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Discrete & Continuous Dynamical Systems - B

2013 , Volume 18 , Issue 5

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Constrained energy minimization and ground states for NLS with point defects
Riccardo Adami, Diego Noja and Nicola Visciglia
2013, 18(5): 1155-1188 doi: 10.3934/dcdsb.2013.18.1155 +[Abstract](103) +[PDF](837.8KB)
We investigate the ground states of the one-dimensional nonlinear Schrödinger equation with a defect located at a fixed point. The nonlinearity is focusing and consists of a subcritical power. The notion of ground state can be defined in several (often non-equivalent) ways. We define a ground state as a minimizer of the energy functional among the functions endowed with the same mass. This is the physically meaningful definition in the main fields of application of NLS. In this context we prove an abstract theorem that revisits the concentration-compactness method and which is suitable to treat NLS with inhomogeneities. Then we apply it to three models, describing three different kinds of defect: delta potential, delta prime interaction, and dipole. In the three cases we explicitly compute ground states and we show their orbital stability.
Optimally swimming stokesian robots
François Alouges, Antonio DeSimone, Luca Heltai, Aline Lefebvre-Lepot and Benoît Merlet
2013, 18(5): 1189-1215 doi: 10.3934/dcdsb.2013.18.1189 +[Abstract](100) +[PDF](1025.1KB)
We study self-propelled stokesian robots composed of assemblies of balls, in dimensions 2 and 3, and prove that they are able to control their position and orientation. This is a result of controllability, and its proof relies on applying Chow's theorem in an analytic framework, similar to what has been done in [4] for an axisymmetric system swimming along the axis of symmetry. We generalize the analyticity result given in [4] to the situation where the swimmers can move either in a plane or in three-dimensional space, hence experiencing also rotations. We then focus our attention on energetically optimal strokes, which we are able to compute numerically. Some examples of computed optimal strokes are discussed in detail.
The existence of weak solutions to immiscible compressible two-phase flow in porous media: The case of fields with different rock-types
Brahim Amaziane, Leonid Pankratov and Andrey Piatnitski
2013, 18(5): 1217-1251 doi: 10.3934/dcdsb.2013.18.1217 +[Abstract](112) +[PDF](657.8KB)
We study a model describing immiscible, compressible two-phase flow, such as water-gas, through heterogeneous porous media taking into account capillary and gravity effects. We will consider a domain made up of several zones with different characteristics: porosity, absolute permeability, relative permeabilities and capillary pressure curves. This process can be formulated as a coupled system of partial differential equations which includes a nonlinear parabolic pressure equation and a nonlinear degenerate diffusion-convection saturation equation. Moreover the transmission conditions are nonlinear and the saturation is discontinuous at interfaces separating different media. There are two kinds of degeneracy in the studied system: the first one is the degeneracy of the capillary diffusion term in the saturation equation, and the second one appears in the evolution term of the pressure equation. Under some realistic assumptions on the data, we show the existence of weak solutions with the help of an appropriate regularization and a time discretization. We use suitable test functions to obtain a priori estimates. We prove a new compactness result in order to pass to the limit in nonlinear terms. This passage to the limit is nontrivial due to the degeneracy of the system.
Analysis and numerical approximations of equations of nonlinear poroelasticity
Yanzhao Cao, Song Chen and A. J. Meir
2013, 18(5): 1253-1273 doi: 10.3934/dcdsb.2013.18.1253 +[Abstract](113) +[PDF](418.4KB)
The equations of quasi-static poroelasticity which model flow through elastic porous media are considered. It is assumed that the hydraulic conductivity depends nonlinearly on the displacement (the dilatation) of the medium. The existence of a weak solution is proved using the modified Rothe's method. Numerical approximations of solutions by the finite element method are considered. Error estimates are obtained and numerical experiments are conducted to illustrate the theoretical results, and the efficiency and accuracy of the numerical method.
Finite-time quenching of competing species with constrained boundary evaporation
Shu Dai, Dong Li and Kun Zhao
2013, 18(5): 1275-1290 doi: 10.3934/dcdsb.2013.18.1275 +[Abstract](96) +[PDF](4611.3KB)
We consider a class of SKT type reaction-cross diffusion models with vanishing random diffusion coefficients. For homogeneous Dirichlet boundary conditions we prove non-existence of global-in-time non-trivial non-negative smooth solutions. Some numerical results are also presented, suggesting the possibility of finite-time extinction.
