Dimitri Mugnai, Angela Pistoia
We study local bifurcation from an eigenvalue with multiplicity greater than one for a class of semilinear elliptic equations. We evaluate the exact number of bifurcation branches of non trivial solutions and we compute the Morse index of the solutions in those branches.
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We discuss an approach to the computer assisted proof of the existence of branches of stationary and periodic solutions for dissipative PDEs, using the Brussellator system with diffusion and Dirichlet boundary conditions as an example, We also consider the case where the branch of periodic solutions emanates from a branch of stationary solutions through a Hopf bifurcation.
In this paper we study the bifurcation of branches of non-symmetric solutions from the symmetric branch of solutions to the Euler-Lagrange equations satisfied by optimal functions in functional inequalities of Caffarelli-Kohn-Nirenberg type. We establish the asymptotic behavior of the branches for large values of the bifurcation parameter. We also perform an expansion in a neighborhood of the first bifurcation point on the branch of symmetric solutions, that characterizes the local behavior of the non-symmetric branch. These results are compatible with earlier numerical and theoretical observa…
In this paper we extend existing results concerning generalized eigenvalues of Pucci's extremal operators. In the radial case, we also give a complete description of their spectrum, together with an equivalent of Rabinowitz's Global Bifurcation Theorem. This allows us to solve equations involving Pucci's operators.
In this paper, we analyze an eigenvalue problem for nonlinear elliptic operators involving homogeneous Dirichlet boundary conditions in a open smooth bounded domain. We prove bifurcation results from trivial solutions and from infinity for the considered nonlinear eigenvalue problem. We also show the existence of multiple solutions of the nonlinear problem using variational methods.
We consider several different bidirectional Whitham equations that have recently appeared in the literature. Each of these models combine the full two-way dispersion relation from the incompressible Euler equations with a canonical shallow water nonlinearity, providing nonlocal model equations that may be expected to exhibit some of the interesting high-frequency phenomena present in the Euler equations that standard "long-wave" theories fail to capture. Of particular interest here is the existence and stability of periodic traveling wave solutions in such models. Using numerical bifurcation t…
In this paper we study for the incompressible Euler equations the global structure of the bifurcation diagram for the rotating doubly connected patches near the degenerate case. We show that the branches with the same symmetry merge forming a small loop provided that they are close enough. This confirms the numerical observations done in the recent work [10]
In this paper we prove the existence of countable branches of rotating patches bifurcating from the ellipses at some implicit angular velocities.
In this short note, we consider the elliptic problem $$ λϕ+ Δϕ= η|ϕ|^σϕ,\quad ϕ\big|_{\partial Ω}=0,\quad λ, η\in \mathbb{C}, $$ on a smooth domain $Ω\subset \mathbb{R}^N$, $N\ge 1$. The presence of complex coefficients, motivated by the study of complex Ginzburg-Landau equations, breaks down the variational structure of the equation. We study the existence of nontrivial solutions as bifurcations from the trivial solution. More precisely, we characterize the bifurcation branches starting from eigenvalues of the Dirichlet-Laplacian of arbitrary multiplicity. This allows us to discuss the nature…
In this paper, we shall study global bifurcation phenomenon for the following Kirchhoff type problem \begin{equation} \left\{ \begin{array}{l} -\left(a+b\int_Ω\vert \nabla u\vert^2\,dx\right)Δu=λu+h(x,u,λ)\,\,\text{in}\,\, Ω,\\ u=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{on}\,\,Ω. \end{array} \right.\nonumber \end{equation} Under some natural hypotheses on $h$, we show that $\left(aλ_1,0\right)$ is a bifurcation point of the above problem. As applications of the above result, we shall determine the interval of $λ$, in which there exist positive solutions for the above problem wit…
Noise-driven bifurcations in a nonlinear Fokker-Planck system describing stochastic neural fields
研究随机神经场的非线性非局部 Fokker–Planck 方程，证明并刻画噪声降低时齐次稳态的多次分岔，并推广至含噪网格细胞系统及六边形模式稳定性。
Bifurcation of double eigenvalues for Aharonov-Bohm operators with a moving pole
该研究针对含原点平面区域内带Dirichlet边界的Aharonov-Bohm算子，分析半整数环量下极点沿原点邻域直线移动时双重特征值的分岔行为，证明正测度锥内直线移动会引发分岔，并给出对称区域尤其是圆盘情形的精确结果。
