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We provide sufficient conditions for the existence of periodic solutions of the Lorentz force equation, which models the motion of a charged particle under the action of an electromagnetic fields. The basic assumptions cover relevant models with singularities like Coulomb-like electric potentials or the magnetic dipole.

In this paper we study the existence and multiplicity of solutions of the Lorentz force equation $$\left(\frac{q'}{\sqrt{1-|q'|^2}}\right)'=E(t,q) + q'\times B(t,q)$$ with periodic or Dirichlet boundary conditions. In Special Relativity, this equation models the motion of a slowly accelerated electron under the influence of an electric field E and a magnetic field B. We provide a rigourous critical point theory by showing that the solutions are the critical points in the Szulkin’s sense of the corresponding Poincare non-smooth Lagrangian action. By using a novel minimax principle, we prove a variety of existence and multiplicity results. Based on the associated Planck relativistic Hamiltonian, an alternative result is proved for the periodic case by means of a minimax theorem for strongly indefinite functionals due to Felmer.

In this paper, we study, using critical point theory for strongly indefinite functionals, the Neumann problem associated to some prescribed mean curvature problems in a FLRW spacetime with one spatial dimension. We assume that the warping function is even and positive and the prescribed mean curvature function is odd and sublinear. Then, we show that our problem has infinitely many solutions. The keypoint is that our problem has a Hamiltonian formulation. The main tool is an abstract result of Clark type for strongly indefinite functionals.

We study the existence of spacelike graphs for the prescribed mean curvature equation in the Friedmann–Lemaitre–Robertson–Walker (FLRW) spacetime. By using a conformal change of variable, this problem is translated into an equivalent problem in the Lorentz–Minkowski spacetime. Then, by using Rabinowitz's global bifurcation method, we obtain the existence and multiplicity of positive solutions for this equation with 0-Dirichlet boundary condition on a ball. Moreover, the global structure of the positive solution set is studied.

In this paper we introduce new Lusternik–Schnirelman type methods for nonsmooth functionals including the action integral associated to the relativistic Lagrangian of a test particle under the action of an electromagnetic field $$\begin{aligned} {\mathcal {L}}(t,q,q')=1-\sqrt{1-|q'|^2}+q'\cdot W(t,q) - V(t,q), \end{aligned}$$where $$V:[0,T]\times {\mathbb {R}}^3\rightarrow {\mathbb {R}}$$ and $$W:[0,T]\times {\mathbb {R}}^3\rightarrow {\mathbb {R}}^3$$ are two $$C^1$$-functions with V even and W odd in the second variable. By applying them, we obtain various multiplicity results concerning T-periodic solutions of the relativistic Lorentz force equation in Special Relativity, $$\begin{aligned} \left( \frac{q'}{\sqrt{1-|q'|^2}}\right) '=E(t,q) + q'\times B(t,q), \end{aligned}$$where $$ E=-\nabla _q V-\frac{\partial W}{\partial t}, B=\hbox {curl}_q\, W. $$ The zero Dirichlet boundary value conditions are considered as well.

We study the existence/nonexistence and multiplicity of spacelike graphs for the following mean curvature equation in a standard static spacetime div (a del u/root 1-a(2)vertical bar del u vertical bar(2)) + g (del u, del a)/root 1-a(2)vertical bar del u vertical bar(2) = lambda NH with 0-Dirichlet boundary condition on the unit ball. According to the behavior of H near 0, we obtain the global structure of one-sign radial spacelike graphs for this problem. Moreover, we also obtain the existence and multiplicity of entire spacelike graphs., National Natural Science Foundation of China (NSFC) 11871129, Xinghai Youqing funds from Dalian University of Technology, Spanish MINECO, European Union (EU) MTM2016-78807-C2-1-P MTM2017-82348-C2-1-P

We identify a family of Friedmann–Lemaitre–Robertson–Walker (FLRW) spacetimes such that the radially symmetric prescribed curvature problem with Neumann boundary condition is solvable on a ball of small radius. Such family contains some examples of interest in Cosmology.

We study the existence of twist periodic solutions of second order differential equations with an attractive–repulsive singularity. Such twist periodic solutions are stable in the sense of Lyapunov. The proof is based on the third order approximation method in combination with some location information obtained by the averaging method and the method of upper and lower solutions on the reversed order.