Traveling wave solutions for a diffusive sis epidemic model
Wei Ding, Wenzhang Huang and Siroj Kansakar
2013, 18(5): 1291-1304 doi: 10.3934/dcdsb.2013.18.1291 +[Abstract](153) +[PDF](645.6KB)
In this paper, we study the traveling wave solutions of an SIS reaction-diffusion epidemic model. The techniques of qualitative analysis has been developed which enable us to show the existence of traveling wave solutions connecting the disease-free equilibrium point and an endemic equilibrium point. In addition, we also find the precise value of the minimum speed that is significant to study the spreading speed of the population towards to the endemic steady state.
Analytical and numerical results on the positivity of steady state solutions of a thin film equation
Daniel Ginsberg and Gideon Simpson
2013, 18(5): 1305-1321 doi: 10.3934/dcdsb.2013.18.1305 +[Abstract](106) +[PDF](1100.4KB)
We consider an equation for a thin film of fluid on a rotating cylinder and present several new analytical and numerical results on steady state solutions. First, we provide an elementary proof that both weak and classical steady states must be strictly positive so long as the speed of rotation is nonzero. Next, we formulate an iterative spectral algorithm for computing these steady states. Finally, we explore a non-existence inequality for steady state solutions from the recent work of Chugunova, Pugh & Taranets.
The evolution of the orbit distance in the double averaged restricted 3-body problem with crossing singularities
Giovanni F. Gronchi and Chiara Tardioli
2013, 18(5): 1323-1344 doi: 10.3934/dcdsb.2013.18.1323 +[Abstract](87) +[PDF](784.1KB)
We study the long term evolution of the distance between two Keplerian confocal trajectories in the framework of the averaged restricted 3-body problem. The bodies may represent the Sun, a solar system planet and an asteroid. The secular evolution of the orbital elements of the asteroid is computed by averaging the equations of motion over the mean anomalies of the asteroid and the planet. When an orbit crossing with the planet occurs the averaged equations become singular. However, it is possible to define piecewise differentiable solutions by extending the averaged vector field beyond the singularity from both sides of the orbit crossing set [8],[5]. In this paper we improve the previous results, concerning in particular the singularity extraction technique, and show that the extended vector fields are Lipschitz-continuous. Moreover, we consider the distance between the Keplerian trajectories of the small body and of the planet. Apart from exceptional cases, we can select a sign for this distance so that it becomes an analytic map of the orbital elements near to crossing configurations [11]. We prove that the evolution of the `signed' distance along the averaged vector field is more regular than that of the elements in a neighborhood of crossing times. A comparison between averaged and non-averaged evolutions and an application of these results are shown using orbits of near-Earth asteroids.
On the Mountain-Pass algorithm for the quasi-linear Schrödinger equation
Christopher Grumiau, Marco Squassina and Christophe Troestler
2013, 18(5): 1345-1360 doi: 10.3934/dcdsb.2013.18.1345 +[Abstract](105) +[PDF](2090.6KB)
We discuss the application of the Mountain Pass Algorithm to the so-called quasi-linear Schrödinger equation, which is naturally associated with a class of nonsmooth functionals so that the classical algorithm cannot directly be used. A change of variable allows us to deal with the lack of regularity. We establish the convergence of a mountain pass algorithm in this setting. Some numerical experiments are also performed and lead to some conjectures.
Numerical study of blow-up in the Davey-Stewartson system
Christian Klein, Benson Muite and Kristelle Roidot
2013, 18(5): 1361-1387 doi: 10.3934/dcdsb.2013.18.1361 +[Abstract](112) +[PDF](6512.4KB)
Nonlinear dispersive partial differential equations such as the nonlinear Schrödinger equations can have solutions that blow up. We numerically study the long time behavior and potential blow-up of solutions to the focusing Davey-Stewartson II equation by analyzing perturbations of the lump and the Ozawa solutions. It is shown in this way that both are unstable to blow-up and dispersion, and that blow-up in the Ozawa solution is generic.
