Nonlinear Sci Research Articles

Investigation of solution behavior of Differential Equations by Sumudu methods of random complex Partial Differential Equations

In this study, solutions of random complex partial differential equations were found using the two-dimensional Sumudu transformation method(STM). The initial conditions of a deterministic equation or the non-homogeneous part of the equation are transformed into random variables to obtain a random complex partial differential equation. With the help of the properties of two-dimensional Sumudu and inverse Sumudu transformation, an approximate analytical solution of a complex partial differential equation with random constant coefficients was obtained by selecting a random variable with an initial condition of Normal and Gamma distribution. The probability characteristics of the resulting solutions, such as expected value and variance, were obtained and graphically shown with the help of the Maple package program.

Dual CAR-T cell therapy for glioblastoma: strategies to cure tumour diseases based on a mathematical model

AbstractThe CAR-T cell immunotherapy entails the genetic reprogramming of T-lymphocytes, which then engage with cancer cells, triggering an anti-tumour immune response. While this treatment has gained approval for hematological cancers, addressing solid tumours presents new obstacles. Challenges include the heterogeneity of antigen expression within solid tumours, encompassing antigen-positive non-tumoural cells, the presence of immune inhibitory molecules, and the difficulty of CAR-T cell trafficking within the tumour microenvironment. In this article, we analytically study a generalisation of a mathematical model proposed by León-Triana et al. (Cancers 13(4):703, 2021a. https://doi.org/10.3390/cancers13040703, Commun Nonlinear Sci Numer Simul 94:105570). This model focuses on the dynamics of glioblastoma, the most aggressive brain tumour, and its response to CAR-T cell treatment. We study the basic properties of the model, the dynamics of the solutions of the model when the treatment is not sustained during the time, and finally we study analytically the model when the therapy is constant, periodic and/or impulsive. We derive sufficient conditions for global stability of tumour-free equilibrium, as well as necessary and sufficient conditions for local stability of the equilibrium obtaining conditions for an effective treatment. Finally, we perform different numerical simulations to find the strategies to keep the tumour under control. The obtained results are based on a combination of different analytical techniques in differential equations, dynamical systems and numerical simulations.

Pointwise estimates on the fundamental solution to nonlocal diffusion equation

In this article we study the pointwise behavior of the fundamental solution S ( t , x ) to the nonlocal diffusion model u t = D ( J ∗ u − u ) on R , S ( t , x ) can be written as e − Dt δ ( x ) + S 0 ( t , x ) , where S 0 ( t , x ) is called the regularizing part. Under some conditions on J and its Fourier transform J ^ , we obtain some new pointwise estimates on S 0 ( t , x ) . Then we apply our results to study the spatial dynamics of the nonlocal diffusion population model in a shifting environment considered in Li et al. [Spatial dynamics of a nonlocal dispersal population model in a shifting environment. J Nonlinear Sci. 2018;28:1189–1219]. In our study the convolution kernel J can be assumed to be more general, i.e. not necessarily of compact support, also the proof is greatly simplified. It is our believe that these pointwise estimates possess their own interest in further studying nonlocal diffusion models.

Spreading speed of an impulsive reaction–diffusion model with non-monotone birth function and age structure

In this paper, we study the spreading speed and traveling wave of an impulsive reaction–diffusion model with non-monotone birth function and age structure, which models the evolution of annually synchronized emergence of adult population with maturation. Our result is an extension to that recently established in Bai, Lou and Zhao (J. Nonlinear Sci. 2022). Numerical simulations are also performed to illustrate our result.

Energy scaling laws for microstructures: from helimagnets to martensites

We consider scalar-valued variational models for pattern formation in helimagnetic compounds and in shape memory alloys. Precisely, we consider a non-convex multi-well bulk energy term on the unit square, which favors four gradients (±α,±β)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\\pm }\\,\\alpha ,{\\pm }\\,\\beta )$$\\end{document}, regularized by a singular perturbation in terms of the total variation of the second derivative. We derive scaling laws for the minimal energy in the case of incompatible affine boundary conditions in terms of the singular perturbation parameter as well as the ratio α/β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha /\\beta $$\\end{document} and the incompatibility of the boundary condition. We discuss how well-studied models for martensitic microstructures in shape-memory alloys arise as a limiting case, and relations between the different models in terms of scaling laws. In particular, we show that scaling regimes arise in which an interpolation between the rather different branching-type constructions in the spirit of Kohn and Müller (Commun Pure Appl Math 47(4):405–435, 1994) and Ginster and Zwicknagl (J Nonlinear Sci 33:20, 2023), respectively, occurs. A particular technical difficulty in the lower bounds arises from the fact that the energy scalings involve various logarithmic terms that we capture in matching upper and lower scaling bounds.