For non-monotone single and two-populations time-dependent Mean-Field Game systems we obtain the existence of an infinite number of branches of non-trivial solutions. These non-trivial solutions are in particular shown to exhibit an oscillatory behaviour when they are close to the trivial (constant) one. The existence of such branches is derived using local and global bifurcation methods, that rely on the analysis of eigenfunction expansions of solutions to the associated linearized problem. Numerical analysis is performed on two different models to observe the oscillatory behaviour of solutio…
We construct the global bifurcation curves, solutions versus level of harvesting, for the steady states of a diffusive logistic equation on a bounded domain, under Dirichlet boundary conditions and other appropriate hypotheses, when $a$, the linear growth rate of the population, is below $λ_2+δ$. Here $λ_2$ is the second eigenvalue of the Dirichlet Laplacian on the domain and $δ>0$. Such curves have been obtained before, but only for $a$ in a right neighborhood of the first eigenvalue. Our analysis provides the exact number of solutions of the equation for $a\leqλ_2$ and new information on the…
A celebrated result in bifurcation theory is that global connected sets of non-trivial solutions bifurcate from trivial solutions at non-zero eigenvalues of odd algebraic multiplicity of the linearized problem when the operators involved are compact. In this paper a simple example is constructed which satisfies the regularity hypotheses of the global bifurcation theorem, and the eigenvalue has algebraic multiplicity one, yet all the path-connected components of the connected sets that bifurcate are singletons. Another example shows that even when the operators are everywhere infinitely differe…
Eigenvalue Bifurcation in Doubly Nonlinear Problems with an Application to Surface Plasmon Polaritons
本文研究解与特征值参数均非线性的非自伴特征值问题，利用Lyapunov-Schmidt约化证明从简单孤立特征值分岔的结果，并应用于证明PT对称构型中横电表面等离子体极化激元的存在性。
In this paper we study the following eigenvalue boundary value problem for Monge-Ampère equations: {equation} \{{array}{l} \det(D^2u)=λ^N f(-u)\,\, \text{in}\,\, Ω, u=0,\,\text{on}\,\, \partial Ω. {array}. {equation} We establish the unilateral global bifurcation results for the problem with $f(u)=u^N+g(u)$ and $Ω$ being the unit ball of $\mathbb{R}^N$. More precisely, under some natural hypotheses on the perturbation function $g:\mathbb{R}\rightarrow\mathbb{R}$, we show that $(λ_1,0)$ is a bifurcation point of the problem and there are two distinct unbounded continua of one-sign solutions, wh…
In this paper, we analyze an eigenvalue problem for a quasi-linear elliptic operators involving Dirichlet boundary condition in an open smooth bounded set of $\mathbb{R}^N$. We investigate a bifurcation results (from trivial solution and from infinity) of an eigenvalue problem involving the $(p,2)$-Laplace operator, from the Fu\v cik spectrum of the Laplacian.
In this paper, we analyze an eigenvalue problem for quasi-linear elliptic operators involving homogeneous Dirichlet boundary conditions in a open smooth bounded domain. We show that the eigenfunctions corresponding to the eigenvalues belong to $L^{\infty}$, which implies $C^{1,α}$ smoothness, and the first eigenvalue is simple. Moreover, we investigate the bifurcation results from trivial solutions using the Krasnoselski bifurcation theorem and from infinity using the Leray-Schauder degree. We also show the existence of multiple critical points using variational methods and the Krasnoselski ge…
Bifurcation from multiple eigenvalues of rotating traveling waves on a capillary liquid drop
本文研究毛细作用下近似球形液滴自由边界问题中旋转行波剖面的分岔，证明线性化问题的任意重数特征值均为分岔点，并利用哈密顿结构和变分方法克服二维退化困难。
We give a quite simple approach how to prove the absence of loops in bifurcation branches of water waves in rotational case with arbitrary vorticity distribution. The supporting flow may contain stagnation points and critical layers and water surface is allowed to be overhanging. Monotonicity properties of the free surface are presented. Especially simple criterium of absence of loops is given for bifurcation branches when the bifurcation parameter is the water wave period. We show that there are no loops if you start from a water wave with a positive/negative vertical component of velocity on…
We consider singular perturbations of eigenvalue problems. We prove that to these problems correspond simple eigenvalues and we study their asymptotic behavior. As a result, we prove global bifurcation results for non uniformly and fully nonlinear elliptic equations. These equations are defined on smooth enough bounded domains and the solutions belonging in these branches are smooth and positive.
This work deals with the focusing Nonlinear Schrodinger Equation in one dimension with pure-power nonlinearity near cubic. We consider the spectrum of the linearized operator about the soliton solution. When the nonlinearity is exactly cubic, the linearized operator has resonances at the edges of the essential spectrum. We establish the degenerate bifurcation of these resonances to eigenvalues as the nonlinearity deviates from cubic. The leading-order expression for these eigenvalues is consistent with previous numerical computations.