We prove the presence of chaotic dynamics for the classical two-body Kepler problem with a time-periodic gravitational coefficient oscillating between two fixed values. The set of chaotic solutions we detect is coded by the number of revolutions in each period. The chaotic dynamics is obtained for large period T as well as for small angular momentum μ. In particular, we provide an explicit lower bound on T and explicit upper bound on μ which guarantee the existence of complex dynamics. We get our results by applying a simple and well-known topological method, the stretching along the path technique. Our results are robust with respect to small perturbations of the gravitational coefficient and to the addition of a small friction term.

For the Gylden-Meshcherskii-type problem with a periodically cha-nging gravitational parameter, we prove the existence of radially periodic solutions with high angular momentum, which are Lyapunov stable in the radial direction.

We consider a 1D nonlinear Schrodinger equation (NLSE) which describes the mean field dynamics of an elongated Bose–Einstein condensate and prove the existence of modulated amplitude waves with non-trivial phase and minimal spatial period tending to infinite. The proof combines the theory of local continuation of non-degenerate periodic solutions with a property of the Ermakov–Pinney equation.

In this article, we construct novel explicit solutions for nonlinear Schr\"odinger systems with spatially inhomogeneous nonlinearity by means of the Lie symmetry method. We focus the attention to solutions with non-trivial phase, which have been scarcely considered in the related literature. To get started, the theoretical method based on Lie symmetries is exposed, thus reducing the problem to the integrability of an ODE. The non-trivial phase introduces a singular term into the ODE. Then, the method is used to construct new families of analytical solutions. Some illustrative examples are provided., Comment: 21 pages, 11 figures, some corrections made and 5 new figures added

We prove the existence of periodic solutions and solitons in the nonlinear Schrodinger equation with a nonlocal integral term of convolution type. By separating phase and amplitude, the problem is reduced to an integro-differential formulation that can be written as a fixed point problem for a suitable operator on a Banach space. Then a fixed point theorem due to Krasnoselskii can be applied.

We derive some sufficient conditions for the existence of dispersion-managed solitons in a nonlinear Schrodinger equation with periodically varying coefficients. The proof relies on a combination of the variational method with the development of a novel upper and lower function method.

Sufficient conditions for the existence of a solution to the problem u ″ ( t ) = g ( t ) u μ ( t ) − h ( t ) u λ ( t ) + f ( t ) for a.e. t ∈ [ 0 , ω ] , u ( 0 ) = u ( ω ) , u ′ ( 0 ) = u ′ ( ω ) are established.

We study the existence of solution for the one-dimensional $\phi$-laplacian equation $(\phi(u'))'=\lambda f(t,u,u')$ with Dirichlet or mixed boundary conditions. Under general conditions, an explicit estimate $\lambda_0$ is given such that the problem possesses a solution for any $|\lambda

A proof of the existence of stationary dark soliton solutions of a cubic-quintic nonlinear Schrodinger equation with a periodic potential is given. It is based on the interpretation of the dark soliton as a heteroclinic of the Poincare map.

Motivated by the study of the propagation of electromagnetic waves through a multi-layered optical medium, we prove the existence of two different kinds of homoclinic solutions to the origin in a Schrodinger equation with a nonlinear term. We use a Krasnoselskii fixed point theorem together with a compactness criterion due to Zima. The main results are illustrated with concrete examples of practical interest such as self-focusing nonlinearities of Kerr and non-Kerr type.

Th ee xistence of non-trivial periodic solutions of a general family of second order differential equations whose main model is a Hill’s equation with a cubic nonlinear term arising in different physical applications is proved. c

We investigate the existence and multiplicity of nontrivial periodic bouncing solutions for linear and asymptotically linear impact oscillators by applying a generalized version of the Poincare--Birkhoff theorem to an adequate Poincare section called the successor map. The main theorem includes a generalization of a related result by Bonheure and Fabry and provides a sufficient condition for the existence of periodic bouncing solutions for Hill's equation with obstacle at $x\not =0$.

