Homoclinic solutions for p(t)-Laplacian Hamiltonian systems with new conditions

In recent years, starting with works of Bolotin [Bol], Coti-Zelati, Ekeland and Sere [CZES], Coti-Zelati & Rabinowitz [CZR1], [CZR2], Rabinowitz [Ra4], variational methods have been applied to study the existence of homoclinic and heteroclinic solutions of second-order equations and Hamiltonian systems. The search for homoclinic and heteroclinic solutions is a classical problem, originated from the work of Poincare and has been developed from several points of view. Existence of homoclinic solutions can be obtained by analyzing the intersection properties of the stable and unstable manifolds of the fixed points. There is a standard method to find infinitely nearby homoclinics provided that the stable and unstable manifolds of a fixed point intersect transversally. For this approach we refer the reader to Moser [Mo], Guckenheimer & Holmes [GH], Wiggins [Wig].

Periodic and homoclinic solutions generated by impulses for asymptotically linear and sublinear Hamiltonian system

This paper studies a Hamiltonian system possessing a double well potential for which the existence of multitransition heteroclinic and homoclinic solutions that are local minimizers of an associated functional is known. Under an additional mild non-degeneracy condition on the set of all homoclinic and heteroclinic solutions, the existence of further heteroclinic and homoclinic solutions that are of mountain pass type is established. A key tool for the existence arguments is a variant of the Mountain Pass Theorem that is of independent interest.

Chapter 2 Hamiltonian Systems: Periodic and Homoclinic Solutions by Variational Methods

Homoclinic solutions for some second order non-autonomous Hamiltonian systems without the globally superquadratic condition

In this paper, we study the homoclinic solutions of the following second‐order Hamiltonian system urn:x-wiley:0025584X:media:mana201200253:mana201200253-math-0001where , and . Applying the symmetric Mountain Pass Theorem, we establish a couple of sufficient conditions on the existence of infinitely many homoclinic solutions. Our results significantly generalize and improve related ones in the literature. For example, is not necessary to be uniformly positive definite or coercive; through is still assumed to be superquadratic near , it is not assumed to be superquadratic near .

In this paper we consider the existence of homoclinic solutions for the following second order non-autonomous Hamiltonian system $${\ddot q}-L(t)q+\nabla W(t,q)=0, \quad\quad\quad\quad\quad\quad\quad (\rm HS)$$ where \({L\in C({\mathbb R},{\mathbb R}^{n^2})}\) is a symmetric and positive definite matrix for all \({t\in {\mathbb R}}\), W(t, q) = a(t)U(q) with \({a\in C({\mathbb R},{\mathbb R}^+)}\) and \({U\in C^1({\mathbb R}^n,{\mathbb R})}\). The novelty of this paper is that, assuming L is bounded from below in the sense that there is a constant M > 0 such that (L(t)q, q) ≥ M |q|2 for all \({(t,q)\in {\mathbb R}\times {\mathbb R}^n}\), we establish one new compact embedding theorem. Subsequently, supposing that U satisfies the global Ambrosetti–Rabinowitz condition, we obtain a new criterion to guarantee that (HS) has one nontrivial homoclinic solution using the Mountain Pass Theorem, moreover, if U is even, then (HS) has infinitely many distinct homoclinic solutions. Recent results from the literature are generalized and significantly improved.

The research for Hamilton system is a classical problem, it has valuable applications in celestial mechanics, plasma physics, and biological engineering. But in actual management, Hamilton system can be changed into solving homoclinic solutions of differential equation or system. This paper studies the existence of homoclinic solutions for a class of second order differential equation, we will prove this equation exists at least one nontrivial homoclinic solution.

In this paper we state the existence of positive homoclinic solutions of the second order equation $$ u - \alpha (x)u + \beta (x)u^2 + \gamma (x)u^3 = 0, x \in \mathbb{R}, $$ (I) where the coefficient functions a(x)s(x) and y(x) are continuous, positive and 27-periodic. We obtain, in some sense, generalizations of results contained in [10] and [6], where s(x) is assumed identically zero. The homoclinic solution u of equation (I) is obtained as the limit of 2n7-periodic solutions of (I). We establish the fact that the quadratic form associated to the linear operator is positive definite and the particular type of the nonlinearity considered introduces simplicity and clearness in the proof, namely when we use the mountain pass lemma to study some periodic approximating problems. We present only the main ideas and sketch the proofs briefly. In Section 2, we study equation (I). The approximating procedure used in the proofs appears in several papers concerning the existence of homoclinics, namely in the case of Hamiltonian systems. We refer to Rabinowitz [10], Ambrosetti and Bertotti [1], Korman and Lazer [6], Arioli and Szulkin [2]. However those results do not apply to equation (I). Essentially, not only does the nonlinearity we consider not satisfy the hypotheses assumed there, but also [1], [10] and [2] do not concern positive solutions. For further details concerning Section 2, see[5].