Accretion–ablation mechanics

In this paper, we formulate a geometric nonlinear theory of the mechanics of accreting–ablating bodies. This is a generalization of the theory of accretion mechanics of Sozio & Yavari (Sozio & Yavari 2019 J. Nonlinear Sci. 29 , 1813–1863 (doi:10.1007/s00332-019-09531-w)). More specifically, we are interested in large deformation analysis of bodies that undergo a continuous and simultaneous accretion and ablation on their boundaries while under external loads. In this formulation, the natural configuration of an accreting–ablating body is a time-dependent Riemannian 3 -manifold with a metric that is an unknown a priori and is determined after solving the accretion–ablation initial-boundary-value problem. In addition to the time of attachment map, we introduce a time of detachment map that along with the time of attachment map, and the accretion and ablation velocities, describes the time-dependent reference configuration of the body. The kinematics, material manifold, material metric, constitutive equations and the balance laws are discussed in detail. As a concrete example and application of the geometric theory, we analyse a thick hollow circular cylinder made of an arbitrary incompressible isotropic material that is under a finite time-dependent extension while undergoing continuous ablation on its inner cylinder boundary and accretion on its outer cylinder boundary. The state of deformation and stress during the accretion–ablation process, and the residual stretch and stress after the completion of the accretion–ablation process, are computed. This article is part of the theme issue ‘Foundational issues, analysis and geometry in continuum mechanics’.

Rigorous computation in dynamics based on topological methods for multivector fields

Motivated by the theoretical results of Mrozek et al. (Commun Nonlinear Sci Numer Simul 108:106–226, 2022) we present an algorithmic construction of a transversal cellular decomposition for a planar ODE. We then use the associated combinatorial multivector field to algorithmically detect the existence of an isolated invariant set with the Conley index of a periodic orbit and admitting a combinatorial Poincaré section. This construction combined with the theoretical results of Mrozek et al. (2022) leads to a method for automatized computer assisted proofs of the existence of periodic solutions in ODE’s.

Theoretical analysis and numerical approximation for the stochastic thermal quasi-geostrophic model

This paper investigates the mathematical properties of a stochastic version of the balanced 2D thermal quasigeostrophic (TQG) model of potential vorticity dynamics. This stochastic TQG model is intended as a basis for parametrization of the dynamical creation of unresolved degrees of freedom in computational simulations of upper ocean dynamics when horizontal buoyancy gradients and bathymetry affect the dynamics, particularly at the submesoscale (250 m–10 km). Specifically, we have chosen the Stochastic Advection by Lie Transport (SALT) algorithm introduced in [D. D. Holm, Variational principles for stochastic fluid dynamics, Proc. Roy. Soc. A: Math. Phys. Eng. Sci. 471 (2015) 20140963, http://dx.doi.org/10.1098/rspa.2014.0963 ] and applied in [C. Cotter, D. Crisan, D. Holm, W. Pan and I. Shevchenko, Modelling uncertainty using stochastic transport noise in a 2-layer quasi-geostrophic model, Found. Data Sci. 2 (2020) 173, https://doi.org/10.3934/fods.2020010 ; Numerically modeling stochastic lie transport in fluid dynamics, SIAM Multiscale Model. Simul. 17 (2019) 192–232, https://doi.org/10.1137/18M1167929 ] as our modeling approach. The SALT approach preserves the Kelvin circulation theorem and an infinite family of integral conservation laws for TQG. The goal of the SALT algorithm is to quantify the uncertainty in the process of up-scaling, or coarse-graining of either observed or synthetic data at fine scales, for use in computational simulations at coarser scales. The present work provides a rigorous mathematical analysis of the solution properties of the thermal quasigeostrophic (TQG) equations with SALT [D. D. Holm and E. Luesink, Stochastic wave-current interaction in thermal shallow water dynamics, J. Nonlinear Sci. 31 (2021), https://doi.org/10.1007/s00332-021-09682-9 ; D. D. Holm, E. Luesink and W. Pan, Stochastic mesoscale circulation dynamics in the thermal ocean, Phys. Fluids 33 (2021) 046603, https://doi.org/10.1063/5.0040026 ].