In this paper, we will give, for the periodic solution of the scalar Newtonian equation, some twist criteria which can deal with the fourth order resonant case. These are established by developing some new estimates for the periodic solution of the Ermakov–Pinney equation, for which the associated Hill equation may across the fourth order resonances. As a concrete example, the least amplitude periodic solution of the forced pendulum is proved to be twist even when the frequency acroses the fourth order resonances. This improves the results in Lei et al. (2003). Twist character of the least amplitude periodic solution of the forced pendulm. SIAM J. Math. Anal. 35, 844–867.

We demonstrate that stabilization of solitons of the multidimensional Schrodinger equation with a cubic nonlinearity may be achieved by a suitable periodic control of the nonlinear term. The effect of this control is to stabilize the unstable solitary waves which belong to the frontier between expanding and collapsing solutions and to provide an oscillating solitonic structure, some sort of breather-type solution. We obtain precise conditions on the control parameters to achieve the stabilization and compare our results with accurate numerical simulations of the nonlinear Schrodinger equation. Because of the application of these ideas to matter waves these solutions are some sort of matter breathers.

This paper is devoted to study the existence of periodic solutions of the second-order equation x 00 ¼ f ðt; xÞ; where f is a Caratheodory function, by combining some new properties of Green's function together with Krasnoselskii fixed point theorem on compression and expansion of cones. As applications, we get new existence results for equations with jumping nonlinearities as well as equations with a repulsive or attractive singularity. In this latter case, our results cover equations with weak singularities and are compared with some recent results by I. Rachunkova´ , M. Tvrdyand I. Vrkoc˘ . r 2002 Elsevier Science (USA). All rights reserved.

The nonlinear Schrodinger equation (NLSE) is one of the most important models of Mathematical Physics and plays a central role in a variety of contexts, especially those related with Quantum Mechanics. Dynamics of Bose-Einstein Condensates and signal transmission in Optical Fibers are the most important examples. In this chapter, we show that the study of the NLSE with cubic nonlinearity (Gross-Pitaevskii equation) is closely related to the analysis of differential equations with singularities.

A force field in \({\mathbb R}^N\) is called central or radially symmetric if its magnitude only depends on the distance to the origin and its direction is proportional to the vector position, that is, it always points to the origin. The study of a radially symmetric system is reduced to a second-order scalar equation with a repulsive singularity. We review some of the most interesting models with radial symmetry.

In this chapter, we consider the dynamics of an electron beam guided by an axially-symmetric periodic magnetic field.

We study a higher order parabolic partial differential equation that arises in the context of condensed matter physics. It is a fourth order semilinear equation whose nonlinearity is the determinant of the Hessian matrix of the solution. We consider this model in a bounded domain of the real plane and study its stationary solutions both when the geometry of this domain is arbitrary and when it is the unit ball and the solution is radially symmetric. We also consider the initial-boundary value problem for the full parabolic equation. We summarize our results on existence of solutions in these cases and propose an open problem related to the existence of self-similar solutions., To appear in Mathematica Bohemica

This work presents the construction of the existence theory of radial solutions to the elliptic equation \begin{equation}\nonumber \Delta^2 u = (-1)^k S_k[u] + \lambda f(x), \qquad x \in B_1(0) \subset \mathbb{R}^N, \end{equation} provided either with Dirichlet boundary conditions \begin{eqnarray}\nonumber u = \partial_n u = 0, \qquad x \in \partial B_1(0), \end{eqnarray} or Navier boundary conditions \begin{equation}\nonumber u = \Delta u = 0, \qquad x \in \partial B_1(0), \end{equation} where the $k-$Hessian $S_k[u]$ is the $k^{\mathrm{th}}$ elementary symmetric polynomial of eigenvalues of the Hessian matrix and the datum $f \in L^1(B_1(0))$ while $\lambda \in \mathbb{R}$. We prove the existence of a Carath\'eodory solution to these boundary value problems that is unique in a certain neighborhood of the origin provided $|\lambda|$ is small enough. Moreover, we prove that the solvability set of $\lambda$ is finite, giving an explicity bound of the extreme value.

The paper deals with the existence of twist solutions of a Hill's equation with singular term (often called Brillouin equation in the related literature) for a given region of parameters involved on the equation. Solutions of twist type are in particular Lyapunov stable and present interesting dynamical features around them. The techniques of proof include upper and lower solutions, topological degree and classical tools on normal forms of area preserving maps.

