Lyapunov Schmidt Reduction Method Research Articles

Stability of solitary wave solutions in the Lugiato–Lefever equation

We analyze the spectral and dynamical stability of solitary wave solutions to the Lugiato–Lefever equation on R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}$$\\end{document}. Our interest lies in solutions that arise through bifurcations from the phase-shifted bright soliton of the nonlinear Schrödinger equation. These solutions are highly nonlinear, localized, far-from-equilibrium waves, and are the physical relevant solutions to model Kerr frequency combs. We show that bifurcating solitary waves are spectrally stable when the phase angle satisfies θ∈(0,π)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta \\in (0,\\pi )$$\\end{document}, while unstable waves are found for angles θ∈(π,2π)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta \\in (\\pi ,2\\pi )$$\\end{document}. Furthermore, we establish asymptotic orbital stability of spectrally stable solitary waves against localized perturbations. Our analysis exploits the Lyapunov–Schmidt reduction method, the instability index count developed for linear Hamiltonian systems, and resolvent estimates.

Non-degeneracy of multi-peak solutions for the Schrödinger-Poisson problem

Abstract In this article, we consider the following Schrödinger-Poisson problem: − ε 2 Δ u + V ( y ) u + Φ ( y ) u = ∣ u ∣ p − 1 u , y ∈ R 3 , − Δ Φ ( y ) = u 2 , y ∈ R 3 , \left\{\begin{array}{ll}-{\varepsilon }^{2}\Delta u+V(y)u+\Phi (y)u={| u| }^{p-1}u,& y\in {{\mathbb{R}}}^{3},\\ -\Delta \Phi (y)={u}^{2},& y\in {{\mathbb{R}}}^{3},\end{array}\right. where ε > 0 \varepsilon \gt 0 is a small parameter, 1 < p < 5 1\lt p\lt 5 , and V ( y ) V(y) is a potential function. We construct multi-peak solution concentrating at the critical points of V ( y ) V(y) through the Lyapunov-Schmidt reduction method. Moreover, by using blow-up analysis and local Pohozaev identities, we prove that the multi-peak solution we construct is non-degenerate. To our knowledge, it seems be the first non-degeneracy result on the Schödinger-Poisson system.

Bifurcation and stability of a diffusive predator-prey model with the fear effect and time delay.

The predator-prey system can induce wealth properties with fear effects. In this paper, we propose a diffusive predator-prey model where the influence of fear effects and time delay is considered, under the Dirichlet boundary condition. It follows from the Lyapunov-Schmidt reduction method that there exists a non-homogeneous steady-state solution of the system and the specific expressions are also given. By the aid of bifurcation theory and eigenvalue theory, we also investigate the existence/non-existence and the stability of Hopf bifurcation under three different conditions of bifurcation parameters. Furthermore, the effects of the fear on population density, stability, and Hopf bifurcation are also considered and the results show that the increase of fear effects will reduce the population density, and Hopf bifurcation is more likely difficult to undergo as k increases under some conditions.

Infinitely many solutions for nonlinear fourth-order Schrödinger equations with mixed dispersion

ABSTRACT In this paper, we first show the nondegeneracy and asymptotic behavior of ground states for the nonlinear fourth-order Schrödinger equation with mixed dispersion: δ Δ 2 u − Δu + u = | u | 2 σ u , u ∈ H 2 ( R N ) , where δ > 0 is sufficiently small, 0 < σ < 2 ( N − 2 ) + , 2 ( N − 2 ) + = 2 N − 2 for N ≥ 3 and 2 ( N − 2 ) + = + ∞ for N=2,3. This work extends some results in Bonheure, Casteras, Dos Santos, and Nascimento [Orbitally stable standing waves of a mixed dispersion nonlinear Schrödinger equation. SIAM J Math Anal. 2018;50:5027–5071]. Next, suppose P ( x ) and Q ( x ) are two positive, radial and continuous functions satisfying that as r = | x | → + ∞ , P ( r ) = 1 + a 1 r m 1 + O ( 1 r m 1 + θ 1 ) , Q ( r ) = 1 + a 2 r m 2 + O ( 1 r m 2 + θ 2 ) , where a 1 , a 2 ∈ R , m 1 , m 2 > 1 , θ 1 , θ 2 > 0 . We use the Lyapunov–Schmidt reduction method developed by Wei and Yan [Infinitely many positive solutions for the nonlinear Schrödinger equations in RN. Calc Var. 2010;37:423–439] to construct infinitely many nonradial positive and sign-changing solutions with arbitrary large energy for the following equation: δ Δ 2 u − Δu + P ( x ) u = Q ( x ) | u | 2 σ u , u ∈ H 2 ( R N ) .

