Pub Date : 2026-02-01Epub Date: 2025-12-16DOI: 10.1016/j.physd.2025.135073
Zhengwu Miao , Yong Chen
The Schrödinger spectral problem is a central topic in mathematical physics. In numerical inverse scattering transform (NIST), the reflection coefficient R(k) contained in the scattering data must be repeatedly computed by solving the spectral problem at discrete wave number k. We propose a novel neural operator framework, the Jost operator network (JostONet), for fast inference of the Jost solution and its associated R(k), offering a promising alternative for computing R(k) in the NIST. JostONet is composed of three specialized modules: (i) High-energy region : inspired by the asymptotic behavior of the Jost solution, a novel amplitude decomposition is derived, based on which a variable amplitude operator network is constructed. Normalization conditions and conservation property are embedded in the loss function, and hard boundary constraints are imposed. (ii) Intermediate-energy region : The wave number k is treated as a degenerate functional variable, and a wave function operator network is constructed based on the multi-input operator network. (iii) Low-energy region : a perturbation-wave function operator network is introduced, which exploits the perturbation expansion of the Jost solution with respect to k and is composed of a sequence of Deep Operator Networks. During training, a novel function space is constructed based on Hermite polynomials to generate potential functions with Gaussian decay, which serve as inputs to the neural operators. JostONet achieves satisfactory predictive accuracy across all energy regions, with an inference speed at least an order of magnitude faster than traditional methods, and it is capable of generalizing to higher-order potentials in the space . In addition, we provide theoretical support and extensive numerical validation for the partitioning of k, along with detailed numerical analysis of each module.
{"title":"JostONet: A neural operator architecture for solving the Jost solution and scattering coefficients of the Schrödinger spectral problem","authors":"Zhengwu Miao , Yong Chen","doi":"10.1016/j.physd.2025.135073","DOIUrl":"10.1016/j.physd.2025.135073","url":null,"abstract":"<div><div>The Schrödinger spectral problem is a central topic in mathematical physics. In numerical inverse scattering transform (NIST), the reflection coefficient <em>R</em>(<em>k</em>) contained in the scattering data <span><math><mi>S</mi></math></span> must be repeatedly computed by solving the spectral problem at discrete wave number <em>k</em>. We propose a novel neural operator framework, the Jost operator network (JostONet), for fast inference of the Jost solution and its associated <em>R</em>(<em>k</em>), offering a promising alternative for computing <em>R</em>(<em>k</em>) in the NIST. JostONet is composed of three specialized modules: (i) High-energy region <span><math><msub><mi>R</mi><mi>h</mi></msub></math></span>: inspired by the asymptotic behavior of the Jost solution, a novel amplitude decomposition is derived, based on which a variable amplitude operator network is constructed. Normalization conditions and conservation property are embedded in the loss function, and hard boundary constraints are imposed. (ii) Intermediate-energy region <span><math><msub><mi>R</mi><mi>m</mi></msub></math></span>: The wave number <em>k</em> is treated as a degenerate functional variable, and a wave function operator network is constructed based on the multi-input operator network. (iii) Low-energy region <span><math><msub><mi>R</mi><mi>l</mi></msub></math></span>: a perturbation-wave function operator network is introduced, which exploits the perturbation expansion of the Jost solution with respect to <em>k</em> and is composed of a sequence of Deep Operator Networks. During training, a novel function space <span><math><mrow><msup><mover><mrow><mi>H</mi></mrow><mo>˜</mo></mover><mrow><mi>κ</mi><mo>,</mo><mi>η</mi></mrow></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span> is constructed based on Hermite polynomials to generate potential functions with Gaussian decay, which serve as inputs to the neural operators. JostONet achieves satisfactory predictive accuracy across all energy regions, with an inference speed at least an order of magnitude faster than traditional methods, and it is capable of generalizing to higher-order potentials in the space <span><math><mrow><msup><mover><mrow><mi>H</mi></mrow><mo>˜</mo></mover><mrow><mi>κ</mi><mo>,</mo><mi>η</mi></mrow></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>. In addition, we provide theoretical support and extensive numerical validation for the partitioning of <em>k</em>, along with detailed numerical analysis of each module.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"486 ","pages":"Article 135073"},"PeriodicalIF":2.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145840767","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-01Epub Date: 2025-11-20DOI: 10.1016/j.physd.2025.135052
