Multivalued Solutions Research Articles

Soliton interactions between multivalued localized waveguide channels within ferrites.

In this paper, we investigate both analytically and numerically the localized multivalued waveguide channels-the loop solitons-dynamics within a ferrite slab. In the starting point of the work, we solve in detail the initial value problem of the system while unveiling the existence of multivalued waveguide channels solutions. Paying particular interest to the nonlinear scattering among these excitations, we study extensively the different kinds of interacting features between these localized waves alongside the depiction of their energy densities. As a result, we find that the interactions can be attractive or repulsive depending strongly on the ratio of the amplitudes of the interacting structures. In the wake of these results, we address some physical implications, accordingly.

Effects of stiffness and load imperfection on the isolation performance of a high-static-low-dynamic-stiffness non-linear isolator under base displacement excitation

The dynamic characteristics of a high-static-low-dynamic-stiffness (HSLDS) non-linear isolator, built by combining an Euler beams formed negative stiffness corrector and a traditional linear isolator are investigated. The system imperfection caused by detuning of the stiffness and the load are considered. The frequency response curves (FRCs) for the non-linear isolator with load and stiffness imperfection under base displacement excitation are calculated by using Harmonic Balance Method (HBM). The stability is studied by the Floquet theory. The effect of the imperfection on the dynamic response as well as the displacement transmissibility of the non-linear isolator is explored and discussed. The results show that both the stiffness and load imperfection can affect the performance of the non-linear isolator significantly. The HSLDS isolator with load and stiffness imperfection for different base excitation amplitudes can demonstrate pure softening, softening-to-hardening and pure hardening characteristics with multi-valued solutions. The occurrence of jump phenomena is observed and explained by the stiffness variation. When the isolator is subjected to load imperfection, keeping a small positive stiffness, rather than making the minimum dynamic stiffness as zero, is preferred in order to obtain the best isolation performance. The performance of the non-linear isolator can outperform the linear one provided that the base displacement excitation is not too large. The non-linear system may undergo unbounded responses for very large base displacement excitation.

On the Dirichlet problemfor the Beltrami equation

We show that homeomorphic Wloc1,1 solutions of the Beltrami equations \(\overline \partial f = \mu \partial f\) satisfy certain modular inequalities. On this basis, we develop the theory of the boundary behavior of such solutions and prove a series of criteria for the existence of regular, pseudoregular and multi-valued solutions for the Dirichlet problem to the Beltrami equation in Jordan domains and finitely connected domains, respectively. These results have important applications to various problems of mathematical physics.

Multi-valued solutions to a class of parabolic Monge-Ampère equations

In this paper, we investigate the multi-valued solutions of a class of parabolic Monge-Ampere equation $-u_{t}\det(D^{2}u)=f$. Using the Perron method, we obtain the existence of finitely valued and infinitely valued solutions to the parabolic Monge-Ampere equations. We generalize the results of elliptic Monge-Ampere equations and Hessian equations.

Existence of solutions for quadratic integral inclusions

In this paper we study the existence of continuous solutions of a quadratic integral inclusion: we will look for solutions of that equation in a Banach space. The theory of quadratic integral inclusions has many useful ap- plications in mathematical physics, economics, biology, as well as in describing real world problems. The main tool used in our investiga- tions is a xed point result for the multivalued solution's map with acyclic values.

The onset of multi-valued solutions of a prescribed mean curvature equation with singular non-linearity

The existence and multiplicity of solutions to a quasilinear, elliptic partial differential equation with singular non-linearity is analysed. The partial differential equation is a recently derived variant of a canonical model used in the modelling of micro-electromechanical systems. It is observed that the bifurcation curve of solutions terminates at singledead-endpoint, beyond which no classical solutions exist. A necessary condition for the existence of solutions is developed, revealing that this dead-end point corresponds to a blow-up in the solution's gradient at a point internal to the domain. By employing a novel asymptotic analysis in terms of two small parameters, an accurate characterization of this dead-end point is obtained. An arc length parameterization of the solution curve can be employed to continue solutions beyond the dead-end point; however, all extra solutions are found to be multi-valued. This analysis therefore suggests that the dead-end is a bifurcation point associated with the onset of multi-valued solutions for the system.