Trees, trunks, and branches -- bifurcation structure of time-periodic solutions to $u_{tt}-u_{xx}\pm u^{3}=0$
该研究提出系统性方法，基于稀疏模式相互作用分析带Dirichlet边界的三次波动方程时间周期解的分岔结构，补充了已有存在性证明，证实任意频率下均存在解且幅值可任意大。
Bifurcation from bubbles in nonconvex cones
研究非凸锥中临界半线性椭圆方程的气泡分支问题，通过特征值分析和分支定理证明非径向解的全局存在性。
We study the one-dimensional Kirchhoff type equation $$ -(b + a\Vert u'\Vert^{2}) u''(x) = λu(x)^p, x \in I:= (-1,1), \enskip u(x) > 0, \enskip x\in I, \enskip u(\pm 1) = 0, $$ where $\Vert u'\Vert = \left(\int_I u'(x)^2 dx\right)^{1/2}$, $a > 0, b > 0, p> 0$ are given constants and $λ> 0$ is a bifurcation parameter. We establish the exact solution $u_λ(x)$ and complete shape of the bifurcation curves $λ= λ(ξ)$, where $ξ:= \Vert u_λ\Vert_\infty$. We also study the nonlinear eigenvalue problem $$ -\Vert u'\Vert^{p-1} u''(x) = μu(x)^p, x \in I, \enskip u(x) > 0, x\in I, \enskip u(\pm 1) = 0, $$ …
We analyze global bifurcations along the family of radially symmetric vortices in the Gross--Pitaevskii equation with a symmetric harmonic potential and a chemical potential $μ$ under the steady rotation with frequency $Ω$. The families are constructed in the small-amplitude limit when the chemical potential $μ$ is close to an eigenvalue of the Schrödinger operator for a quantum harmonic oscillator. We show that for $Ω$ near $0$, the Hessian operator at the radially symmetric vortex of charge $m_{0}\in\mathbb{N}$ has $m_{0}(m_{0}+1)/2$ pairs of negative eigenvalues. When the parameter $Ω$ is i…
A codimension-three bifurcation, characterized by a pair of purely imaginary eigenvalues and a nonsemisimple double zero eigenvalue, arises in the study of a pair of weakly coupled nonlinear oscillators with Z_2 + Z_2 symmetry. The methodology is based on Arnold's ideas of versal deformations of matrices for the linear analysis, and Poincaré normal forms for the nonlinear analysis of the system. The stratified subvariety of primary bifurcations of codimensions one and two is identified in the parameter space. The analysis reveals different types of solutions in the state space, including equil…
This manuscript extends the analysis of a much studied singularly perturbed three-component reaction-diffusion system for front dynamics in the regime where the essential spectrum is close to the origin. We confirm a conjecture from a preceding paper by proving that the triple multiplicity of the zero eigenvalue gives a Jordan chain of length three. Moreover, we simplify the center manifold reduction and computation of the normal form coefficients by using the Evans function for the eigenvalues. Finally, we prove the unfolding of a Bogdanov-Takens bifurcation with symmetry in the model. This l…
Two-dimensional capillary liquid drop: Craig-Sulem formulation on $\mathbb{T}^1$ and bifurcations from multiple eigenvalues of rotating waves
该研究针对二维近圆形毛细液滴自由边界问题，推导了圆周及一维平坦环面上的Craig-Sulem公式，证明其哈密顿结构与运动常数，通过多重特征值分岔得到角动量参数化的唯一对称旋转波解轨道。
In this paper, we use rigorous numerics to compute several global smooth branches of steady states for a system of three reaction-diffusion PDEs introduced by Iida et al. [J. Math. Biol., {\bf 53}, 617--641 (2006)] to study the effect of cross-diffusion in competitive interactions. An explicit and mathematically rigorous construction of a global bifurcation diagram is done, except in small neighborhoods of the bifurcations. The proposed method, even though influenced by the work of van den Berg et al. [Math. Comp., {\bf 79}, 1565--1584 (2010)], introduces new analytic estimates, a new {\em glu…
Consider a branch of unstable solitons of NLS whose linearized operators have one pair of simple real eigenvalues in addition to the zero eigenvalue. Under radial symmetry and standard assumptions, solutions to initial data from a neighbourhood of the branch either converge to a soliton, or exit a larger neighbourhood of the branch transversally. The qualitative dynamic near a branch of unstable solitons is irrespective of whether blowup eventually occurs, which has practical implications for the description of blowup of NLS with supercritical nonlinearity.
