Schmidt Reduction Research Articles

Response solutions for elliptic-hyperbolic equations with nonlinearities and periodic external forces

In this paper, we focus on the existence of response solutions, i.e. periodic solutions with the same frequencies as the external forces, for elliptic-hyperbolic partial differential equations with nonlinearities and periodic forces. The main tools are Lyapunov–Schmidt reduction and Nash–Moser iteration scheme, both of which have demonstrated success in hyperbolic scenarios. At each step of the iteration, the Galerkin approximation of the equation is solved. The new issue is that the spectral theory of the generalized Sturm–Liouville problem is employed, which also introduces new difficulties for estimations at each step. Under appropriate non-resonance conditions on the frequency, the existence of response solutions for the model will be established.

Single‐peak solution for a fractional slightly subcritical problem with non‐power nonlinearity

AbstractWe consider the following fractional problem involving slightly subcritical non‐power nonlinearity, where is a smooth bounded domain in , , , is the fractional critical Sobolev exponent and is a small parameter, is the spectral fractional Laplacian operator. We construct a positive bubbling solution, which concentrates at a nondegenerate critical point of the Robin function by Lyapunov–Schmidt reduction procedure.

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.

Infinitely many solutions for linearly coupled Schrödinger systems in ℝ3

In this paper, we consider the following linearly coupled nonlinear Schrödinger system: where , and are radial, positive, and continuous and satisfying that When and satisfy some weak power decay condition at infinity; we show that problem has infinitely many nonradial positive solutions by using the Lyapunov–Schmidt reduction method.

Branching patterns of wave trains in mass-in-mass lattices

We investigate the existence and branching patterns of wave trains in the mass-in-mass (MiM) lattice, which is a variant of the Fermi–Pasta–Ulam (FPU) lattice. In contrast to FPU lattice, we have to solve coupled advance-delay differential equations, which are reduced to a finite-dimensional bifurcation equation with an inherited Hamiltonian structure by applying a Lyapunov–Schmidt reduction and invariant theory. We establish a link between the MiM lattice and the monatomic FPU lattice. That is, the monochromatic and bichromatic wave trains persist near $\mu =0$ in the nonresonance case and in the resonance case $p:q$ where $q$ is not an integer multiple of $p$ . Furthermore, we obtain the multiplicity of bichromatic wave trains in $p:q$ resonance where $q$ is an integer multiple of $p$ , based on the singular theorem.

Classification and non-degeneracy of positive radial solutions for a weighted fourth-order equation and its application

This paper is devoted to radial solutions of the following weighted fourth-order equation div(|x|α∇(div(|x|α∇u)))=|u|2α∗∗−2uinRN,where N≥2, 4−N2<α<2 and 2α∗∗=2NN−4+2α. It is obvious that the solutions of above equation are invariant under the scaling λN−4+2α2u(λx) while they are not invariant under translation when α≠0. We characterize all the solutions to the related linearized problem about radial solutions, and obtain the conclusion of that if α satisfies (2−α)(2N−2+α)≠4k(N−2+k) for all k∈N+ the radial solution is non-degenerate, otherwise there exist new solutions to the linearized problem that “replace” the ones due to the translations invariance. As applications, firstly we investigate the remainder terms of some inequalities related to above equation. Then when N≥5 and 0<α<2, we establish a new type second-order Caffarelli–Kohn–Nirenberg inequality ∫RN|div(|x|α∇u)|2dx≥C∫RN|u|2α∗∗dx22α∗∗,for allu∈C0∞(RN),and in this case we consider a prescribed perturbation problem by using Lyapunov–Schmidt reduction.

Stability and bifurcation in a reaction–diffusion model with nonlinear boundary conditions

In this paper, we investigate the existence, stability, local and global bifurcation of the steady state solutions for a diffusive logistic model with nonlinear boundary conditions. Supercritical and subcritical bifurcation of steady state solutions are obtained by virtue of the Crandall–Rabinowitz theorem and the Lyapunov–Schmidt reduction. The local stability and the possibility of obtaining an Allee effect are also analyzed. Furthermore, the result on global bifurcation is obtained as well by employing the Rabinowitz theorem.

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.

Local space time constant mean curvature and constant expansion foliations

Inspired by the small sphere-limit for quasi-local energy we study local foliations of surfaces with prescribed mean curvature. Following the strategy used by Ye in [23] to study local constant mean curvature foliations we use a Lyapunov Schmidt reduction in an n+1 dimensional manifold equipped with a symmetric 2-tensor to construct the foliations around a point, prove their uniqueness and show their nonexistence conditions. To be specific, we study two foliation conditions. First we consider constant space-time mean curvature surfaces. They are used to characterizing the center of mass in general relativity [4]. Second, we study local foliations of constant expansion surfaces [18].

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.

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.

Period-Doubling Bifurcation of Cycles in Retarded Functional Differential Equations

AbstractA rigorous description of a period-doubling bifurcation of limit cycles in retarded functional differential equations based on tools of functional analysis and singularity theory is presented. Particularly, sufficient conditions for its occurrence and its normal form coefficients are expressed in terms of derivatives of the operator defining given equations. We also prove the exchange of stability in the case of a non-degenerate period-doubling bifurcation. The approach concerns Fredholm operators, Lyapunov–Schmidt reduction and recognition problem for pitchfork bifurcation.

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.

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.

Vortex patch problem for steady lake equation

We study the vortex patch problem for the steady lake equation in a bounded domain and construct three different kinds of solutions where the vorticity concentrates in the domain or near the boundary. Our approach is based on the Lyapunov–Schmidt reduction, which transforms the construction into a problem of seeking critical points for a function related to the kinetic energy. The method in this paper has a wide applicability and can be used to deal with general elliptic equations in divergence form with Heaviside nonlinearity.