Schmidt Reduction Research Articles

On Post-Critical Behavior of a Beam on an Elastic Foundation

In this paper, we analyze the nonlinear equilibrium equation corresponding to the two-parameter bifurcation problem arising in the stability analysis of an elastic simply supported beam on the Winkler type elastic foundation for the case when bimodal buckling occurs. We perform the bifurcation analysis of the nonlinear problem, by using Lyapunov–Schmidt reduction, thus obtaining the number of the nontrivial solutions to the nonlinear problem and qualitatively characterizing the solution patterns. We also give the formulation of the problem and bifurcation analysis from the total energy viewpoint and determine the energy of each bifurcating solution. We assert that the solution with the smallest energy is the one that will be observed in the post-critical state. For specific choice of parameters, the bifurcating solution in the form of the second buckling mode has the smallest total energy. The numerical results illustrating the theory are also provided.

Quasi-periodic self-excited travelling waves for damped beam equations

A two dimensional beam equation is considered. It is shown that for a dense set of model parameters, the presence of linear and nonlinear damping terms of opposite sign, produce small amplitude quasi-periodic travelling waves, which are continuations of two branches of rotating waves. Lyapunov–Schmidt reduction and only the classical implicit function theorem are used.

Stability and Hopf Bifurcation in a Reaction–Diffusion Model with Chemotaxis and Nonlocal Delay Effect

Chemotaxis is an observed phenomenon in which a biological individual moves preferentially toward a relatively high concentration, which is contrary to the process of natural diffusion. In this paper, we study a reaction–diffusion model with chemotaxis and nonlocal delay effect under Dirichlet boundary condition by using Lyapunov–Schmidt reduction and the implicit function theorem. The existence, multiplicity, stability and Hopf bifurcation of spatially nonhomogeneous steady state solutions are investigated. Moreover, our results are illustrated by an application to the model with a logistic source, homogeneous kernel and one-dimensional spatial domain.

Periodic Solutions to Nonlinear Wave Equation with X-Dependent Coefficients Under the General Boundary Conditions

In this paper we consider a class of nonlinear wave equations with x-dependent coefficients and prove existence of families of time-periodic solutions under the general boundary conditions. Such a model arises from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. The proof is based on a Lyapunov–Schmidt reduction together with a differentiable Nash–Moser iteration scheme.

Multi-bump solutions for fractional Nirenberg problem

We consider the multi-bump solutions of the following fractional Nirenberg problem (0.1)(−Δ)su=K(x)un+2sn−2s,u>0inRn,where s∈(0,1) and n>2+2s. If K is a periodic function in some k variables with 1≤k<n−2s2, we proved that (0.1) has multi-bump solutions with bumps clustered on some lattice points in Rk via Lyapunov–Schmidt reduction. It is also established that the (0.1) has an infinite-many-bump solutions with bumps clustered on some lattice points in Rn which is isomorphic to Z+k.

Effect of the protection zone on coexistence of the species for a ratio-dependent predator-prey model

This paper deals with the effect of the protection zone Ω0 on coexistence of the species for a ratio-dependent predator-prey model. We obtain a critical value λ1N(bδ(x),Ω) which is less than the well-known critical value λ1D(Ω0) obtained in the previous literatures. Furthermore, we show that if λ>λ1N(bδ(x),Ω), then the prey persists regardless of the growth rates of the predator; while if λ≤λ1N(bδ(x),Ω), then there exists a real number μ⁎, such that the prey is ultimately extinct when μ>μ⁎. As for λ<λ1N(bδ(x),Ω) and μ<μ⁎, the curve of the positive steady state solutions of the model emanating from (λ,0;−c) ends at a singular point (0,0;μ2). Meantime, by using the Lyapunov–Schmidt reduction method, we obtain a fine profile of its bifurcation diagrams, and the uniqueness or multiplicity of its positive steady state solutions. In addition, as generally expected, the chances of survival of the prey will increase with the size of the protection zone.

Dynamics of a Nonlocal Dispersal Model with a Nonlocal Reaction Term

In this paper, we study a class of nonlocal dispersal problem with a nonlocal term arising in population dynamics: [Formula: see text] where [Formula: see text] is a bounded domain, [Formula: see text], [Formula: see text] represents the nonlocal dispersal operator with continuous and non-negative dispersal kernel. The kernel [Formula: see text] is assumed to be non-negative and is allowed to have a degeneracy in a smooth subdomain [Formula: see text] of [Formula: see text]. When [Formula: see text] is either positive or vanishes in a subdomain, we respectively investigate the existence, multiplicity and asymptotical stability of positive steady states under the local/global variation of parameter by means of sub-supersolution method, Lyapunov–Schmidt reduction, and bifurcation theory.