Farangis Mahdizadeh Ghohe, Leila N. Azadani
Turbulent flow of non-Newtonian fluids is common in daily life and engineering practice. Unlike Newtonian fluids, non-Newtonian fluids have a viscosity that depends on time or shear rate, complicating their behavior, especially in turbulent regime. Large Eddy Simulation (LES) with Subgrid Scale (SGS) models such as standard Smagorinsky, dynamic Smagorinsky, and scale-dependent dynamic Smagorinsky has shown promise for simulating turbulent flows of Newtonian fluids. However, the application of these models, particularly the scale-dependent dynamic Smagorinsky model, to turbulent flows of non-Newtonian fluids remains largely unexplored. This paper investigated the robustness of the scale-dependent dynamic Smagorinsky model in LES of the turbulent non-Newtonian Burgers’ equation. Results for velocity and energy spectrum from the standard Smagorinsky, dynamic Smagorinsky, and scale-dependent dynamic Smagorinsky models were compared against Direct Numerical Simulation (DNS). It was demonstrated that the scale-dependent dynamic Smagorinsky model performs better than both the standard and dynamic Smagorinsky models in simulating turbulent flow of non-Newtonian fluids.
{"title":"Applying the scale-dependent dynamic Smagorinsky model in large eddy simulation of the turbulent non-Newtonian Burgers’ equation","authors":"Farangis Mahdizadeh Ghohe, Leila N. Azadani","doi":"10.1016/j.physd.2025.135052","DOIUrl":"10.1016/j.physd.2025.135052","url":null,"abstract":"<div><div>Turbulent flow of non-Newtonian fluids is common in daily life and engineering practice. Unlike Newtonian fluids, non-Newtonian fluids have a viscosity that depends on time or shear rate, complicating their behavior, especially in turbulent regime. Large Eddy Simulation (LES) with Subgrid Scale (SGS) models such as standard Smagorinsky, dynamic Smagorinsky, and scale-dependent dynamic Smagorinsky has shown promise for simulating turbulent flows of Newtonian fluids. However, the application of these models, particularly the scale-dependent dynamic Smagorinsky model, to turbulent flows of non-Newtonian fluids remains largely unexplored. This paper investigated the robustness of the scale-dependent dynamic Smagorinsky model in LES of the turbulent non-Newtonian Burgers’ equation. Results for velocity and energy spectrum from the standard Smagorinsky, dynamic Smagorinsky, and scale-dependent dynamic Smagorinsky models were compared against Direct Numerical Simulation (DNS). It was demonstrated that the scale-dependent dynamic Smagorinsky model performs better than both the standard and dynamic Smagorinsky models in simulating turbulent flow of non-Newtonian fluids.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"485 ","pages":"Article 135052"},"PeriodicalIF":2.9,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145616269","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-01Epub Date: 2025-11-22DOI: 10.1016/j.physd.2025.135041
Roberto Onofrio , Bala Sundaram
We discuss adiabatic strategies for heating a gas of particles harmonically trapped in the presence of another gas of particles which are also harmonically trapped, while also experiencing nonlinear interatomic interactions. These include recently proposed strategies, namely shortcuts to adiabaticity, in which the cooling or heating occurs quickly, at variance with the usual occurrence over times much longer than intrinsic timescales. Using quantitative indicators, we track the extent of adherence to Maxwell-Boltzmann energy distributions of both gases during the compression transient. The model, although aimed at addressing ionized gases, also allows for local and extended interactions, interpolating between the description of neutral atomic gases and ordered condensed matter systems, with plasmas – exhibiting genuine nonlinear behavior – in the intermediate regime.