Existence and multiplicity of solutions for a quasilinear system with locally superlinear condition

Abstract We investigate the existence and multiplicity of weak solutions for a nonlinear Kirchhoff type quasilinear elliptic system on the whole space R N {{\mathbb{R}}}^{N} . We assume that the nonlinear term satisfies the locally super- ( m 1 , m 2 ) \left({m}_{1},{m}_{2}) condition, that is, lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ {\mathrm{lim}}_{| \left(u,v)| \to +\infty }\frac{F\left(x,u,v)}{| u{| }^{{m}_{1}}+| v{| }^{{m}_{2}}}=+\infty for a.e. x ∈ G x\in G , where G G is a domain in R N {{\mathbb{R}}}^{N} , which is weaker than the well-known Ambrosseti-Rabinowitz condition and the naturally global restriction, lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ {\mathrm{lim}}_{| \left(u,v)| \to +\infty }\frac{F\left(x,u,v)}{| u{| }^{{m}_{1}}+| v{| }^{{m}_{2}}}=+\infty for a.e. x ∈ R N x\in {{\mathbb{R}}}^{N} . We obtain that the system has at least one weak solution by using the classical mountain pass theorem. To a certain extent, our theorems extend the results of Tang et al. [Nontrivial solutions for Schrodinger equation with local super-quadratic conditions, J. Dynam. Differ. Equ. 31 (2019), no. 1, 369–383]. Moreover, under the aforementioned naturally global restriction, we obtain that the system has infinitely many weak solutions of high energy by using the symmetric mountain pass theorem, which is different from those results of Wang et al. [Existence and multiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces, J. Nonlinear Sci. Appl. 10 (2017), no. 7, 3792–3814] even if we consider the system on the bounded domain with Dirichlet boundary condition.

The Buckling Load of Cylindrical Shells Under Axial Compression Depends on the Cross-Sectional Curvature

It is known that the famous theoretical formula by Koiter for the critical buckling load of circular cylindrical shells under axial compression does not coincide with the experimental data. Namely, while Koiter’s formula predicts linear dependence of the buckling load $$\lambda (h)$$ of the shell thickness h ( $$h>0$$ is a small parameter), one observes the dependence $$\lambda (h)\sim h^{3/2}$$ in experiments; i.e., the shell buckles at much smaller loads for small thickness. This theoretical prediction failure is believed to be caused by the so-called sensitivity to imperfections phenomenon (both, shape and load). Grabovsky and the first author have rigorously proven in (J Nonlinear Sci 26(1):83–119, 2016) , that in the problem of circular cylindrical shells buckling under axial compression, a small load twist leads to the buckling load scaling $$\lambda (h)\sim h^{5/4},$$ while shape imperfections are likely to result in the scaling $$\lambda (h)\sim h^{3/2}.$$ In this work, we prove that in fact the buckling load $$\lambda (h)$$ of cylindrical (not necessarily circular) shells under vertical compression depends on the curvature of the cross section curve. When the cross section is a convex curve with uniformly positive curvature, then $$\lambda (h)\sim h,$$ and when the cross-sectional curve has positive curvature except at finitely many points, then $$C_1h^{8/5}\le \lambda (h)\le C_2h^{3/2}$$ for h small thickness $$h>0.$$

Global solutions to the damped MHD system

AbstractThe global existence of solutions to a damped MHD system in Besov spaces is obtained. This research is motivated by the open question [Wu et al., J. Nonlinear Sci. 25 (2015), 157–192, Remark 5.1] on the possibility of extension to the existence of global small solutions from a potential Banach space to a Besov space. The present result positively confirms the open question. What is more, the initial data are not required to be uniformly small.

A Hint on the Localization of the Buckling Deformation at Vanishing Curvature Points on Thin Elliptic Shells

The general theory of slender structure buckling by Grabovsky and Truskinovsky (Cont. Mech. Thermodyn. 19(3–4):211-243, 2007), (later extended in J. Nonlinear Sci. 26(1):83–119, 2016 by Grabovsky and the author), predicts that the critical buckling load of a thin shell under dead loading is closely related to the Korn’s constant (in Korn’s first inequality) of the shell under the Dirichlet boundary conditions resulting from the loading program. It is known that under zero Dirichlet boundary conditions on the thin part of the boundary of positive, negative, and zero (one principal curvature vanishing, and one apart from zero) Gaussian curvature shells, the optimal Korn constant in Korn’s first inequality scales like the thickness to the power of −1, \(-4/3\), and \(-3/2\) respectively. In this work we analyse the scaling of the optimal constant in Korn’s first inequality for elliptic shells that contain a finite number of points where both principal curvatures vanish. We prove that the presence of at least one such point on the shell leads to the scaling drop from the thickness to the power of −1 to the thickness to the power of \(-3/2\). To our best knowledge, this is the first result in the direction for constant-sign curvature shells, that do not contain a developable region. In addition, under the assumption that a suitable trivial branch exists, we prove that in fact the buckling deformation of such shells under dead loading, should be localized at the vanishing curvature points, as the shell thickness \(h\) goes to zero.

