Traveling waves in the complex Ginzburg-Landau equation

Summary In this paper we consider a modulation (or amplitude) equation that appears in the nonlinear stability analysis of reversible or nearly reversible systems. This equation is the complex Ginzburg-Landau equation with coefficients with small imaginary parts. We regard this equation as a perturbation of the real Ginzburg-Landau equation and study the persistence of the properties of the stationary solutions of the real equation under this perturbation. First we show that it is necessary to consider a two-parameter family of traveling solutions with wave speedυ and (temporal) frequencyθ; these solutions are the natural continuations of the stationary solutions of the real equation. We show that there exists a two-parameter family of traveling quasiperiodic solutions that can be regarded as a direct continuation of the two-parameter family of spatially quasi-periodic solutions of the integrable stationary real Ginzburg-Landau equation. We explicitly determine a region in the (wave speedυ, frequencyθ)-parameter space in which the weakly complex Ginzburg-Landau equation has traveling quasi-periodic solutions. There are two different one-parameter families of heteroclinic solutions in the weakly complex case. One of them consists of slowly varying plane waves; the other is directly related to the analytical solutions due to Bekki & Nozaki [3]. These solutions correspond to traveling localized structures that connect two different periodic patterns. The connections correspond to a one-parameter family of heteroclinic cycles in an o.d.e. reduction. This family of cycles is obtained by determining the limit behaviour of the traveling quasi-periodic solutions as the period of the amplitude goes to ∞. Therefore, the heteroclinic cycles merge into the stationary homoclinic solution of the real Ginzburg-Landau equation in the limit in which the imaginary terms disappear.     

Qualche osservazione sui problemi totalmente reali

Riassunto Si introduce la nozione di “problema enumerativo complesso totalmente reale”: essa caratterizza problemi enumerativi le cui soluzioni, in un qualche aperto dello spazio dei parametri, sono tutte reali. Si dimostra una condizione necessaria e sufficiente affinchè un problema enumerativo complesso sia totalmente reale e vengono forniti esempi di tali problemi.

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Summary In this paper we give the definition of “totally real enumerative problem”. This notion characterises those complex enumerative problems which have, on some open set of the parameter space, all real solutions. We prove a necessary and sufficient condition, for an enumerative problem, to be a totally real one and we give some examples of problems of this kind.

     

Families of Auslander–Reiten components for simply connected differential graded algebras

Peter Jørgensen introduced the Auslander–Reiten quiver of a simply connected Poincaré duality space. He showed that its components are of the form ${{\mathbb {Z}}A_\infty}Peter J?rgensen introduced the Auslander–Reiten quiver of a simply connected Poincaré duality space. He showed that its components are of the form \mathbb ZA_￥{{\mathbb {Z}}A_\infty} and that the Auslander–Reiten quiver of a d-dimensional sphere consists of d − 1 such components. We show that this is essentially the only case where finitely many components appear. More precisely, we construct families of modules, where for each family, each module lies in a different component. Depending on the cohomology dimensions of the differential graded algebras which appear, this is either a discrete family or an n-parameter family for all n.     

On Sign Conditions Over Real Multivariate Polynomials

We present a new probabilistic algorithm to find a finite set of points intersecting the closure of each connected component of the realization of every sign condition over a family of real polynomials defining regular hypersurfaces that intersect transversally. This enables us to show a probabilistic procedure to list all feasible sign conditions over the polynomials. In addition, we extend these results to the case of closed sign conditions over an arbitrary family of real multivariate polynomials. The complexity bounds for these procedures improve the known ones.     

Superposition principle and composite solutions to coupled nonlinear Schrödinger equations

We show that the superposition principle applies to coupled nonlinear Schrödinger equations with cubic nonlinearity where exact solutions may be obtained as a linear combination of other exact solutions. This is possible due to the cancelation of cross terms in the nonlinear coupling. First, we show that a composite solution, which is a linear combination of the two components of a seed solution, is another solution to the same coupled nonlinear Schrödinger equation. Then, we show that a linear combination of two composite solutions is also a solution to the same equation. With emphasis on the case of Manakov system of two-coupled nonlinear Schrödinger equations, the superposition is shown to be equivalent to a rotation operator in a two-dimensional function space with components of the seed solution being its coordinates. Repeated application of the rotation operator, starting with a specific seed solution, generates a series of composite solutions, which may be represented by a generalized solution that defines a family of composite solutions. Applying the rotation operator to almost all known exact seed solutions of the Manakov system, we obtain for each seed solution the corresponding family of composite solutions. Composite solutions turn out, in general, to possess interesting features that do not exist in the seed solution. Using symmetry reductions, we show that the method applies also to systems of N-coupled nonlinear Schrödinger equations. Specific examples for the three-coupled nonlinear Schrödinger equation are given.     