Segregated vector solutions with multi-scale spikes for nonlinear coupled elliptic systems

In this paper, we consider the following nonlinear coupled elliptic system(Aε){−ε2Δu+P(x)u=μ1u3+βuv2in RN,−ε2Δv+Q(x)v=μ2v3+βu2vin RN,u>0,v>0in RN,u→0,v→0as |x|→+∞, where ε>0 is a parameter, μ1,μ2>0 and β>0 are constants, and P(x) and Q(x) are two nonnegative, smooth functions with different nondegenerate critical points and separated zero sets. Due to the Lyapunov–Schmidt reduction method and the Maximum Principle, we show that when β is less than a small positive number, there exists an ε0>0 such that for any 0<ε<ε0, (Aε) has a segregated vector solution where (uε,vε) and uε is trapped in a neighborhood of the nondegenerate critical points of P(x) as well as the zero sets of P(x), and vε is trapped in a neighborhood of the nondegenerate critical points of Q(x) as well as the zero sets of Q(x). Moreover, the amplitudes of uε (res vε) around the nondegenerate critical points and the zero sets of P(x) (res Q(x)) are of a different order compared with ε. To the best of our knowledge, these multi-scale solutions to the system have not been obtained previously.

Periodic Oscillations and Waves in Nonlinear Weakly Coupled Dispersive Systems

Bifurcations of periodic solutions in autonomous nonlinear systems of weakly coupled equations are studied. A comparative analysis is carried out between the mechanisms of Lyapunov–Schmidt reduction of bifurcation equations for solutions close to harmonic oscillations and cnoidal waves. Sufficient conditions for the branching of orbits of solutions are formulated in terms of the Pontryagin functional depending on perturbing terms.

On Lyapunov- Schmidt Reduction to Reduce Differential Algebraic Equation

On Lyapunov- Schmidt Reduction to Reduce Differential Algebraic Equation

On the nonexistence of degenerate phase-shift discrete solitons in a dNLS nonlocal lattice

We consider a one-dimensional discrete nonlinear Schrödinger (dNLS) model featuring interactions beyond nearest neighbors. We are interested in the existence (or nonexistence) of phase-shift discrete solitons, which correspond to four-site vortex solutions in the standard two-dimensional dNLS model (square lattice), of which this is a simpler variant. Due to the specific choice of lengths of the inter-site interactions, the vortex configurations considered present a degeneracy which causes the standard continuation techniques to be non-applicable.In the present one-dimensional case, the existence of a conserved quantity for the soliton profile (the so-called density current), together with a perturbative construction, leads to the nonexistence of any phase-shift discrete soliton which is at least C2 with respect to the small coupling ϵ, in the limit of vanishing ϵ. If we assume the solution to be only C0 in the same limit of ϵ, nonexistence is instead proved by studying the bifurcation equation of a Lyapunov–Schmidt reduction, expanded to suitably high orders. Specifically, we produce a nonexistence criterion whose efficiency we reveal in the cases of partial and full degeneracy of approximate solutions obtained via a leading order expansion.

Diffusive Stability of Spatially Periodic Solutions of the Brusselator Model

Applying the Lyapunov–Schmidt reduction approach introduced by Mielke and Schneider in their analysis of the fourth-order scalar Swift–Hohenberg equation, we carry out a rigorous small-amplitude stability analysis of Turing patterns for the canonical second-order system of reaction–diffusion equations given by the Brusselator model. Our results confirm that stability is accurately predicted in the small-amplitude limit by the formal Ginzburg–Landau amplitude equations, rigorously validating the standard weakly unstable approximation and the Eckhaus criterion.

Patterns in a nonlocal time-delayed reaction–diffusion equation

In this paper, the existence, stability, and multiplicity of nontrivial (spatially homogeneous or nonhomogeneous) steady-state solution and periodic solutions for a reaction–diffusion model with nonlocal delay effect and Dirichlet/Neumann boundary condition are investigated by using Lyapunov–Schmidt reduction. Moreover, we illustrate our general results by applications to population models with one-dimensional spatial domain.

Existence of the solutions of a reaction cross-diffusion model for two species

In this paper, we investigate a two-cooperative species model with reaction cross-diffusion under the Dirichlet boundary value condition. By applying Lyapunov–Schmidt reduction method and some important formulas, we obtain the existence of coexistence solutions under some given conditions.

