In this paper, the quasi-periodic wave solutions are systematically investigated for a supersymmetric, high-dimensional integrable model, namely the supersymmetric Boiti-Leon-Manna-Pempinelli (sBLMP) equation, which can describe quantum fluid fluctuations involving Bose-Fermi coupling. By employing Hirota’s bilinear method and the super Riemann-theta function, the quasi-periodic wave solutions of the sBLMP equation are constructed explicitly. The constraint equations for deriving the quasi-periodic waves are reformulated as an over-determined nonlinear system, in which the number of equations doubles. The quasi-periodic wave solvability problem is successfully transformed into a least squares problem and the global Levenberg-Marquardt (gLM) method is used to obtain its optimal solution, marking the first application of this method to high-dimensional supersymmetric equations. The asymptotic relationship between the quasi-periodic wave solutions and <italic>N</italic>-soliton solutions is rigorously established under the small-amplitude limit. In addition, dynamical properties such as periodicity, boundedness, influence band, bandwidth and the degradation mechanism are analyzed based on the characteristic line analysis method. The present approach provides an effective framework for studying quasi-periodic waves in higher-dimensional supersymmetric integrable systems and can be extended to other related models.

China

Qi Li

Zhonglong Zhao

Zhaohua Li

Numerical evaluations of quasi-periodic wave solutions of the supersymmetric Boiti-Leon-Manna-Pempinelli equation

We consider a cosmological model consisting of two scalar fields defined in the hyperbolic plane known as hyperbolic inflation. For the background space, we consider a homogeneous and isotropic spacetime with nonzero curvature. We study the asymptotic behaviour of solutions and search for attractors in the expanding regime. We prove that two hyperbolic inflationary stages are stable solutions that can solve the flatness problem and describe acceleration for both open and closed models, and additionally, we obtain a Milne-like attractor solution for the open model. We also investigate the contracting branch obtaining mirror solutions with the opposite dynamical behaviours.

South Africa

Chile

Andronikos Paliathanasis

Genly Leon

Hyperbolic inflationary model with nonzero curvature

This research focuses on scalar field cosmologies with a generalized harmonic potential. Our attention is centred on the anisotropic LRS Bianchi I and III metrics, Bianchi V metrics, and their isotropic limits. We provide a comprehensive overview of the first two metrics classes and offer new findings for Bianchi V metrics. We show that the Hubble parameter is a time-dependent perturbation parameter that controls the magnitude of the error between full-system and time-averaged solutions as it decreases, such that those complete and time-averaged systems have the same asymptotic behaviour. Therefore, oscillations entering the system can be controlled and smoothed out, simplifying the problem.

Alfredo Millano

Averaging generalized scalar field cosmologies IV: locally rotationally symmetric Bianchi V model

We use the recently modified Friedmann equations obtained from string T-duality effects that encode the zero-point length (Jusufi and Sheykhi (2023) [132]) to study the phase space analyses of a bouncing early universe. An important implication of such stringy effects is that they can alleviate the initial singularity since the Raychaudhuri equation is modified. We investigate if the Universe can undergo an accelerated expansion phase for a specific domain of the equation of state parameter. The stringy effects are encoded in the parameter Γ, which depends not only on the zero-point length but also on the state parameter ω. We construct two dynamical systems depending on whether <math><mo>−</mo><mn>1</mn><mo>&lt;</mo><mi>ω</mi><mo>≤</mo><mo>−</mo><mn>1</mn><mo>/</mo><mn>3</mn></math> and <math><mi>ω</mi><mo>≥</mo><mo>−</mo><mn>1</mn><mo>/</mo><mn>3</mn></math>, and we classify the equilibrium points of each system. Exact solutions and cosmological implications are discussed.

Alfredo D. Millano

North Macedonia

Kimet Jusufi

Phase space analysis of the bouncing universe with stringy effects

We investigate the inflation driven by a nonlinear electromagnetic field based on an NLED lagrangian density  $${\mathscr {L}}_{\text {nled}} = - {F} f \left( {F} \right) $$  <math> <mrow> <msub> <mi>L</mi> <mtext>nled</mtext> </msub> <mo>=</mo> <mo>-</mo> <mi>F</mi> <mi>f</mi> <mfenced> <mi>F</mi> </mfenced> </mrow> </math>  , where  $$f \left( {F}\right) $$  <math> <mrow> <mi>f</mi> <mfenced> <mi>F</mi> </mfenced> </mrow> </math>   is a general function depending on F. We first formulate an f-NLED cosmological model with a more general function  $$f \left( {F}\right) $$  <math> <mrow> <mi>f</mi> <mfenced> <mi>F</mi> </mfenced> </mrow> </math>   and show that all NLED models can be expressed in this framework; then, we investigate in detail two interesting examples of the function  $$f \left( {F}\right) $$  <math> <mrow> <mi>f</mi> <mfenced> <mi>F</mi> </mfenced> </mrow> </math>  . We present our phenomenological model based on a new Lagrangian for NLED. Solutions to the field equations with the physical properties of the cosmological parameters are obtained. We show that the early Universe had no Big-Bang singularity, which accelerated in the past. We also investigate the qualitative implications of NLED by studying the inflationary parameters, like the slow-roll parameters, spectral index  $$n_s$$  <math> <msub> <mi>n</mi> <mi>s</mi> </msub> </math>  , and tensor-to-scalar ratio r, and compare our results with observational data. Detailed phase-space analysis of our NLED cosmological model is performed with and without matter source. As a first approach, we consider the motion of a particle of unit mass in an effective potential. Our systems correspond to fast-slow systems for physical values of the electromagnetic field and the energy densities at the end of inflation. We analyze a complementary system using Hubble-normalized variables to investigate the cosmological evolution before the matter-dominated Universe.