{"title":"Adiabatic heating strategies in many-body classical systems and plasmas","authors":"Roberto Onofrio , Bala Sundaram","doi":"10.1016/j.physd.2025.135041","DOIUrl":"10.1016/j.physd.2025.135041","url":null,"abstract":"<div><div>We discuss adiabatic strategies for heating a gas of particles harmonically trapped in the presence of another gas of particles which are also harmonically trapped, while also experiencing nonlinear interatomic interactions. These include recently proposed strategies, namely shortcuts to adiabaticity, in which the cooling or heating occurs quickly, at variance with the usual occurrence over times much longer than intrinsic timescales. Using quantitative indicators, we track the extent of adherence to Maxwell-Boltzmann energy distributions of both gases during the compression transient. The model, although aimed at addressing ionized gases, also allows for local and extended interactions, interpolating between the description of neutral atomic gases and ordered condensed matter systems, with plasmas – exhibiting genuine nonlinear behavior – in the intermediate regime.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"485 ","pages":"Article 135041"},"PeriodicalIF":2.9,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145616272","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-01Epub Date: 2025-11-15DOI: 10.1016/j.physd.2025.135040
Man-Hong Fan , Jun-Hao Zhao , Lin Ding , Xiao-Ying Ma
The traditional methods for forecasting nonlinear dynamics problems rely mainly on experimental means and numerical simulations; however, both methods struggle to address high-order complex dynamic issues. Physics-informed neural networks (PINNs) have been extensively applied to predict partial differential equations (PDEs) and can be used to simulate physical systems. Nevertheless, when their solutions exhibit high-dimensional nonlinear characteristics, the accuracy of PINNs can decrease significantly. To enhance the predictive capability of PINNs for high-order complex dynamical systems, this study proposes a novel PINNs architecture integrated with Residual Network (ResNet) blocks. The framework addresses critical challenges such as gradient vanishing through identity mappings when employing deep network structures, thereby enabling effective capture of rapidly varying solutions in physical fields. To validate the performance of the PINNs with ResNet blocks, numerical experiments are conducted on the chaotic Lorenz system, the Kuramoto-Sivashinsky equation in a chaotic state, and the Navier-Stokes equation. These results are compared with those obtained using a PINNs framework that is based on multilayer perceptrons (MLPs). The results indicate that the PINNs with ResNet blocks exhibit stronger prediction capabilities and robustness than the PINNs framework based on MLPs.
{"title":"Forecasting of spatiotemporal nonlinear dynamic systems by Physics-informed neural networks with ResNet blocks","authors":"Man-Hong Fan , Jun-Hao Zhao , Lin Ding , Xiao-Ying Ma","doi":"10.1016/j.physd.2025.135040","DOIUrl":"10.1016/j.physd.2025.135040","url":null,"abstract":"<div><div>The traditional methods for forecasting nonlinear dynamics problems rely mainly on experimental means and numerical simulations; however, both methods struggle to address high-order complex dynamic issues. Physics-informed neural networks (PINNs) have been extensively applied to predict partial differential equations (PDEs) and can be used to simulate physical systems. Nevertheless, when their solutions exhibit high-dimensional nonlinear characteristics, the accuracy of PINNs can decrease significantly. To enhance the predictive capability of PINNs for high-order complex dynamical systems, this study proposes a novel PINNs architecture integrated with Residual Network (ResNet) blocks. The framework addresses critical challenges such as gradient vanishing through identity mappings when employing deep network structures, thereby enabling effective capture of rapidly varying solutions in physical fields. To validate the performance of the PINNs with ResNet blocks, numerical experiments are conducted on the chaotic Lorenz system, the Kuramoto-Sivashinsky equation in a chaotic state, and the Navier-Stokes equation. These results are compared with those obtained using a PINNs framework that is based on multilayer perceptrons (MLPs). The results indicate that the PINNs with ResNet blocks exhibit stronger prediction capabilities and robustness than the PINNs framework based on MLPs.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"485 ","pages":"Article 135040"},"PeriodicalIF":2.9,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145578350","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-01Epub Date: 2025-11-26DOI: 10.1016/j.physd.2025.135029