In this paper we establish a unilateral bifurcation result for a class of quasilinear elliptic system strongly coupled, extending a bifurcation theorem due to J. López-Gómez. To this aim, we use several results, such that Fredholm operator of index $0$ theory, eigenvalues of elliptic operators and the Krein-Rutman theorem. Lastly, we apply this result to some particular systems arising from population dynamics and determine a region of existence of coexistence states.
Numerical Bifurcation for the Capillary Whitham Equation
本文研究了将表面张力纳入Whitham方程后的分岔现象，提出了一种谱插值方法计算周期性行波，并发现丰富的分岔结构，包括亚谐波分岔、连接与交叉分支等。
We numerically study solitary waves in the coupled nonlinear Schrödinger equations. We detect pitchfork bifurcations of the fundamental solitary wave and compute eigenvalues and eigenfunctions of the corresponding eigenvalue problems to determine the spectral stability of solitary waves born at the pitchfork bifurcations. Our numerical results demonstrate the theoretical ones which the authors obtained recently. We also compute generalized eigenfunctions associated with the zero eigenvalue for the bifurcated solitary wave exhibiting a saddle-node bifurcation, and show that it does not change i…
Bifurcations of unstable eigenvalues for Stokes waves derived from conserved energy
针对无限深流体无旋重力波的Euler方程，严格证明Stokes波线性化运动方程的零特征值分岔出现在能量对陡度的极值点，且新不稳定特征值仅沿陡度增大方向产生。
We consider the Neumann problem for the equation $u_{xx}+λf(u)=0$ in the punctured interval $(-1,1) \setminus \{0\}$, where $λ>0$ is a bifurcation parameter and $f(u)=u-u^3$. At $x=0$, we impose the conditions $u(-0)+au_x(-0)=u(+0)-au_x(+0)$ and $u_x(-0)=u_x(+0)$ for a constant $a>0$ (the symbols $+0$ and $-0$ stand for one-sided limits). The problem appears as a limiting equation for a semilinear elliptic equation in a higher dimensional domain shrinking to the interval $(-1,1)$. First we prove that odd solutions and even solutions form families of branches $\{ \mathcal{C}^o_k\}_{k \in \mathb…
Local and global bifurcation of electron-states
本文研究一维周期 Vlasov-Poisson 方程中从平坦速度条带分岔出的行波电子态，利用 Crandall-Rabinowitz 定理分析对称与不对称情形下的局部分岔结构及分支全局延拓。
We consider the focusing (attractive) nonlinear Schrödinger (NLS) equation with an external, symmetric potential which vanishes at infinity and supports a linear bound state. We prove that the symmetric, nonlinear ground states must undergo a symmetry breaking bifurcation if the potential has a non-degenerate local maxima at zero. Under a generic assumption we show that the bifurcation is either subcritical or supercritical pitchfork. In the particular case of double-well potentials with large separation, the power of nonlinearity determines the subcritical or supercritical character of the bi…
This paper is devoted to the investigation of the following nonlocal dispersal equation $$ u_{t}(t,x)=\frac{D}{σ^m}\left[\int_ΩJ_σ(x-y)u(t,y)dy-u(t,x)\right]+f(t,x,u(t,x)), \quad t>0,\quad x\in\overlineΩ, $$ where $Ω\subset\mathbb{R}^{N}$ is a bounded and connected domain with smooth boundary, $m\in[0,2)$, $D>0$ is the dispersal rate, $σ>0$ characterizes the dispersal range, $J_σ=\frac{1}{σ^{N}} J\left(\frac{\cdot}σ\right)$ is the scaled dispersal kernel, and $f$ is a time-periodic nonlinear function of generalized KPP type. We first study the principal spectral theory of the linear operator a…
In this paper we give the complete classification of solitons for a cubic NLS equation on the simplest network with a non-trivial topology: the tadpole graph, i.e. a ring with a half-line attached to it and free boundary conditions at the junction. The model, although simple, exhibits a surprisingly rich behavior and in particular we show that it admits: 1) a denumerable family of continuous branches of embedded solitons bifurcating from linear eigenstates and threshold resonances of the system; 2) a continuous branch of edge solitons displaying a pitchfork symmetry breaking bifurcation at the…
Numerical Approximation and Bifurcation Results for an Elliptic Problem with Superlinear Subcritical Nonlinearity on the Boundary
针对边界上超线性次临界非线性椭圆问题，提出有限差分与延拓结合的方法，通过主特征值分析确定分支点并追踪完整分支曲线，验证了理论结果。
A stochastic perturbation approach to nonlinear bifurcating problems
本文提出利用多项式混沌展开方法，通过随机扰动检测参数系统的分支，无需先验信息并提高计算效率。
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