We provide sufficient conditions for the existence of a uniparametric family of entire spacelike graphs with prescribed mean curvature in a Friedmann–Lemaître–Robertson–Walker spacetime with flat fiber. The proof is based on the analysis of the associated homogeneous Dirichlet problem on a Euclidean ball together with suitable bounds for the gradient which permit the prolongability of the solution to the whole space.

The first author is partially supported by a GENIL grant YTR-2011-7 (Spain) and by the grant PN-II-RU-TE-2011-3-0157 (Romania). The second author is partially supported by the grant PN-II-RU-TE-2011-3-0157 (Romania). The third author is partially supported by Ministerio de Economia y Competitividad, Spain, project MTM2011-23652., In this paper, by using Leray-Schauder degree arguments and critical point theory for convex, lower semicontinuous perturbations of C1-functionals, we obtain existence of classical positive radial solutions for Dirichlet problems of type div ( √1 − |∇ ∇v v|2 ) + f(|x|; v) = 0 in B(R); v = 0 on @B(R): Here, B(R) = {x ∈ RN : |x| < R} and f : [0; R] × [0; α) → R is a continuous function, which is positive on (0; R] × (0; α), GENIL (Spain) YTR-2011-7, Ministerio de Economia y Competitividad, Spain MTM2011-23652, PN-II-RU-TE-2011-3-0157

We consider a Kepler problem, with an additional rotating external force, and study the existence of periodic solutions when a small perturbative term is introduced. Surprisingly enough, we always get at least one of such solutions. Moreover, if a nonresonance assumption is added, then the existence of a second solution is also proved.

The dynamics of one species chemical kinetics is studied. Chemical reactions are modelled by means of continuous time Markov processes whose probability distribution obeys a suitable master equation. A large deviation theory is formally introduced, which allows developing a Hamiltonian dynamical system able to describe the system dynamics. Using this technique we are able to show that the intrinsic fluctuations, originated in the discrete character of the reagents, may sustain oscillations and chaotic trajectories which are impossible when these fluctuations are disregarded. An important point is that oscillations and chaos appear in systems whose mean-field dynamics has too low a dimensionality for showing such a behavior. In this sense these phenomena are purely induced by noise, which does not limit itself to shifting a bifurcation threshold. On the other hand, they are large deviations of a short transient nature which typically appear only after long waiting times. We also discuss the implications of our results in understanding extinction events in population dynamics models expressed by means of stoichiometric relations, This work was partially supported by the MICINN (Spain) through Project MTM2008-02502

Motivated by the study of matter waves in Bose-Einstein condensates and coupled nonlinear optical systems, we study a system of two coupled nonlinear Schrodinger equations with inhomogeneous parameters, including a linear coupling. For that system we prove the existence of two different kinds of homoclinic solutions to the origin describing solitary waves of physical relevance. We use a Krasnoselskii fixed point theorem together with a suitable compactness criterion., Comment: 16 pages

Using Lie group theory and canonical transformations, we construct explicit solutions of nonlinear Schrödinger equations with spatially inhomogeneous nonlinearities. We present the general theory, use it to show that localized nonlinearities can support bound states with an arbitrary number solitons, and discuss other applications of interest to the field of nonlinear matter waves.

We study an example of exact parametric resonance in a extended system ruled by nonlinear partial differential equations of nonlinear Schr\"odinger type. It is also conjectured how related models not exactly solvable should behave in the same way. The results have applicability in recent experiments in Bose-Einstein condensation and to classical problems in Nonlinear Optics., Comment: 1 figure

In this paper, we give a proof of the existence of stationary d ark soliton solutions of the cubic �

Using Lie group theory and canonical transformations, we construct explicit solutions of nonlinear Schrodinger equations with spatially inhomogeneous nonlinearities. We present the general theory, use it to study different examples and use the qualitative theory of dynamical systems to obtain some properties of these solutions.

We consider the prescribed mean curvature problem of spacelike graphs in Robertson- Walker spacetimes of flat fiber with homogeneous Dirichlet conditions on an Euclidean ball. Under reasonable assumptions, it is shown that every possible solution must be radially symmetric. Besides, an existence result for a singular nonlinear equation is proved by making use of the classical Schauder fixed point Theorem. The results are applied to realistic examples of Robertson-Walker spacetimes.