Separation of internal and interaction dynamics for NLS-described wave packets with different carrier waves

We give a detailed analysis of the interaction of two NLS-described wave packets with different carrier waves for a nonlinear wave equation. By separating the internal dynamics of each wave packet from the dynamics caused by the interaction we prove that there is almost no interaction of such wave packets. We also prove the validity of a formula for the envelope shift caused by the interaction of the wave packets.     

Relaxation of solitons in nonlinear Schrödinger equations with potential

In this paper we study dynamics of solitons in the generalized nonlinear Schrödinger equation (NLS) with an external potential in all dimensions except for 2. For a certain class of nonlinearities such an equation has solutions which are periodic in time and exponentially decaying in space, centered near different critical points of the potential. We call those solutions which are centered near the minima of the potential and which minimize energy restricted to L²-unit sphere, trapped solitons or just solitons. In this paper we prove, under certain conditions on the potentials and initial conditions, that trapped solitons are asymptotically stable. Moreover, if an initial condition is close to a trapped soliton then the solution looks like a moving soliton relaxing to its equilibrium position. The dynamical law of motion of the soliton (i.e. effective equations of motion for the soliton's center and momentum) is close to Newton's equation but with a dissipative term due to radiation of the energy to infinity.     

Asymptotic regularity and attractors of the reaction-diffusion equation with nonlinear boundary condition

We consider the dynamical behavior of the reaction-diffusion equation with nonlinear boundary condition for both autonomous and non-autonomous cases. For the autonomous case, under the assumption that the internal nonlinear term f is dissipative and the boundary nonlinear term g is non-dissipative, the asymptotic regularity of solutions is proved. For the non-autonomous case, we obtain the existence of a compact uniform attractor in H¹(Ω) with dissipative internal and boundary nonlinearities.     

Energy Scattering Theory for Electromagnetic NLS in Dimension Two

We study the well-posedness and long-time behavior of solution to both defocusing and focusing nonlinear Schr?dinger equations with scaling critical magnetic potentials in dimension two.In the defocusing case, and under the assumption that the initial data is radial, we prove interaction Morawetz-type inequalities and show the scattering holds in the energy space. The magnetic potential considered here is the Aharonov–Bohm potential which decays likely the Coulomb potential |x|~(-1).     

Nonlinear PDE based numerical methods for cell tracking in zebrafish embryogenesis

The paper presents numerical algorithms leading to an automated cell tracking and reconstruction of the cell lineage tree during the first hours of animal embryogenesis. We present results obtained for large-scale 3D+time two-photon laser scanning microscopy images of early stages of zebrafish (Danio rerio) embryo development. Our approach consists of three basic steps – the image filtering, the cell centers detection and the cell trajectories extraction yielding the lineage tree reconstruction. In all three steps we use nonlinear partial differential equations. For the filtering the geodesic mean curvature flow in level set formulation is used, for the cell center detection the motion of level sets by a constant speed regularized by mean curvature flow is used and the solution of the eikonal equation is essential for the cell trajectories extraction. The core of our new tracking method is an original approach to cell trajectories extraction based on finding a continuous centered paths inside the spatio-temporal tree structures representing cell movement and divisions. Such paths are found by using a suitably designed distance function from cell centers detected in all time steps of the 3D+time image sequence and by a backtracking in the steepest descent direction of a potential field based on this distance function. We also present efficient and naturally parallelizable discretizations of the aforementioned nonlinear PDEs and discuss properties and results of our new tracking method on artificial and real 4D data.     

Travelling wave and similarity solutions to nonlinear evolution type equations through inverse variational and symmetry methods

Shallow water equations are usually modelled by nonlinear KdV type equations of which various generalisations now exist. For example there are vector versions of the modified KdV equation and shallow water equations with nonlinear internal waves. We discuss the reduction and solutions of these and other large classes of such type of equations using inverse variational and symmetry methods.     

Nonlinear Internal Damping of Wave Equations with Variable Coefficients

For wave equations with variable coefficients on regions which are not necessarily smooth, we study the energy decay rate when a nonlinear damping is applied on a suitable subrigion.     

Directional fields algebraic non-linear solution equations for mobile robot planning

In this study, a novel approach to robot navigation/planning by using half-cell electrochemical potentials is presented. The half-cell electrode’s potential is modelled by the Nernst equation to yield automatic search/detection of pipeline flaws by using the direct current voltage gradient (DCVG) technique. We introduce a theory of spherical volumetric electric density in the soil to sustain our postulates for navigational potential fields. The Nernst potential is correlated with the distance to a pipe’s flaw by proposing a fitted theoretical-empirical nonlinear regression model. From this, volumetric derivatives are solved as gradient-based fields to control wheeled robot’s motion. A nonlinear system for trajectory planning is proposed, and analytically solved by an algebraic solution. This solution directly adjust robot’s speed kinematic values to lead it toward the flaw. The inverse/forward kinematic constraints are non-holonomic, and are recursively integrated into the general potential equation. Analytical modelling is reported, and a set of numerical simulations are presented to prove the feasibility of the proposed formulations.     

