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 problem in @cite_19 is formulated in terms of increasing traveling waves of negative speed; a change of variables is required to convert to our setting. In terms of , the main hypotheses in @cite_19 are @math ; @math . @math for @math and @math . The first and third conditions above are analogous to our (G1.2) and (G1.3). The second condition above is stronger; it should be thought of as analogous to and allows for the application of monotone iteration techniques. Under these conditions, there is some @math such that a decreasing front exists for all @math (Theorem 1.1 in @cite_19 , reformulated for our setting). In @cite_19 and the companion paper @cite_28 the authors also consider with a piecewise linear @math for which the monotonicity condition above fails, but which is simple enough to admit detailed analysis. In results analogous to ours, the authors describe condititions under which, as @math drops below a critical level, monotone fronts give way to non-monotone fronts that overshoot" and oscillate about their limit at @math .
@cite_28 @cite_19
2046114643 2071564573
In this paper, we study the structure of traveling wave solutions of Cellular Neural Networks of the advanced type. We show the existence of monotone traveling wave, oscillating wave and eventually periodic wave solutions by using shooting method and comparison principle. In addition, we obtain the existence of periodic wave train solutions. 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.