Multiple mixed interior and boundary peaks synchronized solutions for nonlinear coupled elliptic systems

This paper is devoted to a class of singularly perturbed nonlinear Schrödinger systems defined on a smooth bounded domain in RN(N=2,3). We use the Lyapunov–Schmidt reduction method to construct synchronized vector solutions with multiple spikes both on the boundary and in the interior of the domain. For each vector solution that has been constructed, we point out that the interior spikes locate near sphere packing points in the domain, and the boundary spikes locate near the critical points of the mean curvature function related to the boundary of the domain.

Doubling the equatorial for the prescribed scalar curvature problem on {{mathbb {S}}}^N

We consider the prescribed scalar curvature problem on {{mathbb {S}}}^N ΔSNv-N(N-2)2v+K~(y)vN+2N-2=0onSN,v>0inSN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\Delta _{{{\\mathbb {S}}}^N} v-\\frac{N(N-2)}{2} v+{\ ilde{K}}(y) v^{\\frac{N+2}{N-2}}=0 \\quad \ ext{ on } \\ {{\\mathbb {S}}}^N, \\qquad v >0 \\quad {\\quad \\hbox {in } }{{\\mathbb {S}}}^N, \\end{aligned}$$\\end{document}under the assumptions that the scalar curvature {tilde{K}} is rotationally symmetric, and has a positive local maximum point between the poles. We prove the existence of infinitely many non-radial positive solutions, whose energy can be made arbitrarily large. These solutions are invariant under some non-trivial sub-group of O(3) obtained doubling the equatorial. We use the finite dimensional Lyapunov–Schmidt reduction method.

Stability and bifurcation of a reaction-diffusion-advection model with nonlinear boundary condition

The dynamics of the reaction-diffusion-advection population models with linear boundary condition has been widely studied. This paper is devoted to the dynamics of a reaction-diffusion-advection population model with nonlinear boundary condition. Firstly, the stability of the trivial steady state is investigated by studying the corresponding eigenvalue problem. Secondly, the existence and stability of nontrivial steady states are proved by applying the Crandall-Rabinowitz bifurcation Theorem, the Lyapunov-Schmidt reduction method and perturbation method, in which bifurcation from simple eigenvalue and that from degenerate simple eigenvalue are both possible. The general results are applied to a parabolic equation with monostable nonlinear boundary condition, and to a parabolic equation with sublinear growth and superlinear boundary condition. Our theoretical results show that the nonlinear boundary condition can lead to the occurrence of various steady state bifurcations. Meanwhile, compared with the linear boundary condition, the nonlinear boundary condition can induce the multiplicity and growing-up property of positive steady-state solutions for the model with logistic interior growth. Finally, the numerical results show that the advection can change the bifurcation direction of some bifurcation, and affect the density distribution of the species.

New concentrated solutions for the nonlinear Schrödinger–Newton system

ABSTRACT In this paper, we study the following nonlinear Schrödinger–Newton system { Δ u − V ( x ) u + Ψ u = 0 , x ∈ R 3 , Δ Ψ + 1 2 u 2 = 0 , x ∈ R 3 , which is a nonlinear system obtained by coupling the linear Schrödinger equation of quantum mechanics with the gravitation law of Newtonian mechanics. Assuming that V ( y ) is radial and satisfies some algebraic decay at the infinity, we construct infinitely many non-radial positive solutions which have polygonal symmetry with respect to y 1 and y 2 and are even in y 2 for the system above by the Lyapunov-Schmidt reduction method. Moreover, we have overcame some new difficulties caused by the non-local term.

Infinitely many positive multi-bump solutions for fractional Kirchhoff equations

The purpose of this paper is to study the existence of infinitely many non-radial positive solutions to the following Kirchhoff equation involving the fractional Laplacian in RN(K)(a+b∫RN×RN|u(x)−u(y)|2|x−y|N+2sdxdy)(−Δ)su+V(|x|)u=up,u>0,inRN where a,b>0, 0<s<1, N>2s, 1<p<2s⁎−1, 2s⁎=2NN−2s is the critical Sobolev exponent and V(|x|) is a positive function. Here we are interested in the case N≥3. For the case N=3, by reducing (K) to a finite dimensional problem by the Lyapunov-Schmidt reduction method, we show that (K) has infinitely solutions whose energy tends to infinity. In contrast to most existing multiplicity results in the literature, we do not need the restriction p>3. For the case N≥4, we use a scaling technique to prove that the energy on the set of solutions to (K) is bounded. It turns out that N≤3 is necessary for the existence of solutions with large energy.