Nonlinear quasistatic problems of gradient type in inelastic deformations theory

In this paper we study nonlinear quasistatic problems from inelastic deformations theory. Only strictly monotone, gradient-type constitutive equations are considered. We prove existence for both coercive and non-coercive models, using energy estimates and Young measures. For non-coercive models we use the L² self-controlling property.     

Global existence of small classical solutions to nonlinear Schrödinger equations

We study the global Cauchy problem for nonlinear Schrödinger equations with cubic interactions of derivative type in space dimension n?3$n?3$. The global existence of small classical solutions is proved in the case where every real part of the first derivatives of the interaction with respect to first derivatives of wavefunction is derived by a potential function of quadratic interaction. The proof depends on the energy estimate involving the quadratic potential and on the endpoint Strichartz estimates.     

Lattice Approximations for Stochastic Quasi-Linear Parabolic Partial Differential Equations driven by Space-Time White Noise II

We approximate quasi-linear parabolic SPDEs substituting the derivatives with finite differences. We investigate the resulting implicit and explicit schemes. For the implicit scheme we estimate the rate of L^p convergence of the approximations and we also prove their almost sure convergence when the nonlinear terms are Lipschitz continuous. When the nonlinear terms are not Lipschitz continuous we obtain convergence in probability provided pathwise uniqueness for the equation holds. For the explicit scheme we get these results under an additional condition on the mesh sizes in time and space.     

Global existence of solutions to a coupled coagulation system

We consider a model equations describing the coagulation process of a gas on a surface. The problem is modeled by two coupled equations. The first one is a nonlinear transport equation with bilinear coagulation operator while the second one is a nonlinear ordinary differential equation. The velocity and the boundary condition of the transport equation depend on the supersaturation function satisfying the nonlinear ode. We first prove global existence and uniqueness of solution to the nonlinear transport equation then, we consider the coupled problem and prove existence in the large of solutions to the full coagulation system.     

Nonlinear responses from the interaction of two progressing waves at an interface

For scalar semilinear wave equations, we analyze the interaction of two (distorted) plane waves at an interface between media of different nonlinear properties. We show that new waves are generated from the nonlinear interactions, which might be responsible for the observed nonlinear effects in applications. Also, we show that the incident waves and the nonlinear responses determine the location of the interface and some information of the nonlinear properties of the media. In particular, for the case of a jump discontinuity at the interface, we can determine the magnitude of the jump.     

Coherent States for Systems of L 2-Supercritical Nonlinear Schrödinger Equations

We consider the propagation of wave packets for a nonlinear Schrödinger equation, with a matrix-valued potential, in the semi-classical limit. For a matrix-valued potential, Strichartz estimates are available under long range assumptions. Under these assumptions, for an initial coherent state polarized along an eigenvector, we prove that the wave function remains in the same eigenspace, at leading order, in a scaling such that nonlinear effects cannot be neglected. We also prove a nonlinear superposition principle for these nonlinear wave packets.     

Notes on a paper considering nonlinear equations

In abstract of the paper [A. Rafiq, A note on “A family of methods for solving nonlinear equations”, Appl. Math. Comput. 195 (2008) 819-821] we can find the following sentences. We cite: Ujevi? et al. introduced a family of methods for solving nonlinear equations. However the main Algorithm 1 put forward by Ujevi? et al. (p. 7) is wrong. This is the main aim of this note. We also point out some major bugs in the results of Ujevi? et al. - the end of the citation. Here it is shown that all of the mentioned assertions are not true. In other words, the Algorithm 1 is correct (up to an obvious misprint, which is not mentioned in the above paper) and there are no major bugs in the paper by Ujevi? et al. In fact, these observations, which will be given in this note, show that the main aim of the paper by Rafiq is wrong.     

Instability of standing waves for a class of nonlinear Schrödinger equations

This paper discusses a class of nonlinear Schrödinger equations with different power nonlinearities. We first establish the existence of standing wave associated with the ground states by variational calculus. Then by the potential well argument and the concavity method, we get a sharp condition for blowup and global existence to the solutions of the Cauchy problem and answer such a problem: how small are the initial data, the global solutions exist? At last we prove the instability of standing wave by combing those results.     

Convergence of Equilibria of Thin Elastic Plates – The Von Kármán Case

We study the behaviour of thin elastic bodies of fixed cross-section and of height h, with h → 0. We show that critical points of the energy functional of nonlinear three-dimensional elasticity converge to critical points of the von Kármán functional, provided that their energy per unit height is bounded by Ch ⁴ (and that the stored energy density function satisfies a technical growth condition). This extends recent convergence results for absolute minimizers.     

On global small amplitude solutions to systems of cubic nonlinear Klein-Gordon equations with different mass terms in one space dimension

We consider the Cauchy problem for systems of cubic nonlinear Klein-Gordon equations with different mass terms in one space dimension. We prove some result concerning the global existence of small amplitude solutions and their asymptotic behavior. As a consequence, we see that the condition for small data global existence is actually influenced by the difference of masses in some cases.     

A multi-iterate method to solve systems of nonlinear equations

We propose an extension of secant methods for nonlinear equations using a population of previous iterates. Contrarily to classical secant methods, where exact interpolation is used, we prefer a least squares approach to calibrate the linear model. We propose an explicit control of the numerical stability of the method.