We consider a laser beam incident normally upon a waveguide with a Kerr nonlinearity. Although the laser is uniform in the plane of the waveguide, we show that the waveguide fields can be unstable to growth, forming coherent periodic structures. The initial, exponential time dependence is determined, and we find a steady-state solution. Finally, we note that the mechanism necessary for this ins...

where β is an appropriate positive constant, see [1, 2, 4, 6, 7]. The question of uniqueness of ground states for (1.1) when f(u) has such exponential behavior has not previously been treated; the purpose of this paper is to make a beginning on this problem. Appropriate assumptions on the operator A will be given in the next section. On the other hand, for the nonlinearity f we consider the spe...

In this paper we present two methods for constructing highly nonlinear injective s-boxes. Both of these methods, which are based on exponential sums, outperform previously proposed methods. In particular, we are able to obtain injective 8 32 s-boxes with nonlinearity equal to 80 and maximum XOR table entry of 2. We also re-evaluate the resistance of the CAST-like encryption algorithms construct...

Morrill and Valent́ın in the paper “Computational coverage of TLG: Nonlinearity” considered two extensions of the Lambek calculus with socalled “exponential” modalities. These calculi serve as a basis for describing fragments of natural language via categorial grammars. In this paper we show undecidability of derivability problems in these two calculi. On the other hand, restricted fragments of ...

The aim of this work is to perform a complete Lie symmetry classification of a generalized LaneEmden type system in two dimensions which models many physical phenomena in biological and physical sciences. The classical approach of group classification is employed for classification. We show that several cases arise in classifying the arbitrary parameters, the forms of which include amongst othe...

A nonlinear integro-differential equation of convolution type with order of nonlinearity more than one and a stable trivial solution is considered. The integral in this equation has an exponential kernel and polynomial integrand. The difference analogue of the equation considered is constructed in the form of a difference equation with continuous time and it is shown that this difference analog...

Dissipative wave equations with critical quintic nonlinearity and damping term involving the fractional Laplacian are considered. The additional regularity of energy solutions is established by constructing the new Lyapunov-type functional and based on this, the global well-posedness and dissipativity of the energy solutions as well as the existence of a smooth global and exponential attractors...

We show the existence of (H2 0 (Ω)×L2(Ω), H2 0 (Ω)×H2 0 (Ω))-global attractors for plate equations with critical nonlinearity when g ∈ H−2(Ω). Furthermore we prove that for each fixed T > 0, there is an (H2 0 (Ω) × L2(Ω), H2 0 (Ω) × H2 0 (Ω))T -exponential attractor for all g ∈ L2(Ω), which attracts any H2 0 (Ω)×L2(Ω)-bounded set under the stronger H2(Ω)×H2(Ω)-norm for all t ≥ T .

We study a modified version of the stochastic susceptible-infected-refractory-susceptible (SIRS) model by employing a nonlinear (exponential) reinforcement in the contagion rate and no diffusion. We run simulations for complete and random graphs as well as d-dimensional hypercubic lattices (for d=3,2,1). For weak nonlinearity, a continuous nonequilibrium phase transition between an absorbing an...