On topologically distinct solutions of the Dirichlet problem for Yang-Mills connections

We prove that for generic Dirichlet boundary data there exist infinitely many topologically distinct solutions to the Dirichlet problem for -Yang-Mills equations over . These are absolute Yang-Mills minimizers in topologically distinct connected components of the space of connections considered. There is a special case for which only finitely many topologically distinct solutions can be found by our method. This corresponds to the simultaneous existence of self dual and anti-self dual solutions, for the given boundary data. If the boundary data is non-flat there exists always more than one solution. This paper generalizes to Yang-Mills fields an important result by Brezis and Coron, who show existence of more than one minimizing harmonic map for non-constant Dirichlet data in two dimensions. Received February 1, 1996 / Accepted March 15, 1996     

Dispersionless Limit of Hirota Equations in Some Problems of Complex Analysis

We study the integrable structure recently revealed in some classical problems in the theory of functions in one complex variable. Given a simply connected domain bounded by a simple analytic curve in the complex plane, we consider the conformal mapping problem, the Dirichlet boundary problem, and the 2D inverse potential problem associated with the domain. A remarkable family of real-valued functionals on the space of such domains is constructed. Regarded as a function of infinitely many variables, which are properly defined moments of the domain, any functional in the family gives a formal solution of the above problems. These functions satisfy an infinite set of dispersionless Hirota equations and are therefore tau-functions of an integrable hierarchy. The hierarchy is identified with the dispersionless limit of the 2D Toda chain. In addition to our previous studies, we show that within a more general definition of the moments, this connection pertains not to a particular solution of the Hirota equations but to the hierarchy itself.     

On generalized Nash equilibrium problems with linear coupling constraints and mixed-integer variables

ABSTRACT

We define and discuss different enumerative methods to compute solutions of generalized Nash equilibrium problems with linear coupling constraints and mixed-integer variables. We propose both branch-and-bound methods based on merit functions for the mixed-integer game, and branch-and-prune methods that exploit the concept of dominance to make effective cuts. We show that under mild assumptions the equilibrium set of the game is finite and we define an enumerative method to compute the whole of it. We show that our branch-and-prune method can be suitably modified in order to make a general equilibrium selection over the solution set of the mixed-integer game. We define an application in economics that can be modelled as a Nash game with linear coupling constraints and mixed-integer variables, and we adapt the branch-and-prune method to efficiently solve it.     

带约束条件集值优化问题近似Henig有效解集的连通性

研究了带约束条件集值优化问题近似Henig有效解集的连通性.在实局部凸Hausdorff空间中,讨论了可行域为弧连通紧的,目标函数为C-弧连通的条件下,带约束条件集值优化问题近似Henig有效解集的存在性和连通性.并给出了带约束条件集值优化问题近似Henig有效解集的连通性定理.     

Branch and win: OR tree search algorithms for solving combinatorial optimisation problems

Currently, most combinatorial optimisation problems have to be solved, if the optimum solution is sought, using general techniques to explore the space of feasible solutions and, more specifically, through exploratory enumerative procedures in trees and search graphs. We propose Branch and Win, a general formulation for understanding and synthesising the different tree search procedures that have been presented in the literature of operations research as well as in that of artificial intelligence. Several general ideas are also presented, whose application allows designing new hybrid search algorithms, in order to implement the procedure.     

The Generic Stability and Existence of Essentially Connected Components of Solutions for Nonlinear Complementarity Problems

The aim of this paper is to develop the general generic stability theory for nonlinear complementarity problems in the setting of infinite dimensional Banach spaces. We first show that each nonlinear complementarity problem can be approximated arbitrarily by a nonlinear complementarity problem which is stable in the sense that the small change of the objective function results in the small change of its solution set; and thus we say that almost all complementarity problems are stable from viewpoint of Baire category. Secondly, we show that each nonlinear complementarity problem has, at least, one connected component of its solutions which is stable, though in general its solution set may not have good behaviour (i.e., not stable). Our results show that if a complementarity problem has only one connected solution set, it is then always stable without the assumption that the functions are either Lipschitz or differentiable.     

Remarks on nonlocal boundary value problems at resonance

We show that it is important to allow the nonlinear term to change sign when discussing existence of a positive solution for multipoint, or more general nonlocal, boundary value problems in the resonant case. When the nonlinear term has a fixed sign we obtain simple necessary and sufficient conditions for the existence of positive solutions.     

A methodology to create robust job rotation schedules

This research proposes a methodology for developing robust job rotation schedules to reduce the likelihood of low back injury due to lifting. We consider settings that have uncertain task demands and different worker profiles in order to simulate real settings. We begin by considering deterministic versions of the problem and solve these using mathematical programming. Because mathematical programming cannot be readily applied to stochastic versions of the problem, heuristic solution methods are developed. The effectiveness of these methods is demonstrated by comparing the results with provably optimal solutions from the deterministic problems and with an enumerative approach that is applied to the stochastic version of the problem. Across the test problems, the proposed heuristics are effective at finding good job rotation solutions. The proposed methods could also be applied to solve other job rotation objectives such as maximizing productivity and reducing exposure to other work environmental factors such as excessive noise.     

