves by this law is a long-standing open question that has only been answered for the special case of a circle travelling with constant speed along its axis, the thin vortex-rings. We provide what appears to be t…
Existence and stability for the travelling waves of the Benjamin equation
本文针对Benjamin模型行波，提出全参数范围的约束最大化程序以证明其谱稳定性，并将构造扩展至所有L²次临界情形，同时基于Sobolev嵌入不等式提出适用于更广幂次但可能产生不稳定波的新方法。
We consider the homogeneous integro-differential equation$\partial \_t u=J*u-u+f(u)$ with a monostable nonlinearity $f$. Our interest is twofold: we investigate the existence/non existence of travelling waves, and the propagation properties of the Cauchy problem.When the dispersion kernel $J$ is exponentially bounded, travelling waves are known to exist and solutions of the Cauchy problem typically propagate at a constant speed \cite{Schumacher1980}, \cite{Weinberger1982}, \cite{Carr2004}, \cite{Coville2007a}, \cite{Coville2008a}, \cite{Yagisita2009}. %When the dispersion kernel $J$ is exponen…
We are revisiting the topic of travelling fronts for the food-limited (FL) model with spatio-temporal nonlocal reaction. These solutions are crucial for understanding the whole model dynamics. Firstly, we prove the existence of monotone wavefronts. In difference with all previous results formulated in terms of `sufficiently small parameters', our existence theorem indicates a reasonably broad and explicit range of the model key parameters allowing the existence of monotone waves. Secondly, numerical simulations realized on the base of our analysis show appearance of non-oscillating and non-mon…
We consider the semi linear Fisher-Kolmogorov-Petrovski-Piscounov equation for the advance of an advantageous gene in biology. Its non-smooth reaction function $f(u)$ allows for the introduction of travelling waves with a new profile. We study existence, uniqueness, and long-time asymptotic behavior of the solutions $u(x,t)$, $(x,t)\in \mathbb{R}\times \mathbb{R}_+$. We prove also the existence and uniqueness (up to a spatial shift) of a travelling wave $U$. Our main result is the uniform convergence (for $x\in \mathbb{R}$) of every solution $u(x,t)$ of the Cauchy problem to a single traveling…
We consider the equation $u_t=u_{xx}+b(x)u(1-u),$ $x\in\mathbb R,$ where $b(x)$ is a nonnegative measure on $\mathbb R$ that is periodic in $x.$ In the case where $b(x)$ is a smooth periodic function, it is known that there exists a travelling wave with speed $c$ for any $c\geq c^*(b),$ where $c^*(b)$ is a certain positive number depending on $b.$ Such a travelling wave is often called a \lq\lq pulsating travelling wave" or a \lq\lq periodic travelling wave", and $c^*(b)$ is called the \lq\lq minimal speed". In this paper, we first extend this theory by showing the existence of the minimal spe…
We consider space-periodic evolutionary and travelling-wave solutions to the regularised long-wave equation (RLWE) with damping and forcing. We establish existence, uniqueness and smoothness of the evolutionary solutions for smooth initial conditions, and global in time spatial analyticity of such solutions for analytical initial conditions. The width of the analyticity strip decays at most polynomially. We prove existence of travelling-wave solutions and uniqueness of travelling waves of a sufficiently small norm. The importance of damping is demonstrated by showing that the problem of findin…
This paper has various goals: first, we develop a local and global well-posedness theory for the regularized Benjamin-Ono equation in the periodic setting, second, we show that the Cauchy problem for this equation (in both periodic and non-periodic case) cannot be solved by an iteration scheme based on the Duhamel formula for negative Sobolev indices, third, a proof of the existence of a smooth curve of periodic travelling wave solutions, for the regularized Benjamin-Ono equation, with fixed minimal period 2L, is given. It is also shown that these solutions are nonlinearly stable in the energy…
Periodic travelling wave solutions of discrete nonlinear Klein-Gordon lattices
该研究针对离散非线性Klein-Gordon系统，分别用结合Schauder不动点定理与压缩映射原理的不动点方法、基于山路定理的变分方法，证明了硬、软在位势下周期行波解的存在性，并给出对应阈值与速度上界。
Construction of minimizing travelling waves for the Gross-Pitaevskii equation on $\mathbb{R} \times \mathbb{T}$
本研究针对$\mathbb{R} \times \mathbb{T}$上的Gross-Pitaevskii方程，通过适用于周期情形的新型对称化论证得到极小化序列紧性，构造出固定动量下金兹堡-朗道能量极小的有限能量行波分支，明确了不同横向长度下极小解的形态。
We are interested in the existence of travelling waves for the Benjamin-Bona-Mahony equation on a network. First we construct an explicit wave, defined in $\mathbb{R}$. Then, we use this wave to derive some conditions on the coefficients appearing in the equations and on the geometry of the network to ensure the existence of travelling waves on the network.
We investigate the properties of finite energy travelling waves to the nonlinear Schrodinger equation with nonzero conditions at infinity for a wide class of nonlinearities. In space dimension two and three we prove that travelling waves converge in the transonic limit (up to rescaling) to ground states of the Kadomtsev-Petviashvili equation. Our results generalize an earlier result of F. Bethuel, P. Gravejat and J-C. Saut for the two-dimensional Gross-Pitaevskii equation, and provide a rigorous proof to a conjecture by C. Jones and P. H. Roberts about the existence of an upper branch of trave…
We are concerned with the asymptotic behaviour of classical solutions of systems of the form u_t = Au_xx + f(u, u_x), x in R, t>0, u(x,t) a vector in RN, with u(x,0)= U(x), where A is a positive-definite diagonal matrix and f is a 'bistable' nonlinearity satisfying conditions which guarantee the existence of a comparison principle. Suppose that there is a travelling-front solution w with velocity c, that connects two stable equilibria of f. We show that if U is bounded, uniformly continuously differentiable and such that w(x) - U(x) is small when the modulus of x is large, then there exists y …
The Fisher-KPP equation with general nonlinear diffusion and arbitrary kinetic orders in the reaction terms is considered. The existence of oscillatory travelling wave solutions is proved for this model. Conditions for the existence of such solutions are provided.
On the instability of travelling wave solutions for the transport-Stokes equation
本文研究了 $\mathbb{R}^3$ 中 Transport-Stokes 系统球形行波解的不稳定性，通过尺度缩放论证、轮廓动力学方程线性化系统的正特征值数值分析以及 FreeFem 有限元求解，从理论和数值上验证了其不稳定性。
Travelling breather solutions in waveguides for cubic nonlinear Maxwell equations with retarded material laws
本文研究含时间迟滞三次非线性极化的 Maxwell 方程，利用变分结构与山路定理证明平板及圆柱波导中局域 TE 模行呼吸子的存在性。
Existence of non-negative weak solutions is shown for a full curvature thin-film model of a liquid thin film flowing down a vertical fibre. The proof is based on the application of a priori estimates derived for energy-entropy functionals. Long-time behaviour of these weak solutions is analysed and, under some additional constraints for the model parameters and initial values, convergence towards a travelling wave solution is obtained. Numerical studies of energy minimizers and travelling waves are presented to illustrate analytical results.
Lower Bounds for Admissible Values of the Travelling Wave Speed in Asymmetrically Supported Beam
本文研究了不对称支撑梁中行波解存在的波速下限，利用山脊定理和Fučík谱提出猜想。
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