Vector solutions with clustered peaks for nonlinear fractional Schrödinger systems in $$\mathbb {R}^{N}$$ R N

We consider the fractional nonlinear Schrodinger system $$\begin{aligned} \left\{ \begin{array}{ll} \epsilon ^{2s}(-\Delta )^s u +P_1( x)u=\mu _1 |u|^{2p-2}u+\beta |v|^p|u|^{p-2}u, \quad x\in \mathbb {R}^N,\\ \epsilon ^{2s}(-\Delta )^s v +P_2( x)v=\mu _2 |v|^{2p-2}v+\beta |u|^p|v|^{p-2}v, \quad \; x\in \mathbb {R}^N,\\ \end{array} \right. \end{aligned}$$ where $$\epsilon >0$$ is a small parameter, $$0<s<1,$$ $$P_1$$ and $$P_2$$ are positive potentials, $$\mu _1>0,~\mu _2>0$$ , and $$\beta \in \mathbb {R}$$ is a coupling constant. To construct solutions to this system, we use the Lyapunov–Schmidt reduction that takes advantage of the variational structure of the problem. For any positive integer $$k\ge 2$$ , we construct k interacting spikes concentrating near the local maximum point $$x_{0}$$ of $$P_1$$ and $$P_2$$ when $$P_{1}(x_{0})=P_{2}(x_{0})$$ in the attractive case. For any two positive integers $$k,m\ge 2$$ , we construct k interacting spikes for u near the local maximum point $$x_{1,0}$$ of $$P_1$$ and m interacting spikes for v near the local maximum point $$x_{2,0}$$ of $$P_2$$ , respectively, when $$x_{1,0}\ne x_{2,0}$$ . For $$s = 1$$ , this corresponds to the system studied by Peng and Pi (Discrete Contin Dyn Syst 36:2205–2227, 2016) for the classical nonlinear Schrodinger system.

Multi-bump bound states for a Schrödinger system via Lyapunov–Schmidt Reduction

Motivated by problems arising in nonlinear optics and Bose–Einstein condensates, we consider in $$\mathbb R^N$$ ( $$N \le 3$$ ) the following $$n \times n$$ system of coupled Schrodinger equations where $$\varepsilon >0$$ is a parameter, $$\beta _{ij}$$ are constants satisfying $$\beta _{ii} > 0$$ , and $$V_i$$ are positive potentials that admit some common critical points $$a_1, \ldots ,a_k$$ satisfying certain non-degenerate assumption. Then for any subsets $$J\subset \{1,2,\ldots ,k\}$$ , using a Lyapunov–Schmidt reduction method, we prove the existence of multi-bump bound solutions which as $$\varepsilon \rightarrow 0$$ concentrate on $$\cup _{j\in J}a_j$$ .

Bifurcation Analysis and Application for Impulsive Systems with Delayed Impulses

In this article, we present a systematic approach to bifurcation analysis of impulsive systems with autonomous or periodic right-hand sides that may exhibit delayed impulse terms. Methods include Lyapunov–Schmidt reduction and center manifold reduction. Both methods are presented abstractly in the context of the stroboscopic map associated to a given impulsive system, and are illustrated by way of two in-depth examples: the analysis of a SIR model of disease transmission with seasonality and unevenly distributed moments of treatment, and a scalar logistic differential equation with a delayed census impulsive harvesting effort. It is proven that in some special cases, the logistic equation can exhibit a codimension two bifurcation at a 1:1 resonance point.

"On Lyapunov- Schmidt Reduction to Reduce Differential Algebraic Equation "

"In this paper , we transformation singularity differential – Algebraic equation to an ordinary differential equations by use Lyapunov– Schmidt reduction and constructing Lyapunov functions depending on Reiss and Geiss method."

On a free boundary problem and minimal surfaces

From minimal surfaces such as Simons' cone and catenoids, using refined Lyapunov–Schmidt reduction method, we construct new solutions for a free boundary problem whose free boundary has two components. In dimension 8, using variational arguments, we also obtain solutions which are global minimizers of the corresponding energy functional. This shows that the theorem of Valdinoci et al. [41,42] is optimal.

Persistence of periodic solutions for higher order perturbed differential systems via Lyapunov–Schmidt reduction

In this work we first provide sufficient conditions to assure the persistence of some zeros of functions having the formfor sufficiently small. Here , for , are smooth functions being an open bounded set. Then we use this result to compute the bifurcation functions which allow us to study the periodic solutions of the following T-periodic smooth differential systemIt is assumed that the unperturbed differential system has a sub-manifold of periodic solutions , . We also study the case when the bifurcation functions have a continuum of zeros. Finally we provide the explicit expressions of the bifurcation functions up to order 5.