New synchronized solutions for linearly coupled Schrödinger systems

In this paper, we consider the following linearly coupled Schrödinger system:(Pε){−Δu+P(|y|)u=u3+λ(|y|)vinR3,−Δv+Q(|y|)v=v3+λ(|y|)uinR3, where P(|y|), Q(|y|) and λ(|y|) are positive radial potentials such that λ(|y|)<min⁡{P(|y|),Q(|y|)}. Motivated by the work of Duan and Musso [9], we use the Lyapunov–Schmidt reduction method to construct new synchronized solutions of the problem (Pε) with more complex concentration structure than the results in Wei, Yan [26] and Peng, Wang [23], when P(|y|), Q(|y|) and λ(|y|) satisfy some decay assumptions at infinity.

Multi-peak solutions for the Schnakenberg model with heterogeneity on star shaped graphs

In this paper, the Schnakenberg model with heterogeneity function is considered on star shaped metric graphs. We establish the existence and the linear stability of N-peak stationary solutions. In particular, we reveal that the location, amplitude, and stability are decided by the effects of the heterogeneity function and the geometry of the graph, represented by the associated Green’s function. The existence theorem is shown by using Lyapunov–Schmidt reduction method, and the stability is analyzed by investigating the associated linearized eigenvalue problem. Also, by considering several concrete examples, we describe how the location and the stability are decided by the interaction of the geometry of the graph with the heterogeneity function in detail. Moreover, we give the classification of the lengths of edges for the case of one spike per one edge. The case of a one-dimensional interval case without heterogeneity function case was studied by Iron et al. (2004). Although the proof of our main results is based on their strategy, we present all key estimates needed the analysis for the case of a star graph with heterogeneity function in detail.

Stability and Bifurcation of a Delayed Reaction–Diffusion Model with Robin Boundary Condition in Heterogeneous Environment

In this paper, we investigate a reaction–diffusion model with delay and Robin boundary condition in heterogeneous environment. The existence, multiplicity and stability of spatially nonhomogeneous steady-state solutions and periodic solutions are studied by employing the Lyapunov–Schmidt reduction method. Moreover, the Hopf bifurcation direction is derived. It is observed that Robin boundary condition plays a crucial role in the Hopf bifurcation. More precisely, when the boundary effect is stronger than the interaction of the species within the region, there is no Hopf bifurcation no matter how the time delay [Formula: see text] changes. Finally, we illustrate our general theoretical results by an application to the Nicholson’s blowflies model.

The existence and stability of spikes in the one-dimensional Keller-Segel model with logistic growth.

It is well known that Keller-Segel models serve as a paradigm to describe the self aggregation phenomenon, which exists in a variety of biological processes such as wound healing, tumor growth, etc. In this paper, we study the existence of monotone decreasing spiky steady state and its linear stability property in the Keller-Segel model with logistic growth over one-dimensional bounded domain subject to homogeneous Neumann boundary conditions. Under the assumption that chemo-attractive coefficient is asymptotically large, we construct the single boundary spike and next show this non-constant steady state is locally linear stable via Lyapunov-Schmidt reduction method. As a consequence, the multi-symmetric spikes are obtained by reflection and periodic extension. In particular, we present the formal analysis to illustrate our rigorous theoretical results.

Hopf bifurcation in a Lotka-Volterra competition-diffusion-advection model with time delay

In this paper, we mainly investigate a Lotka-Volterra competition-diffusion-advection system with time delay, where the diffusion and advection rates of two competitors are different. By employing the Lyapunov-Schmidt reduction method, we obtain the existence of steady state solution. A weighted inner product has been introduced to study stability and Hopf bifurcation at the spatially nonhomogeneous steady-state. Our results imply that the infinitesimal generator associated with the linearized system have two pairs of purely imaginary eigenvalues, and time delay can make the spatially nonconstant positive steady state unstable for a reaction-diffusion-advection model. In addition, the bifurcation direction and stability of Hopf bifurcating periodic orbits was obtained by means of the center manifold reduction and the normal form theory.

Existence of multi-bump solutions for a nonlinear Kirchhoff-type system

In this paper, we use the Lyapunov–Schmidt reduction method to obtain the existence of multi-bump solutions for a nonlinear Kirchhoff-type system with the parameter ɛ. As a result, when ɛ → 0, the system has more and more multi-bump positive solutions.