Yuan Cai , Xiufang Cui , Fei Jiang , Hao Liu
The small Alfvén number (denoted by ) limit (one type of large parameter limits, i.e. singular limits) in magnetohydrodynamic (abbr. MHD) equations was first proposed by Klainerman–Majda in (Comm. Pure Appl. Math. 34: 481–524, 1981). Recently Ju–Wang–Xu mathematically verified that the local-in-time solutions of three-dimensional (abbr. 3D) ideal (i.e. the absence of the dissipative terms) incompressible MHD equations with general initial data in (i.e. a spatially periodic domain) tend to a solution of 2D ideal MHD equations in the distribution sense as by Schochet’s fast averaging method in (J. Differential Equations, 114: 476–512, 1994). In this paper, we revisit the small Alfvén number limit in with , 3, and develop another approach, motivated by Cai–Lei’s energy method in (Arch. Ration. Mech. Anal. 228: 969–993, 2018), to establish a new conclusion that the global-in-time solutions of incompressible MHD equations (including the viscous resistive case) with general initial data converge to zero as for any given time–space variable with . In addition, we find that the large perturbation solutions and vanishing phenomenon of the nonlinear interactions also exist in the viscous resistive MHD equations for small Alfvén numbers, and thus extend Bardos et al.’s results of the ideal MHD equations in Bardos et al. (1988).
磁流体动力学（简称MHD）方程中的小alfv数极限（用* * *表示）（一种大参数极限，即奇异极限）是由Klainerman-Majda在《Comm. Pure application》中首次提出的。数学。34:481-524,1981)。最近，juwang - xu用Schochet快速平均法在数学上验证了具有一般初始数据在T3（即空间周期域）的三维理想（即不存在耗散项）不可压缩MHD方程的局域解趋向于分布意义上的二维理想MHD方程的解（J.微分方程，14:476-512,1994）。在本文中，我们重新审视了n= 2,3的Rn中的小alfvsamn数极限，并开发了另一种方法，该方法的动机是蔡磊的能量法。配给。动力机械。对于任意给定的时空变量（x,t），当t>；0时，建立了具有一般初始数据的不可压缩MHD方程（包括粘滞阻力情况）的全局时解收敛于0的新结论。此外，我们发现小alfv数的粘阻MHD方程也存在非线性相互作用的大摄动解和消失现象，从而推广了Bardos et al.（1988）中Bardos et al.关于理想MHD方程的结果。
{"title":"Small Alfvén number limit for the global-in-time solutions of incompressible MHD equations with general initial data","authors":"Yuan Cai , Xiufang Cui , Fei Jiang , Hao Liu","doi":"10.1016/j.physd.2025.135029","DOIUrl":"10.1016/j.physd.2025.135029","url":null,"abstract":"<div><div>The small Alfvén number (denoted by <span><math><mi>ɛ</mi></math></span>) limit (one type of large parameter limits, i.e. singular limits) in magnetohydrodynamic (abbr. MHD) equations was first proposed by Klainerman–Majda in (Comm. Pure Appl. Math. 34: 481–524, 1981). Recently Ju–Wang–Xu mathematically verified that the <em>local-in-time</em> solutions of three-dimensional (abbr. 3D) ideal (i.e. the absence of the dissipative terms) incompressible MHD equations with general initial data in <span><math><msup><mrow><mi>T</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> (i.e. a spatially periodic domain) tend to a solution of 2D ideal MHD equations in the distribution sense as <span><math><mrow><mi>ɛ</mi><mo>→</mo><mn>0</mn></mrow></math></span> by Schochet’s fast averaging method in (J. Differential Equations, 114: 476–512, 1994). In this paper, we revisit the small Alfvén number limit in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> with <span><math><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow></math></span>, 3, and develop another approach, motivated by Cai–Lei’s energy method in (Arch. Ration. Mech. Anal. 228: 969–993, 2018), to establish a new conclusion that the <em>global-in-time</em> solutions of incompressible MHD equations (including the viscous resistive case) with general initial data converge to zero as <span><math><mrow><mi>ɛ</mi><mo>→</mo><mn>0</mn></mrow></math></span> for any given time–space variable <span><math><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></math></span> with <span><math><mrow><mi>t</mi><mo>></mo><mn>0</mn></mrow></math></span>. In addition, we find that the large perturbation solutions and vanishing phenomenon of the nonlinear interactions also exist in the <em>viscous resistive</em> MHD equations for small Alfvén numbers, and thus extend Bardos et al.’s results of the <em>ideal</em> MHD equations in Bardos et al. (1988).</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"485 ","pages":"Article 135029"},"PeriodicalIF":2.9,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145616271","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-01Epub Date: 2025-11-17DOI: 10.1016/j.physd.2025.135031