Multidimensional nonlinear wave equations with multivalued solutions

We present the theory of breaking waves in nonlinear systems whose dynamics and spatial structure are described by multidimensional nonlinear hyperbolic wave equations. We obtain a general relation between systems of first-order quasilinear equations and nonlinear hyperbolic equations of higher orders, which, in particular, describe electromagnetic waves in a medium with nonlinear polarization of an arbitrary form. We use this approach to construct exact multivalued solutions of such equations and to study their spatial structure and dynamics. The results are generalized to a wide class of multidimensional equations such as d’Alembert equations, nonlinear Klein-Gordon equations, and nonlinear telegraph equations.

On the Dirichlet problem for the Beltrami equations in finitely connected domains

We establish a series of criteria for the existence of regular solutions of the Dirichlet problem for degenerate Beltrami equations in arbitrary Jordan domains. We also formulate the corresponding criteria for the existence of pseudoregular and multivalued solutions of the Dirichlet problem in the case of finitely connected domains.

Multivalued Geodesic Ray-Tracing for Computing Brain Connections Using Diffusion Tensor Imaging

Diffusion tensor imaging (DTI) is a magnetic resonance technique used to explore anatomical fibrous structures, like brain white matter. Fiber-tracking methods use the diffusion tensor (DT) field to reconstruct the corresponding fibrous structure. A group of fiber-tracking methods trace geodesics on a Riemannian manifold whose metric is defined as a function of the DT. These methods are more robust to noise than more commonly used methods where just the main eigenvector of the DT is considered. Until now, geodesic-based methods were not able to resolve all geodesics, since they solved the Eikonal equation, and therefore were not able to deal with multivalued solutions. Our algorithm computes multivalued solutions using an Euler–Lagrange form of the geodesic equations. The multivalued solutions become relevant in regions with sharp anisotropy and complex geometries, or when the first arrival time does not describe the geodesic close to the anatomical fibrous structure. In this paper, we compare our algorit...

Multi-valued solutions to Hessian quotient equations

Multi-valued solutions to Hessian quotient equations

Multi-valued solutions to fully nonlinear uniformly elliptic equations

In this paper, we discuss the existence and regularity of multi-valued viscosity solutions to fully nonlinear uniformly elliptic equations. We use the Perron method to prove the existence of bounded multi-valued viscosity solutions.

Dynamics of an AMB-rotor with time varying stiffness and mixed excitations

A rotor- active magnetic bearing (AMB) system with a periodically time-varying stiffness subjected to multi- external, -parametric and -tuned excitations is studied and solved. The method of multiple scales is applied to analyze the response of the two modes of the system near the simultaneous sub-harmonic, super-harmonic and combined resonance case. The stability of the steady state solution near this resonance case is determined and studied applying Lyapunov’s first method. Also, the system exhibits many typical nonlinear behaviors including multi-valued solutions, jump phenomenon, softening nonlinearities. The effects of the different parameters on the steady state solutions are investigated and discussed. Simulation results are achieved using MATLAB 7.0 program.

Conditional quadrature method of moments for kinetic equations

Kinetic equations arise in a wide variety of physical systems and efficient numerical methods are needed for their solution. Moment methods are an important class of approximate models derived from kinetic equations, but require closure to truncate the moment set. In quadrature-based moment methods (QBMM), closure is achieved by inverting a finite set of moments to reconstruct a point distribution from which all unclosed moments (e.g. spatial fluxes) can be related to the finite moment set. In this work, a novel moment-inversion algorithm, based on 1-D adaptive quadrature of conditional velocity moments, is introduced and shown to always yield realizable distribution functions (i.e. non-negative quadrature weights). This conditional quadrature method of moments (CQMOM) can be used to compute exact N-point quadratures for multi-valued solutions (also known as the multi-variate truncated moment problem), and provides optimal approximations of continuous distributions. In order to control numerical errors arising in volume averaging and spatial transport, an adaptive 1-D quadrature algorithm is formulated for use with CQMOM. The use of adaptive CQMOM in the context of QBMM for the solution of kinetic equations is illustrated by applying it to problems involving particle trajectory crossing (i.e. collision-less systems), elastic and inelastic particle–particle collisions, and external forces (i.e. fluid drag).