Dynamics of a predator–prey system with nonlinear prey-taxis

In this paper, we investigate a predator–prey system with nonlinear prey-taxis under Neumann boundary condition. For a class of chemotactic sensitive functions, we obtain the existence and boundedness of global classical solutions for initial boundary value problems in arbitrary dimensional space. In addition, we also study the local stability of the constant steady state solution, and obtain the global asymptotic stability of the steady state solution under different predation intensity by constructing appropriate Lyapunov functions. Furthermore, the steady state bifurcation, Hopf bifurcation and fold-Hopf Singularity are analysed in detail by using Lyapunov–Schmidt reduction method.

Spatiotemporal patterns and bifurcations with degeneration in a symmetry glycolysis model

In this paper, we consider the glycolysis system with symmetry and degeneration. Based on the detailed stability of constant steady state solution and symmetry of nonlinear glycolysis system, for the Hopf bifurcations, we obtain the existence, stability and bifurcation direction not only for the homogeneous and inhomogeneous periodic solutions, but also for the degenerate bifurcating periodic solutions. The approach we adopted is the Lyapunov–Schmidt reduction method instead of the center manifold theory. On the other hand, we derive the nonexistence of the steady state solutions for small input flux of substrate. Additionally, for the steady state bifurcations, main object is to discuss degenerate bifurcations by means of Lyapunov–Schmidt procedure and singularity theory because the classical Crandall–Rabinowitz theorem cannot be applied. Some numerical simulations are achieved to illustrate the analytical results, especially the degenerate bifurcation results. It is interesting to noticed that the degenerate bifurcations can lead to richer spatiotemporal patterns, including the meeting of two Hopf bifurcation branches and the coupling of two eigenfunction models.

Existence and stability of symmetric and asymmetric patterns for the half-Laplacian Gierer–Meinhardt system in one-dimensional domain

In this paper, we study the existence and stability of multiple spikes pattern to the fractional Gierer–Meinhardt model with periodic boundary conditions and the fractional power [Formula: see text]. Specifically, we rigorously establish the existence of symmetric multiple spikes and asymmetric two-spikes solutions by the classical Lyapunov–Schmidt reduction method. We also investigate the stability of the constructed solution by studying its associated large and small eigenvalue problems, where we need to consider two nonlocal eigenvalue problems in their fractional versions. In the study of the large eigenvalue problem, the quantity [Formula: see text] is the critical threshold which determines the stability of [Formula: see text]-peaked solutions. For the symmetric two-spikes pattern we obtain the asymptotic expansion for the critical threshold [Formula: see text] up to the second order. Moreover, we provide some elementary properties of the Green’s function, including the first and second derivatives, they are linked to the location of the spikes and the stability. Among these properties on the Green’s function, we find out that the polygamma function [Formula: see text] plays a crucial role.

Non-degeneracy of Positive Solutions for Fractional Kirchhoff Problems: High Dimensional Cases

In this paper, we establish the nondegeneracy of positive solutions to the fractional Kirchhoff problem (a+b∫RN|(-Δ)s2u|2dx)(-Δ)su+u=up,inRN,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\Big (a+b{\\int _{\\mathbb {R}^{N}}}|(-\\Delta )^{\\frac{s}{2}}u|^2\\mathrm{{d}}x\\Big )(-\\Delta )^su+u=u^p,\\quad \\text {in}\\ \\mathbb {R}^{N}, \\end{aligned}$$\\end{document}where a,b>0, 0<s<1, 1<p<frac{N+2s}{N-2s} and (-Delta )^s is the fractional Laplacian. In particular, we prove that uniqueness breaks down for dimensions N>4s, i.e., we show that there exist two non-degenerate positive solutions which seem to be completely different from the result of the fractional Schrödinger equation or the low dimensional fractional Kirchhoff equation. As one application, combining this nondegeneracy result and Lyapunov-Schmidt reduction method, we can derive the existence of solutions to the singularly perturbation problems.

A singular perturbed problem with critical Sobolev exponent

This paper deals with the following nonlinear elliptic problem \begin{equation}\label{eq0.1} -\varepsilon^2\Delta u+\omega V(x)u=u^{p}+u^{2^{*}-1},\quad u&gt; 0\quad\text{in}\ \R^N, \end{equation} where $\omega\in\R^{+}$, $N\geq 3$, $p\in (1,2^{*}-1)$ with $2^{*}={2N}/({N-2})$, $\varepsilon&gt; 0$ is a small parameter and $V(x)$ is a given function. Under suitable assumptions, we prove that problem (\ref{eq0.1}) has multi-peak solutions by the Lyapunov-Schmidt reduction method for sufficiently small $\varepsilon$, which concentrate at local minimum points of potential function $V(x)$. Moreover, we show the local uniqueness of positive multi-peak solutions by using the local Pohozaev identity.