Amadeu Delshams , Mercè Ollé , Juan Ramon Pacha , Óscar Rodríguez
We consider the Rydberg electron in a circularly polarized microwave field, whose dynamics is described by a 2 d.o.f. Hamiltonian, which is a perturbation of size of the standard rotating Kepler problem. In a rotating frame, the largest chaotic region of this system lies around a saddle–center equilibrium point and its associated invariant manifolds. We compute the distance between stable and unstable manifolds of by means of a semi-analytical method, which consists of combining normal form, Melnikov, and averaging methods with numerical methods performed with multiple precision computations. Also, we introduce a new family of Hamiltonians, which we call Toy CP systems, to be able to compare our numerical results with the existing theoretical results in the literature. It should be noted that the distance between these stable and unstable manifolds is exponentially small in the perturbation parameter (in analogy with the libration point of the R3BP).
{"title":"Breakdown of homoclinic orbits to L1 of the hydrogen atom in a circularly polarized microwave field","authors":"Amadeu Delshams , Mercè Ollé , Juan Ramon Pacha , Óscar Rodríguez","doi":"10.1016/j.physd.2025.135031","DOIUrl":"10.1016/j.physd.2025.135031","url":null,"abstract":"<div><div>We consider the Rydberg electron in a circularly polarized microwave field, whose dynamics is described by a 2 d.o.f. Hamiltonian, which is a perturbation of size <span><math><mrow><mi>K</mi><mo>></mo><mn>0</mn></mrow></math></span> of the standard rotating Kepler problem. In a rotating frame, the largest chaotic region of this system lies around a saddle–center equilibrium point <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and its associated invariant manifolds. We compute the distance between stable and unstable manifolds of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> by means of a semi-analytical method, which consists of combining normal form, Melnikov, and averaging methods with numerical methods performed with multiple precision computations. Also, we introduce a new family of Hamiltonians, which we call <em>Toy CP systems</em>, to be able to compare our numerical results with the existing theoretical results in the literature. It should be noted that the distance between these stable and unstable manifolds is exponentially small in the perturbation parameter <span><math><mi>K</mi></math></span> (in analogy with the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> libration point of the R3BP).</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"485 ","pages":"Article 135031"},"PeriodicalIF":2.9,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145578349","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-01Epub Date: 2025-11-15DOI: 10.1016/j.physd.2025.135038
Iacopo P. Longo , Rafael Obaya , Ana M. Sanz
Starting from a classical Budyko–Sellers–Ghil energy balance model for the average surface temperature of the Earth, a nonautonomous version is designed by allowing the solar irradiance and the cloud cover coefficients to vary with time on a fast timescale, and to exhibit chaos in a precise sense. The dynamics of this model is described in terms of three existing nonautonomous equilibria, the upper one being attracting and representing the present temperature profile. The theory of averaging is used to compare the nonautonomous model and its time-averaged version. We analyse the influence of the qualitative properties of the time-dependent coefficients and obtain reasonable approximations close to the upper hyperbolic solution. Furthermore, previous concepts of two-point response and sensitivity functions are adapted to the nonautonomous context and used to value the increase in temperature when a forcing caused by CO and other emissions intervenes.
