Influence of coupling on the dynamics of three delayed oscillators

The purpose of this study is to construct the asymptotics of the relaxation regimes of a system of differential equations with delay, which simulates three diffusion-coupled oscillators with nonlinear compactly supported delayed feedback under the assumption that the factor in front of the feedback function is large enough. Also, the purpose is to study the influence of the coupling between the oscillators on the nonlocal dynamics of the model. Methods. We construct the asymptotics of solutions of the considered model with initial conditions from a special set. From the asymptotics of the solutions, we obtain an operator of the translation along the trajectories that transforms the set of initial functions into a set of the same type. The main part of this operator is described by a finite-dimensional mapping. The study of its dynamics makes it possible to refine the asymptotics of the solutions of the original model and draw conclusions about its dynamics. Results. It follows from the form of the constructed mapping that for positive coupling parameters of the original model, starting from a certain moment of time, all three generators have the same main part of the asymptotics — the generators are “synchronized”. At negative values of the coupling parameter, both inhomogeneous relaxation cycles and irregular regimes are possible. The connection of these modes with the modes of the constructed finite-dimensional mapping is described. Conclusion. From the results of the work it follows that the dynamics of the model under consideration is fundamentally influenced by the value of the coupling parameter between the generators.

Modified scattering for the nonlinear nonlocal Schrödinger equation in one-dimensional case

We study the large time asymptotics of solutions to the Cauchy problem for the nonlinear nonlocal Schrödinger equation with critical nonlinearity $$\begin{aligned} \left\{ \begin{array}{l} i\partial _{t}\left( u-\partial _{x}^{2}u\right) +\partial _{x}^{2}u-a\partial _{x}^{4}u=\lambda \left| u\right| ^{2}u,\text { } t>0,{\ }x\in {\mathbb {R}}\mathbf {,} \\ u\left( 0,x\right) =u_{0}\left( x\right) ,{\ }x\in {\mathbb {R}}\mathbf {,} \end{array} \right. \end{aligned}$$ i ∂ t u - ∂ x 2 u + ∂ x 2 u - a ∂ x 4 u = λ u 2 u , t > 0 , x ∈ R , u 0 , x = u 0 x , x ∈ R , where $$a>\frac{1}{5},$$ a > 1 5 , $$\lambda \in {\mathbb {R}}$$ λ ∈ R . We continue to develop the factorization techniques which was started in papers Hayashi and Naumkin (Z Angew Math Phys 59(6):1002–1028, 2008) for Klein–Gordon, Hayashi and Naumkin (J Math Phys 56(9):093502, 2015) for a fourth-order Schrödinger, Hayashi and Kaikina (Math Methods Appl Sci 40(5):1573–1597, 2017) for a third-order Schrödinger to show the modified scattering of solutions to the equation. The crucial points of our approach presented here are based on the $${\mathbf {L}}^{2}$$ L 2 -boundedness of the pseudodifferential operators.