We prove the existence and uniqueness, for wave speeds sufficiently large, of monotone traveling wave solutions connecting stable to unstable spatial equilibria for a class of @math -dimensional lattice differential equations with unidirectional coupling. This class of lattice equations includes some spatial discretizations for hyperbolic conservation laws with a source term as well as a subclass of monotone systems. We obtain a variational characterization of the critical wave speed above which monotone traveling wave solutions are guaranteed to exist. We also discuss non-monotone waves, and the coexistence of monotone and non-monotone waves.
The existence results in @cite_26 use a monotone iteration technique. The same is true of @cite_19 (described further below), where a monotonicity condition on @math is also imposed. Recently such techniques have been extended, in the setting of lattice integro-difference equations, to the case where the nonlinearity is not necessarily monotone but satisfies conditions which are similar to our (G1). In particular @cite_17 and @cite_8 both obtain the existence of (not necessarily monotone) traveling waves as well as a variational characterization of the minimum wave speed guaranteeing monotone fronts. Although our setting and techniques differ, our results here can be regarded as complementing these latter works.
@cite_19 @cite_26 @cite_8 @cite_17
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Abstract This work proves the existence and multiplicity results of monotonic traveling wave solutions for some lattice differential equations by using the monotone iteration method. Our results include the model of cellular neural networks (CNN). In addition to the monotonic traveling wave solutions, non-monotonic and oscillating traveling wave solutions in the delay type of CNN are also obtained. Monotonicity in concurrent systems stipulates that, in any global state, extant system actions remain executable when new processes are added to the state. This concept is not only natural and common in multi-threaded software, but also useful: if every thread’s memory is finite, monotonicity often guarantees the decidability of safety property verification even when the number of running threads is unknown. In this paper, we show that the act of obtaining finite-data thread abstractions for model checking can be at odds with monotonicity: Predicate-abstracting certain widely used monotone software results in non-monotone multi-threaded Boolean programs — the monotonicity is lost in the abstraction. As a result, well-established sound and complete safety checking algorithms become inapplicable; in fact, safety checking turns out to be undecidable for the obtained class of unbounded-thread Boolean programs. We demonstrate how the abstract programs can be modified into monotone ones, without affecting safety properties of the non-monotone abstraction. This significantly improves earlier approaches of enforcing monotonicity via overapproximations. In some applications, it is reasonable to assume that geodesics (rays) have a consistent orientation so that the Helmholtz equation may be viewed as an evolution equation in one of the spatial directions. With such applications in mind, we propose a new Eulerian computational geometrical-optics method, dubbed the fast Huygens sweeping method, for computing Green functions of Helmholtz equations in inhomogeneous media in the high-frequency regime and in the presence of caustics. The first novelty of the new method is that the Huygens–Kirchhoff secondary source principle is used to integrate many locally valid asymptotic solutions to yield a globally valid asymptotic solution so that caustics associated with the usual geometrical-optics ansatz can be treated automatically. The second novelty is that a butterfly algorithm is adapted to carry out the matrix–vector products induced by the Huygens–Kirchhoff integration in O(NlogN) operations, where N is the total number of mesh points, and the proportionality constant depends on the desired accuracy and is independent of the frequency parameter. To reduce the storage of the resulting traveltime and amplitude tables, we compress each table into a linear combination of tensor-product based multivariate Chebyshev polynomials so that the information of each table is encoded into a small number of Chebyshev coefficients. The new method enjoys the following desired features: (1) it precomputes a set of local traveltime and amplitude tables; (2) it automatically takes care of caustics; (3) it constructs Green functions of the Helmholtz equation for arbitrary frequencies and for many point sources; (4) for a specified number of points per wavelength it constructs each Green function in nearly optimal complexity in terms of the total number of mesh points, where the prefactor of the complexity only depends on the specified accuracy and is independent of the frequency parameter. Both two-dimensional (2-D) and three-dimensional (3-D) numerical experiments are presented to demonstrate the performance and accuracy of the new method. In this paper we propose a satisfiability-based approach for enumerating all frequent, closed and maximal patterns with wildcards in a given sequence. In this context, since frequency is the most used criterion, we introduce a new polynomial inductive formulation of the cardinality constraint as a Boolean formula. A nogood-based formulation of the anti-monotonicity property is proposed and dynamically used for pruning. This declarative framework allows us to exploit the efficiency of modern SAT solvers and particularly their clause learning component. The experimental evaluation on real world data shows the feasibility of our proposed approach in practice.