# Registered Data

Contents

- 1 [CT027]
- 1.1 [00284] Stability analysis of solutions of nonlinear Schrodinger equation in presence of PT-symmetric potential
- 1.2 [00822] Dynamics and Diffusion Limit of a Two-Species Chemotaxis Model
- 1.3 [02006] Dynamics of localization patterns in some nonlocal evolution equations
- 1.4 [02184] Oscillatory Translational Instability of Localized Spot Patterns in the Schnakenberg Reaction-Diffusion System in Defected 3D Domains
- 1.5 [00956] Inner Structure of Attractors for a Nonlocal Chafee-Infante Problem

# [CT027]

## [00284] Stability analysis of solutions of nonlinear Schrodinger equation in presence of PT-symmetric potential

**Session Date & Time**: 1C (Aug.21, 13:20-15:00)**Type**: Contributed Talk**Abstract**: We extract exact stationary solutions of nonlinear Schrödinger equation in the presence of complex deformed supersymmetric potential (PT -symmetric Scarf potential) in terms of bright soliton and dark soliton. The corresponding spectrum of linear Schrödinger equation has been investigated and the PT broken and unbroken regions of linear Schrödinger equation have been delineated analytically. The stability of these solutions is corroborated by means of linear stability analysis and validated by direct numerical simulations.**Classification**:__35B35__,__35Q51__**Author(s)**:**Amiya Das**(University of Kalyani)

## [00822] Dynamics and Diffusion Limit of a Two-Species Chemotaxis Model

**Session Date & Time**: 1C (Aug.21, 13:20-15:00)**Type**: Contributed Talk**Abstract**: The traveling wave solutions to a two-species chemotaxis model with logarithmic sensitivity, which describes the initiation of angiogenesis using reinforced random walks theory and the chemotactic response of two-interacting species to a chemical stimulus are examined. The existence, asymptotic decay rates, stability, wave speed, and convergence, as the chemical diffusion coefficient goes to zero, of the traveling wave solutions are discussed.**Classification**:__35B35__,__35B40__,__35F50__,__35K40__**Author(s)**:**Annapoorani N**(Bharathiar University)

## [02006] Dynamics of localization patterns in some nonlocal evolution equations

**Session Date & Time**: 1C (Aug.21, 13:20-15:00)**Type**: Contributed Talk**Abstract**: Recently, studies have been proposed to simplify biological pattern formation problems by using nonlocal evolution equations to capture the self-organization caused by complex interactions with many factors. Especially, it has been reported that linear reaction-diffusion networks reduce to some nonlocal evolution equations reproducing patterns. Also, nonlocal effects are derived to reduce the structure of the network. In this talk, we report the influence of nonlocal effects on pattern dynamics for this reduced equation.**Classification**:__35B36__,__92C15__,__35K57__**Author(s)**:**Hiroshi Ishii**(Kyoto University)

## [02184] Oscillatory Translational Instability of Localized Spot Patterns in the Schnakenberg Reaction-Diffusion System in Defected 3D Domains

**Session Date & Time**: 1C (Aug.21, 13:20-15:00)**Type**: Contributed Talk**Abstract**: For a two-component reaction-diffusion system in a bounded $3D$ domain, we investigate oscillatory instabilities of $N$-spot equilibrium. An $N$-spot equilibrium consists of localized spots in which the activator concentration is exponentially small everywhere except localized regions. In the stability analysis, we consider the translation mode and obtain the eigenvalue $\lambda$ is $\mathcal{O}(\varepsilon^2)$, which is the same order as the spot dynamics, while $\tau $ is $\mathcal{O}(\varepsilon^{-3})$. As a result, the system which contains the behavior of $\lambda$ and $\tau \lambda$ falls into the $\mathcal{O}(\varepsilon^2)$ correction. We later find that stability of these solutions is governed by a $3N \times 3N$ nonlinear matrix eigenvalue problem. Entries of the $3N \times 3N$ matrix involves terms calculated from certain Green’s function that contains information about the domain’s geometry. In the nonlinear matrix eigenvalue system, the most unstable eigenvalue decides the oscillation frequency at onset while the corresponding eigenvector determines the mode of spot oscillations. Further, we demonstrate the impact of various types of localized heterogeneity on this instability. An example of localized domain defects that we consider is to analyze the effect of perturbing the system by removing a small ball in the domain, which therefore allows a leakage of the chemical species out of the domain. Perturbation techniques is employed to compute Green’s function of near-spherical and near-cubic domains to gain analytic insight into how domain geometry select the dominant mode of oscillation. We show full solutions of the $3$-$D$ Schnakenberg PDE to confirm our asymptotic results.**Classification**:__35B36__,__35B35__,__35B25__**Author(s)**:**Siwen Deng**(Macquarie University)- Justin Tzou (Macquarie University)

## [00956] Inner Structure of Attractors for a Nonlocal Chafee-Infante Problem

**Session Date & Time**: 1C (Aug.21, 13:20-15:00)**Type**: Contributed Talk**Abstract**: The structure of the global attractor for the multivalued semiflow generated by a nonlocal reaction-diffusion equation in which we cannot guarantee the uniqueness of the Cauchy problem is studied. The existence and properties of stationary points are analysed. Also, the study of the stability and connections between them are carried out, establishing that the semiflow is a dynamic gradient. As a consequence, the attractor consists of the stationary points and their heteroclinic connections.**Classification**:__35B40__,__35B41__,__35B51__,__35K55__,__35K57__**Author(s)**:**RUBEN CABALLERO**(UNIVERSIDAD MIGUEL HERNANDEZ DE ELCHE)