Sign-changing solutions of a quasilinear heat equation with a source term
Guirong Liu and Yuanwei Qi
2013, 18(5): 1389-1414 doi: 10.3934/dcdsb.2013.18.1389 +[Abstract](107) +[PDF](191.7KB)
The Cauchy problem of a heat equation with a source term $$ \psi_t=\Delta \left(|\psi|^{m-1}\psi\right)+|\psi|^{\gamma-1}\psi\ \ \mbox{in}\ \ (0, \infty)\times R^n $$ is considered, where $\gamma>m>1$. We are interested in global solutions with H$\ddot{o}$lder continuity which satisfy the equation in the distribution sense, and with a fixed number of sign changes at any given time $ t > 0$. Through detailed analysis of the self-similarity problem, we prove the existence of two type of such solutions, one with compact support and the other decays to zero as $ | x| \rightarrow \infty$ with an algebraic rate determined uniquely by $ n, m$ and $\gamma$. Our results extend previous study on positive self-similar solutions. Moreover, they demonstrate vital difference from the well-studied semi-linear case of $m = 1$.
Analysis of a scalar nonlocal peridynamic model with a sign changing kernel
Tadele Mengesha and Qiang Du
2013, 18(5): 1415-1437 doi: 10.3934/dcdsb.2013.18.1415 +[Abstract](131) +[PDF](436.5KB)
In this paper, a scalar peridynamic model is analyzed. The study extends earlier works in the literature on scalar nonlocal diffusion and nonlocal peridynamic models to include a sign changing kernel. We prove the well-posedness of both variational problems with nonlocal constraints and time-dependent equations with or without damping. The analysis is based on some nonlocal Poincaré type inequalities and compactness of the associated nonlocal operators. It also offers careful characterizations of the associated solution spaces such as compact embedding, separability and completeness along with regularity properties of solutions for different types of kernels.
Dynamics of a limit cycle oscillator with extended delay feedback
Ben Niu and Weihua Jiang
2013, 18(5): 1439-1458 doi: 10.3934/dcdsb.2013.18.1439 +[Abstract](133) +[PDF](942.6KB)
Investigating limit cycle oscillator with extended delay feedback is an efficient way to understand the dynamics of a global coupled ensemble or a large system with periodic oscillation. The stability and bifurcation of the arisen neutral equation are obtained. Stability switches and Hopf bifurcations appear when delay passes through a sequence of critical values. Global continuation of Hopf bifurcating periodic solutions and double--Hopf bifurcation are studied. With the help of the unfolding system near double--Hopf bifurcation obtained by using method of normal forms, quasiperiodic oscillations are found. The number of the coexisted periodic solutions is estimated. Finally, some numerical simulations are carried out.
A relaxation method for one dimensional traveling waves of singular and nonlocal equations
Weiran Sun and Min Tang
2013, 18(5): 1459-1491 doi: 10.3934/dcdsb.2013.18.1459 +[Abstract](106) +[PDF](1046.6KB)
Recent models motivated by biological phenomena lead to non-local PDEs or systems with singularities. It has been recently understood that these systems may have traveling wave solutions that are not physically relevant [19]. We present an original method that relies on the physical evolution to capture the ``stable" traveling waves. This method allows us to obtain the traveling wave profiles and their traveling speed simultaneously. It is easy to implement, and it applies to classical differential equations as well as nonlocal equations and systems with singularities. We also show the convergence of the scheme analytically for bistable reaction diffusion equations over the whole space $\mathbb{R}$.
$L^\infty$ estimation of the LDG method for 1-d singularly perturbed convection-diffusion problems
Huiqing Zhu and Runchang Lin
2013, 18(5): 1493-1505 doi: 10.3934/dcdsb.2013.18.1493 +[Abstract](115) +[PDF](387.2KB)
Pointwise error estimates of the local discontinuous Galerkin (LDG) method for a one-dimensional singularly perturbed problem are studied. Several uniform $L^\infty$ error bounds for the LDG approximation to the solution and its derivative are established on a Shishkin-type mesh. Numerical experiments are presented.
Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises
Xiaoling Zou, Dejun Fan and Ke Wang
2013, 18(5): 1507-1519 doi: 10.3934/dcdsb.2013.18.1507 +[Abstract](157) +[PDF](3259.6KB)
The existence of a stationary distribution and a stochastic Hopf bifurcation phenomenon for a noisy predator-prey system with Beddington-DeAngelis functional response are studied both theoretically and numerically. Considering the qualitative change of the shape of the stationary distribution, the stochastic Hopf bifurcation appears as a change from a peak-like distribution to a crater-like distribution. Results are obtained through the original niosy system rather than approximations based on stochastic averaging or scaling methods.

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