{"title":"Nonautonomous modelling in energy balance models of climate. Limitations of averaging and climate sensitivity","authors":"Iacopo P. Longo , Rafael Obaya , Ana M. Sanz","doi":"10.1016/j.physd.2025.135038","DOIUrl":"10.1016/j.physd.2025.135038","url":null,"abstract":"<div><div>Starting from a classical Budyko–Sellers–Ghil energy balance model for the average surface temperature of the Earth, a nonautonomous version is designed by allowing the solar irradiance and the cloud cover coefficients to vary with time on a fast timescale, and to exhibit chaos in a precise sense. The dynamics of this model is described in terms of three existing nonautonomous equilibria, the upper one being attracting and representing the present temperature profile. The theory of averaging is used to compare the nonautonomous model and its time-averaged version. We analyse the influence of the qualitative properties of the time-dependent coefficients and obtain reasonable approximations close to the upper hyperbolic solution. Furthermore, previous concepts of two-point response and sensitivity functions are adapted to the nonautonomous context and used to value the increase in temperature when a forcing caused by CO<span><math><msub><mrow></mrow><mrow><mn>2</mn></mrow></msub></math></span> and other emissions intervenes.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"485 ","pages":"Article 135038"},"PeriodicalIF":2.9,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145578388","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-01Epub Date: 2025-11-15DOI: 10.1016/j.physd.2025.135037
Rafael de la Rosa, Elena Medina
<div><div>The survival of a population confined within a bounded habitat is a classical problem, traditionally analyzed in terms of the habitat size. In the linear case, persistence is ensured when the domain length exceeds a critical size <span><math><msub><mrow><mi>l</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span>. In nonlinear models, however survival conditions become considerably more complex and may even take less intuitive forms, such as <span><math><mrow><mi>l</mi><mspace></mspace><mo>≤</mo><mspace></mspace><msub><mrow><mi>l</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></math></span>. In this context, Colombo and Anteneodo (2018) studied the power-law reaction–diffusion model <span><math><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mspace></mspace><mo>=</mo><mspace></mspace><mi>D</mi><mspace></mspace><msub><mrow><mrow><mo>(</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>ν</mi><mo>−</mo><mn>1</mn></mrow></msup><mspace></mspace><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>x</mi></mrow></msub><mspace></mspace><mo>+</mo><mspace></mspace><mi>a</mi><mspace></mspace><msup><mrow><mi>u</mi></mrow><mrow><mi>μ</mi></mrow></msup></mrow></math></span>, with <span><math><mrow><mi>μ</mi><mo>,</mo><mi>ν</mi><mo>></mo><mn>0</mn></mrow></math></span>, accompanied by hostile boundary conditions, determining survival thresholds in terms of habitat size for initially homogeneous populations.</div><div>In this paper, we propose a general formulation of the persistence question by rewriting the power-law reaction–diffusion model in terms of suitable nondimensional variables. This approach reveals that persistence can be naturally expressed through a parameter <span><math><mrow><mi>Q</mi><mo>≔</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi>D</mi></mrow></mfrac><msup><mrow><mi>l</mi></mrow><mrow><mo>−</mo><mi>μ</mi><mo>+</mo><mi>ν</mi><mo>+</mo><mn>2</mn></mrow></msup><msubsup><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow><mrow><mi>μ</mi><mo>−</mo><mi>ν</mi></mrow></msubsup></mrow></math></span>. We show that there exists a critical value <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span> depending on <span><math><mi>μ</mi></math></span>, <span><math><mi>ν</mi></math></span> and the initial distribution, such that survival occurs whenever <span><math><mrow><mi>Q</mi><mo>≥</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></math></span>. This more intuitive condition reconciles the various survival criteria within a unified framework.</div><div>To further explore this condition, we analyze two one-parameter families of initial distributions, including the homogeneous case, and apply a finite-difference scheme to estimate <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span>. Conversely, for given model parameters <span><math><mi>μ</mi></math></span>, <span><math><mi>ν</mi></math></span>, <span><math><mi>