Construction of usc Solutions for a Multivalued Iterative Equation of Order n

Without Lipschitz condition, only the second order polynomial-like iterative equation was discussed for monotone usc multivalued solutions under the assumption that the coefficient of the first order iterate is large. In this paper without this assumption we give a general construction of infinitely many increasing usc multivalued solutions for multivalued iterative equation of order n.

Multi-valued solutions to Hessian equations

In this paper, we use the Perron method to prove the existence of bounded multi-valued viscosity solutions to Hessian equations and interior Lipschitz continuity of the multi-valued solutions.

Existence of multi-valued solutions with asymptotic behavior of Hessian equations

In this paper, we use the Perron method to prove the existence of multi-valued solutions with asymptotic behavior at infinity of Hessian equations.

Numerical continuation of high Reynolds number external flows

AbstractA numerical continuation method for the compressible Reynolds‐Averaged Navier–Stokes equation with the Spalart–Allmaras turbulence model is presented and applied to the flow around a 2D airfoil. Using continuation methods it is possible to study the steady flow states of a system as a parameter such as angle of attack is varied. This approach allows unstable solutions to be calculated, which are important for understanding the nonlinear dynamics of the system. Furthermore, this method can be used to find any multivalued solutions that exist at a single parameter value. The eigenvalues of the system are calculated using the Cayley transform to precondition the eigenvalue solver ARPACK. The eigenvalues are important as they show the stability of the solutions as well as accurately detect parameter values at which bifurcations take place. Copyright © 2010 John Wiley & Sons, Ltd.

Recent advances of neural network-based EM-CAD

In this article, we provide an overview of recent advances in computer-aided design techniques using neural networks for electromagnetic (EM) modeling and design applications. Summary of various recent neural network modeling techniques including passive component modeling, design and optimization using the models are discussed. Training data for the models are generated from EM simulations. The trained neural networks become fast and accurate models of EM structures. The models are then incorporated into various optimization methods and commercially available circuit simulators for fast design and optimization. We also provide an overview of recently developed neural network inverse modeling technique. Training a neural network inverse model directly may become difficult due to the nonuniqueness of the input–output relationship in the inverse model. Training data containing multivalued solutions are divided into groups according to derivative information. Multiple inverse submodels are built based on divided data groups and are then combined to form a complete model. Comparison between the conventional EM-based design approach and the inverse design approach has also been discussed. These computer-aided design techniques using neural models provide circuit level simulation speed with EM level accuracy avoiding the high computational cost of EM simulation. © 2010 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2010.

A direct method for solving the generalized sine-Gordon equation II

The generalized sine-Gordon (sG) equation utx = (1 + ν∂2x)sin u was derived as an integrable generalization of the sG equation. In a previous paper (Matsuno 2010 J. Phys. A: Math. Theor. 43 105204) which is referred to as I, we developed a systematic method for solving the generalized sG equation with ν = −1. Here, we address the equation with ν = 1. By solving the equation analytically, we find that the structure of solutions differs substantially from that of the former equation. In particular, we show that the equation exhibits kink and breather solutions and does not admit multi-valued solutions like loop solitons as obtained in I. We also demonstrate that the equation reduces to the short pulse and sG equations in appropriate scaling limits. The limiting forms of the multisoliton solutions are also presented. At last, we provide a recipe for deriving an infinite number of conservation laws by using a novel Bäcklund transformation connecting solutions of the sG and generalized sG equations.

A new algorithm for solving convex parametric quadratic programs based on graphical derivatives of solution mappings

In this paper we derive formulas for computing graphical derivatives of the (possibly multivalued) solution mapping for convex parametric quadratic programs. Parametric programming has recently received much attention in the control community, however most algorithms are based on the restrictive assumption that the so called critical regions of the solution form a polyhedral subdivision, i.e. the intersection of two critical regions is either empty or a face of both regions. Based on the theoretical results of this paper, we relax this assumption and show how we can efficiently compute all adjacent full dimensional critical regions along a facet of an already discovered critical region. Coupling the proposed approach with the graph traversal paradigm, we obtain very efficient algorithms for the solution of parametric convex quadratic programs.