Group analysis, invariance results, exact solutions and conservation laws of the perturbed fractional Boussinesq equation

The scope of this work is based on symmetry analysis of perturbed fractional Boussinesq equation. For beginning the group formalism, invariance properties and conservation laws of the nonlinear perturbed fractional Boussinesq equation have been explored generally. This method was first described by Lukashchuk [Commun. Nonlinear Sci. Numer. Simul. 68 (2019) 147–159]. The key subject is that when the order of fractional derivative in a fractional differential equation (FDE) is nearly integers, we can approximate it to a perturbed integer-order differential equation with a small perturbation parameter. For obtaining the results, perturbed and unperturbed symmetries are computed. Then, the methodology of reduction is applied for finding some new solutions by the symmetry operators of the equation. These solutions are obtained by the similarity transformations of the symmetries. Another exact solution will be found with constructing one-dimensional optimal system of the symmetries. Finally, the meaning of nonlinear self-adjointness concept is attended in order to find conservation laws with informal Lagrangians.

On the mechanism of multiscroll chaos generation in coupled non-oscillatory rayleigh-duffing oscillators

We investigate the dynamics of a pair of coupled non oscillatory Rayleigh-Duffing oscillators (RDOs here after). The RDO serves as a model for a class of nonlinear oscillators including microwave Gunn oscillators [Guin et al Comm. in Nonlinear Sci. Numerical Simulat, 2017]. Here, the coupling between the two oscillators is obtained by superimposing to each one’s amplitude a perturbation proportional to the other one. We demonstrate that the coupling induces more equilibrium points and results in extremely complex nonlinear behaviors including multistability (up to six coexisting attractors), multiple Hopf bifurcations, multi-scroll chaos, and coexisting bifurcation trees. These phenomena are studied in detail by utilizing one-parametric bifurcation plots, bi-parametric Lyapunov exponent diagrams, phase space trajectory plots, and basins of attraction as well. Experimental results captured from an Arduino microcontroller-based realization of the coupled RDOs are included to support the observations made through numerical analysis. We would like to point out that the coupling scheme followed in this work may stimulate the research on multiscroll chaos generation based on coupled nonlinear oscillators.

Eulerian vs Lagrangian Irreversibility in an Experimental Turbulent Swirling Flow.

In a turbulent fluid, the time-reversal symmetry is explicitly broken by viscosity, and spontaneously broken in the inviscid limit. Recently, Drivas [J. Nonlinear Sci. 29, 65 (2019).JNSCEK0938-897410.1007/s00332-018-9476-8] proved the equivalence of two different local indicators of time irreversibility: (i)an Eulerian one, based on regularity properties of the velocity field [Duchon and Robert, Nonlinearity 13, 249 (2000).NONLE50951-771510.1088/0951-7715/13/1/312]; (ii)a Lagrangian one, based on symmetry properties of the trajectories under time reversal [Jucha et al., Phys. Rev. Lett. 113, 054501 (2014).PRLTAO0031-900710.1103/PhysRevLett.113.054501]. We test this equivalence in a turbulent Von Kármán experiment at a resolution of the order of the Kolmogorov scale using a high resolution 4D-PTV technique. We use the equivalence to perform the first joined Eulerian-Lagrangian exploration of the dynamics leading to time irreversibility, and find that it is linked with vortex interaction, suggesting a link between irreversibility and singularity.