{"title":"A general formulation of the survival problem in a power-law reaction–diffusion model: Emergence of a critical parameter","authors":"Rafael de la Rosa, Elena Medina","doi":"10.1016/j.physd.2025.135037","DOIUrl":"10.1016/j.physd.2025.135037","url":null,"abstract":"<div><div>The survival of a population confined within a bounded habitat is a classical problem, traditionally analyzed in terms of the habitat size. In the linear case, persistence is ensured when the domain length exceeds a critical size <span><math><msub><mrow><mi>l</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span>. In nonlinear models, however survival conditions become considerably more complex and may even take less intuitive forms, such as <span><math><mrow><mi>l</mi><mspace></mspace><mo>≤</mo><mspace></mspace><msub><mrow><mi>l</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></math></span>. In this context, Colombo and Anteneodo (2018) studied the power-law reaction–diffusion model <span><math><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mspace></mspace><mo>=</mo><mspace></mspace><mi>D</mi><mspace></mspace><msub><mrow><mrow><mo>(</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>ν</mi><mo>−</mo><mn>1</mn></mrow></msup><mspace></mspace><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>x</mi></mrow></msub><mspace></mspace><mo>+</mo><mspace></mspace><mi>a</mi><mspace></mspace><msup><mrow><mi>u</mi></mrow><mrow><mi>μ</mi></mrow></msup></mrow></math></span>, with <span><math><mrow><mi>μ</mi><mo>,</mo><mi>ν</mi><mo>></mo><mn>0</mn></mrow></math></span>, accompanied by hostile boundary conditions, determining survival thresholds in terms of habitat size for initially homogeneous populations.</div><div>In this paper, we propose a general formulation of the persistence question by rewriting the power-law reaction–diffusion model in terms of suitable nondimensional variables. This approach reveals that persistence can be naturally expressed through a parameter <span><math><mrow><mi>Q</mi><mo>≔</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi>D</mi></mrow></mfrac><msup><mrow><mi>l</mi></mrow><mrow><mo>−</mo><mi>μ</mi><mo>+</mo><mi>ν</mi><mo>+</mo><mn>2</mn></mrow></msup><msubsup><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow><mrow><mi>μ</mi><mo>−</mo><mi>ν</mi></mrow></msubsup></mrow></math></span>. We show that there exists a critical value <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span> depending on <span><math><mi>μ</mi></math></span>, <span><math><mi>ν</mi></math></span> and the initial distribution, such that survival occurs whenever <span><math><mrow><mi>Q</mi><mo>≥</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></math></span>. This more intuitive condition reconciles the various survival criteria within a unified framework.</div><div>To further explore this condition, we analyze two one-parameter families of initial distributions, including the homogeneous case, and apply a finite-difference scheme to estimate <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span>. Conversely, for given model parameters <span><math><mi>μ</mi></math></span>, <span><math><mi>ν</mi></math></span>, <span><math><mi>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"485 ","pages":"Article 135037"},"PeriodicalIF":2.9,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145578389","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-01Epub Date: 2025-11-21DOI: 10.1016/j.physd.2025.135039
Axel Kröner , Heiko Kröner
In this paper we study the evolution of special hypersurfaces in Euclidean space evolving according to a geometric curvature flow which is motivated from the phenomenon of evaporation–condensation and which was introduced in Mullins (1957). The first example is a closed, embedded and convex curve which does not converge to a point under the (smooth) flow. This shows that the flow behaves differently from curve shortening flow. Further we show existence of two examples of hypersurfaces which translate under this evolution. Their shape is motivated from translators as they appear in the theory of mean curvature flow: A complete, unbounded graph over a finite time interval and in the higher dimensional case a rotationally symmetric entire graph over .