Strategies of node selection in search procedures for solving combinatorial optimization problems: A survey and a general formalization

Currently, most combinatorial optimization problems have to be solved, if the optimum solution is sought, using general techniques to explore the space of feasible solutions and, more specifically, through exploratory enumerative procedures in trees and search graphs. The objective of this work is to propose a survey and a general formalization of the selection strategy of the next node to explore, a feature that is common to all these optimization procedures. This research has been partially supported by TAP98-0494 project     

Exact solutions of one-dimensional nonlinear shallow water equations over even and sloping bottoms

We establish an equivalence of two systems of equations of one-dimensional shallow water models describing the propagation of surface waves over even and sloping bottoms. For each of these systems, we obtain formulas for the general form of their nondegenerate solutions, which are expressible in terms of solutions of the Darboux equation. The invariant solutions of the Darboux equation that we find are simplest representatives of its essentially different exact solutions (those not related by invertible point transformations). They depend on 21 arbitrary real constants; after “proliferation” formulas derived by methods of group theory analysis are applied, they generate a 27-parameter family of essentially different exact solutions. Subsequently using the derived infinitesimal “proliferation” formulas for the solutions in this family generates a denumerable set of exact solutions, whose linear span constitutes an infinite-dimensional vector space of solutions of the Darboux equation. This vector space of solutions of the Darboux equation and the general formulas for nondegenerate solutions of systems of shallow water equations with even and sloping bottoms give an infinite set of their solutions. The “proliferation” formulas for these systems determine their additional nondegenerate solutions. We also find all degenerate solutions of these systems and thus construct a database of an infinite set of exact solutions of systems of equations of the one-dimensional nonlinear shallow water model with even and sloping bottoms.     

Subcritical nonlinear heat equation

We study asymptotic behavior in time of small solutions to nonlinear heat equations in subcritical case. We find a new family of self-similar solutions which change a sign. We show that solutions are stable in the neighborhood of these self-similar solutions.     

On regular minimax optimization

We consider functions of maximum type (max functions for short), subject to (in)equality constraints. The space dimension is finite, and the maximum is taken over a compact manifold with boundary. Effective local minimization algorithms based on Newton's method can be derived in the case where a local minimum is nondegenerate (in a two-level sense). In fact, nondegeneracy refers on the one hand to a local (implicit) reduction of the original max function to another one, where the maximum is taken over a finite set. On the other hand, it refers to strict complementarity and nondegeneracy of the underlying quadratic form with respect to the reduced stationary situation. As the main goal, we show that the set ofn-parameter families of functions, for which the stationary points of the corresponding max function are nondegenerate, constitutes an open and dense subset in the space of alln-parameter families (the topology used takes derivatives up to second order into account). An application to approximation problems of Chebyshev type is presented.     

Detection of Hopf points by counting sectors in the complex plane

Summary. Let ( real) be a family of real by matrices. A value of is called a Hopf value if has a conjugate pair of purely imaginary eigenvalues , . We describe a technique for detecting Hopf values based on the evolution of the Schur complement of in a bordered extension of where varies along the positive imaginary axis of the complex plane. We compare the efficiency of this method with more obvious methods such as the use of the QR algorithm and of the determinant function of as well as with recent work on the Cayley transform. In particular, we show the advantages of the Schur complement method in the case of large sparse matrices arising in dynamical problems by discretizing boundary value problems. The Hopf values of the Jacobian matrices are important in this setting because they are related to the Hopf bifurcation phenomenon where steady state solutions bifurcate into periodic solutions. Received September 15, 1994 / Revised version received July 7, 1995     

Null space distributions—A new approach to finite convolution equations with a Hankel kernel

In some earlier publications it has been shown that the solutions of the boundary integral equations for some mixed boundary value problems for the Helmholtz equation permit integral representations in terms of solutions of associated complicated singular algebraic ordinary differential equations. The solutions of these differential equations, however, are required to be known on some infinite interval on the real line, which is unsatisfactory from a practical point of view. In this paper, for the example of one specific boundary integral equation, the relevant solutions of the associated differential equation are expressed by integrals which contain only one unknown generalized function, the support of this generalized function is no longer unbounded but a compact subset of the real line. This generalized function is a distributional solution of the homogeneous boundary integral equation. By this null space distribution the boundary integral equation can be solved for arbitrary right-hand sides, this solution method can be considered of being analogous to the method of variation of parameters in the theory of ordinary differential equations. The nature of the singularities of the null space distribution is worked out and it is shown that the null space distribution itself can be expressed by solutions of the associated ordinary differential equation.