Multiplicity and concentration of homoclinic solutions for some second order Hamiltonian systems

This article presents a rigorous existence theory for three-dimensional gravity-capillary water waves which are uniformly translating and periodic in one spatial directionx and have the profile of a uni- or multipulse solitary wave in the otherz. The waves are detected using a combination of Hamiltonian spatial dynamics and homoclinic Lyapunov-Schmidt theory. The hydrodynamic problem is formulated as an infinite-dimensional Hamiltonian system in whichz is the timelike variable, and a family of pointsP k,k+1, k= 1, 2,… in its two-dimensional parameter space is identified at which a Hamiltonian 0202 resonance takes place (the zero eigenspace and generalised eigenspace are respectively two and four dimensional). The pointP k,k+1is precisely that at which a pair of two-dimensional periodic linear travelling waves with frequency ratiok: k + 1 simultaneously exist (“Wilton ripples”). A reduction principle is applied to demonstrate that the problem is locally equivalent to a four-dimensional Hamiltonian system nearP k,k+1. It is shown that a Hamiltonian real semisimple 1∶1 resonance, where two geometrically double real eigenvalues exist, arises along a critical curveR k,k+1emanating fromP k,k+1.Unipulse transverse homoclinic solutions to the reduced Hamiltonian system at points ofR k,k+1nearP k,k+1are found by a scaling and perturbation argument, and the homoclinic Lyapunov-Schmidt method is applied to construct an infinite family of multipulse homoclinic solutions which resemble multiple copies of the unipulse solutions.

In this paper, we consider of the following second-order Hamiltonian system u ¨ ( t ) − L ( t ) u ( t ) + ∇ W ( t , u ( t ) ) = 0 , ∀ t ∈ R , where W ( t , x ) is subquadratic at infinity. With a competition condition, we establish the existence of homoclinic solutions by using the variational methods. In our theorem, the smallest eigenvalue function l ( t ) of L ( t ) is not necessarily coercive or bounded from above and W ( t , x ) is not necessarily integrable on R with respect to t . Our theorem generalizes many known results in the references.

This paper presents an existence theory for solitary waves at the interface between a thin ice sheet (modelled using the Cosserat theory of hyperelastic shells) and an ideal fluid (of finite depth and in irrotational motion). The theory takes the form of a review of the Kirchgässner reduction to a finite-dimensional Hamiltonian system, highlighting the refinements in the theory over the years and presenting some novel aspects including the use of a higher-order Legendre transformation to formulate the problem as a spatial Hamiltonian system, and a Riesz basis for the phase space to complete the analogy with a dynamical system. The reduced system is to leading order given by the focussing cubic nonlinear Schrödinger equation, agreeing with the result of formal weakly nonlinear theory (which is included for completeness). We give a precise proof of the persistence of two of its homoclinic solutions as solutions to the unapproximated reduced system which correspond to symmetric hydroeleastic solitary waves.

In this article, we investigate a class of impulsive Hamiltonian systems with a p-Laplacian operator. By establishing a series of new sufficient conditions, the existence of homoclinic solutions to such type of systems is revealed. We show the existence of homoclinic orbit induced by impulses by introducing some conditions. To illustrate the applications of the main results in this article, we create an example.

ABSTRACTIn this paper, we are concerned with the multiplicity and concentration of solutions for a Hamiltonian system driven by the fractional Laplace operator with variable order derivative. More precisely, we consider where , , is a nonlocal fractional integro-differential operator with variable order derivative, is a parameter, is a real symmetric matrix and belongs to . Under some suitable assumptions, we show that the system admits at least two distinct homoclinic solutions. Moreover, we investigate the concentration of solutions as . This paper is the first time to deal with Hamiltonian systems with variable order fractional derivatives. Moreover, our system is anisotropic. Thus this is different from other papers in the literature.