{"title":"Special solutions to a geometric curvature flow of evaporation–condensation","authors":"Axel Kröner , Heiko Kröner","doi":"10.1016/j.physd.2025.135039","DOIUrl":"10.1016/j.physd.2025.135039","url":null,"abstract":"<div><div>In this paper we study the evolution of special hypersurfaces in Euclidean space evolving according to a geometric curvature flow which is motivated from the phenomenon of evaporation–condensation and which was introduced in Mullins (1957). The first example is a closed, embedded and convex curve which does not converge to a point under the (smooth) flow. This shows that the flow behaves differently from curve shortening flow. Further we show existence of two examples of hypersurfaces which translate under this evolution. Their shape is motivated from translators as they appear in the theory of mean curvature flow: A complete, unbounded graph over a finite time interval and in the higher dimensional case a rotationally symmetric entire graph over <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"485 ","pages":"Article 135039"},"PeriodicalIF":2.9,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145616189","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The nonlinear Schrödinger equation (NLSE) is used to study the dynamics of electrostatic wave envelopes in a multi-ion, superthermal plasma. The plasma contains Sr, Ti, and O ions with a -distribution of electrons. We use weakly nonlinear analysis to develop analytical formulas for the dispersion () and nonlinear () coefficients, methodically investigating their effect on the superthermality index and ion concentration ratio We found that reducing (stronger suprathermal effects) increases anomalous dispersion () and self-focusing nonlinearity (), whereas increasing increases modulational instability () via increasing ion inertia and charge density. Localized wave structures are predicted universally by . The Peregrine soliton (transient rogue wave) and Akhmediev breather (periodic modulation) show remarkable spatiotemporal localization in NLSE numerical solutions. Plasma density variations cause mechanical strains beyond material cohesion limitations, causing nanoscale surface deformation under ion irradiation. Our results provide a plasma parameter-based paradigm for nanostructure morphology control in plasma-assisted nanofabrication and space plasma settings.
{"title":"Peregrine soliton and Akhmediev Breather in superthermal multi-ion plasmas and their role in ion-induced nanostructuring","authors":"N.A. El-Bedwehy , R. Sabry , W.M. Moslem , I.S. Elkamash","doi":"10.1016/j.physd.2025.135027","DOIUrl":"10.1016/j.physd.2025.135027","url":null,"abstract":"<div><div>The nonlinear Schrödinger equation (NLSE) is used to study the dynamics of electrostatic wave envelopes in a multi-ion, superthermal plasma. The plasma contains Sr<span><math><msup><mrow></mrow><mrow><mn>2</mn><mo>+</mo></mrow></msup></math></span>, Ti<span><math><msup><mrow></mrow><mrow><mn>4</mn><mo>+</mo></mrow></msup></math></span>, and O<span><math><msup><mrow></mrow><mrow><mn>2</mn><mo>−</mo></mrow></msup></math></span> ions with a <span><math><mi>κ</mi></math></span>-distribution of electrons. We use weakly nonlinear analysis to develop analytical formulas for the dispersion (<span><math><mi>P</mi></math></span>) and nonlinear (<span><math><mi>Q</mi></math></span>) coefficients, methodically investigating their effect on the superthermality index <span><math><mi>κ</mi></math></span> and ion concentration ratio <span><math><mrow><mi>β</mi><mo>=</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>T</mi><mn>0</mn></mrow></msub><mo>/</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>s</mi><mn>0</mn></mrow></msub></mrow></math></span> We found that reducing <span><math><mi>κ</mi></math></span> (stronger suprathermal effects) increases anomalous dispersion (<span><math><mrow><mi>P</mi><mo><</mo><mn>0</mn></mrow></math></span>) and self-focusing nonlinearity (<span><math><mrow><mi>Q</mi><mo><</mo><mn>0</mn></mrow></math></span>), whereas increasing <span><math><mi>β</mi></math></span> increases modulational instability (<span><math><mrow><mi>P</mi><mi>Q</mi><mo>></mo><mn>0</mn></mrow></math></span>) via increasing ion inertia and charge density. Localized wave structures are predicted universally by <span><math><mrow><mi>P</mi><mi>Q</mi></mrow></math></span>. The Peregrine soliton (transient rogue wave) and Akhmediev breather (periodic modulation) show remarkable spatiotemporal localization in NLSE numerical solutions. Plasma density variations cause mechanical strains beyond material cohesion limitations, causing nanoscale surface deformation under ion irradiation. Our results provide a plasma parameter-based paradigm for nanostructure morphology control in plasma-assisted nanofabrication and space plasma settings.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"485 ","pages":"Article 135027"},"PeriodicalIF":2.9,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145616188","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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