Determining Map, Data Assimilation and an Observable Regularity Criterion for the Three-Dimensional Boussinesq System

In this paper, we consider the analysis of a nudging based algorithm for data assimilation for the three-dimensional Boussinesq system, which we call the AOT system. A rigorous analysis of this algorithm for well-posed dissipative partial differential equations was first provided by Azouani, Olson and Titi (J Nonlinear Sci 24:277–304, 2014); thus justifying our terminology for the associated nudging system. We provide a sufficient condition, based solely on the observed velocity data obtained from a Leray-Hopf weak solution, for the global well-posedness, regularity and most crucially, the asymptotic tracking property of solutions of the associated (three-dimensional) nudging system. It is to be noted that neither regularity nor any knowledge of a uniform \({\mathbb {H}}^1\)-norm bound is a priori assumed on the solution of the original three-dimensional Boussinesq system from which the observations are obtained. As a corollary of our result, we obtain a novel observable regularity criterion based on finitely many observational data. Our condition also guarantees the construction of the Lipschitz continuous determining map, which is known to play a crucial role in the construction of the so-called determining form and in statistical data assimilation.

Global stability for a SEIQR worm propagation model in mobile internet

Abstract Recently, propagation models of worms in the mobile environment are drawing extensive attention, particularly in the Wi-Fi scenario. Considering that worm-free equilibrium is exponential convergent means that the propagation time and control time of worms are much shorter than for other asymptotic convergence. Besides, the global asymptotic stability of the endemic equilibrium is more important than the local asymptotic stability, which reflects the more global qualitative behavior of the worm propagation. In this paper, we discuss the global dynamics of SEIQR worm propagation model in mobile internet proposed by Xiao et al. [X. Xiao, P. Fu, C. Dou, Q. Li, G. Hu, and S. Xia, “Design and analysis of SEIQR worm propagation model in mobile internet,” Commun. Nonlinear Sci. Numer. Simulat., vol. 43, pp. 341–350, 2017] to improve and complement the related results. Through a series of mathematical derivations, sufficient conditions are derived to ensure the global exponentially stability of worm-free equilibrium, and the exponential convergent rate can be unveiled. Then, by using the classical geometric approach, it is shown that the endemic equilibrium is globally asymptotically stable and the system is persistent when R 0 > 1. Moreover, numerical simulations are given to demonstrate our theoretical results.

New fixed point theorems for θ - ϕ -contraction on b -metric spaces

In this paper, we define \(\theta\)-\(\phi\)-contraction on a \(b\)-metric space into itself by extending \(\theta\)-\(\phi \)-contraction introduced by Zheng {et al.} [D. W. Zheng, Z. Y. Cai, P. Wang, J. Nonlinear Sci. Appl., \(\bf 10\) (2017), 2662--2670] in metric space and also, we prove \(\theta \)-type theorem in the setting of \(b\)-metric spaces as well as \(\theta\)-\(\phi \)-type theorem in the framework of \(b\)-rectangular metric spaces. Moreover, we give some applications to nonlinear integral equations. We also give illustrative examples to exhibit the utility of our results.

Ill-Posedness Issues on (abcd)-Boussinesq System

In this paper, we consider the Cauchy problem for (abcd)-Boussinesq system posed on one- and two-dimensional Euclidean spaces. This model, initially introduced by Bona et al. (J Nonlinear Sci 12:283–318, 2002, Nonlinearity 17:925–952, 2004), describes a small-amplitude waves on the surface of an inviscid fluid, and is derived as a first order approximation of incompressible, irrotational Euler equations. We mainly establish the ill-posedness of the system under various parameter regimes, which generalize the result of one-dimensional BBM–BBM case by Chen and Liu (Anal Math 121:299–316, 2013). Among results established here, we emphasize that the ill-posedness result for two-dimensional BBM–BBM system is optimal. The proof follows from an observation of the high to low frequency cascade present in nonlinearity, motivated by Bejenaru and Tao (J Funct Anal 233:228–259, 2006).

Acute threshold dynamics of an epidemic system with quarantine strategy driven by correlated white noises and Lévy jumps associated with infinite measure.

Several studies have previously been conducted on the dynamics of probabilistic epidemic models driven by Lévy disorder. All of these works have used the Poisson counting process with finite Lévy measures. However, this scope disregards a considerable category of correlated Lévy jump processes governed by an infinite Lévy measure. In this research, we take into consideration this general framework applied to an epidemic model with a quarantine strategy. Under an appropriate hypothetical setting, we infer the exact threshold value between the ergodicity and the disease disappearance. Our analysis completes the work presented by Privault and Wang (J Nonlinear Sci 31(1):1–28, 2021) and puts forward a novel analytical aspect to deal with other stochastic models in several areas. As a numerical application, we implement the algorithm of Rosinski (Stoch Process Appl 117:677–707, 2007) for tempered stable Lévy processes with an infinite Lévy measure.