United Arab Emirates

H. Benaoum

Türkiye

A. Övgün

Mexico

Italy

H. Quevedo

Inflation driven by non-linear electrodynamics

Sebastián Cuéllar

Esteban González

Samuel Lepe

Claudio Michea

Erratum to: Averaging generalized scalar field cosmologies II: locally rotationally symmetric Bianchi I and flat Friedmann–Lemaître–Robertson–Walker models

Erratum to: Averaging generalized scalar field cosmologies I: locally rotationally symmetric Bianchi III and open Friedmann–Lemaître–Robertson–Walker models

Erratum to: Averaging generalized scalar-field cosmologies III: Kantowski–Sachs and closed Friedmann–Lemaître–Robertson–Walker models

In this paper, we study a cosmological model inspired in the axionic matter with two canonical scalar fields  $$\phi _1$$  <math> <msub> <mi>ϕ</mi> <mn>1</mn> </msub> </math>   and  $$\phi _2$$  <math> <msub> <mi>ϕ</mi> <mn>2</mn> </msub> </math>   interacting through a term added to its potential. Introducing novel dynamical variables, and a dimensionless time variable, the resulting dynamical system is studied. The main difficulties arising in the standard dynamical systems approach, where expansion normalized dynamical variables are usually adopted, are due to the oscillations entering the nonlinear system through the Klein–Gordon (KG) equations. This motivates the analysis of the oscillations using methods from the theory of averaging nonlinear dynamical systems. We prove that time-dependent systems, and their corresponding time-averaged versions, have the same late-time dynamics. Then, we study the time-averaged system using standard techniques of dynamical systems. We present numerical simulations as evidence of such behavior.

India

Saikat Chakraborty

Bin Wang

Time-averaging axion-like interacting scalar fields models

Scalar-field cosmologies with a generalized harmonic potential and matter with energy density  $$\rho _m$$  <math> <msub> <mi>ρ</mi> <mi>m</mi> </msub> </math>  , pressure  $$p_m$$  <math> <msub> <mi>p</mi> <mi>m</mi> </msub> </math>  , and barotropic equation of state (EoS)  $$p_m=(\gamma -1)\rho _m, \; \gamma \in [0,2]$$  <math> <mrow> <msub> <mi>p</mi> <mi>m</mi> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mi>γ</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mi>ρ</mi> <mi>m</mi> </msub> <mo>,</mo> <mspace width="0.277778em"></mspace> <mi>γ</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo>]</mo> </mrow> </mrow> </math>   in Kantowski–Sachs (KS) and closed Friedmann–Lemaître–Robertson–Walker (FLRW) metrics are investigated. We use methods from non-linear dynamical systems theory and averaging theory considering a time-dependent perturbation function D. We define a regular dynamical system over a compact phase space, obtaining global results. That is, for KS metric the global late-time attractors of full and time-averaged systems are two anisotropic contracting solutions, which are non-flat locally rotationally symmetric (LRS) Kasner and Taub (flat LRS Kasner) for  $$0\le \gamma \le 2$$  <math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>γ</mi> <mo>≤</mo> <mn>2</mn> </mrow> </math>  , and flat FLRW matter-dominated universe if  $$0\le \gamma \le \frac{2}{3}$$  <math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>γ</mi> <mo>≤</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> </math>  . For closed FLRW metric late-time attractors of full and averaged systems are a flat matter-dominated FLRW universe for  $$0\le \gamma \le \frac{2}{3}$$  <math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>γ</mi> <mo>≤</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> </math>   as in KS and Einstein–de Sitter solution for  $$0\le \gamma &lt;1$$  <math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>γ</mi> <mo>&lt;</mo> <mn>1</mn> </mrow> </math>  . Therefore, a time-averaged system determines future asymptotics of the full system. Also, oscillations entering the system through Klein–Gordon (KG) equation can be controlled and smoothed out when D goes monotonically to zero, and incidentally for the whole D-range for KS and closed FLRW (if  $$0\le \gamma &lt; 1$$  <math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>γ</mi> <mo>&lt;</mo> <mn>1</mn> </mrow> </math>  ) too. However, for  $$\gamma \ge 1$$  <math> <mrow> <mi>γ</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math>   closed FLRW solutions of the full system depart from the solutions of the averaged system as D is large. Our results are supported by numerical simulations.

Averaging generalized scalar-field cosmologies III: Kantowski–Sachs and closed Friedmann–Lemaître–Robertson–Walker models