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.

A note on the p-adic Kozyrev wavelets basis

We present a basis of p-adic wavelets for Sobolev-type spaces consisting of eigenvectors of certain pseudodifferential operators. Our result extends a well-known result due to S. Kozyrev.

Theoretical and methodological approach to the study of technological processes

The article investigates some issues of complex systems analysis and synthesis that contain local, discrete sources of temperature fields. The emphasis of the author's research lies in the calculation and optimization of the parameters of the laser action on the embryo. The biotechnological process is described by the boundary value problem of a non-stationary, multidimensional differential equation of thermal conductivity which satisfies the boundary conditions of heat flux, beginning and end of the laser action process. The author proposes an algorithm for calculation and optimization of the control parameters of laser action on multilayer microbiological material. The object of the study is the embryo under the action of a laser beam for fission. As a first approximation, at the stage of calculation and optimization of the parameters of the biotechnological process, the embryo is considered as a homogeneous body. The values of thermophysical characteristics are calculated by the method of expert evaluation of the parameters of the emitters. The correctness of the boundary value problem of the process of laser action on the embryo is proved by the author using the method of pseudodifferential operators. Seeking the solution of the boundary value problem in the form of the series, the analytical function of the temperature field distribution is obtained using the Fourier method of the separated variables in the article. Using the method of indeterminate coefficients, the author found the temperature of the laser action on the embryo. Using the approximate gradient method of finding local extrema and the method of directed search of local extrema, it is possible to obtain rational values of optimized parameters of the biotechnological process. The author outlines possible approaches for optimization of technical parameters of laser emitters. In his opinion, the results of research can be considered fundamental for the calculation and optimization of the parameters of the laser action on the embryo, taking into account the multilayer structure of the latter.

On the geometry of $Diff(S^1)$-pseudodifferential operators based on renormalized traces.

In this article, we examine the geometry of a group of Fourier-integral operators, which is the central extension of $Diff(S^1)$ with a group of classical pseudo-differential operators of any order. Several subgroups are considered, and the corresponding groups with formal pseudodifferential operators are defined. We investigate the relationship of this group with the restricted general linear group $GL_{res}$, we define a right-invariant pseudo-Riemannian metric on it that extends the Hilbert-Schmidt Riemannian metric by the use of renormalized traces of pseudo-differential operators, and we describe classes of remarkable connections.

On L 2 -Boundedness of h -Pseudodifferential Operators

Let T a h be the h -pseudodifferential operators with symbol a . When a ∈ S ρ , 1 m and m = n ρ − 1 / 2 , it is well known that T a h is not always bounded in L 2 ℝ n . In this paper, under the condition a x , ξ ∈ L ∞ S ρ n ρ − 1 / 2 ω , we show that T a h is bounded on L 2 .

Subexponential decay and regularity estimates for eigenfunctions of localization operators

We consider time-frequency localization operators $$A_a^{\varphi _1,\varphi _2}$$ A a φ 1 , φ 2 with symbols a in the wide weighted modulation space $$ M^\infty _{w}({\mathbb {R}^{2d}})$$ M w ∞ ( R 2 d ) , and windows $$ \varphi _1, \varphi _2 $$ φ 1 , φ 2 in the Gelfand–Shilov space $$\mathcal {S}^{\left( 1\right) }(\mathbb {R}^d)$$ S 1 ( R d ) . If the weights under consideration are of ultra-rapid growth, we prove that the eigenfunctions of $$A_a^{\varphi _1,\varphi _2}$$ A a φ 1 , φ 2 have appropriate subexponential decay in phase space, i.e. that they belong to the Gelfand–Shilov space $$ \mathcal {S}^{(\gamma )} (\mathbb {R^{d}}) $$ S ( γ ) ( R d ) , where the parameter $$\gamma \ge 1 $$ γ ≥ 1 is related to the growth of the considered weight. An important role is played by $$\tau $$ τ -pseudodifferential operators $$Op_{\tau } (\sigma )$$ O p τ ( σ ) . In that direction we show convenient continuity properties of $$Op_{\tau } (\sigma )$$ O p τ ( σ ) when acting on weighted modulation spaces. Furthermore, we prove subexponential decay and regularity properties of the eigenfunctions of $$Op_{\tau } (\sigma )$$ O p τ ( σ ) when the symbol $$\sigma $$ σ belongs to a modulation space with appropriately chosen weight functions. As an auxiliary result we also prove new convolution relations for (quasi-)Banach weighted modulation spaces.