Pub Date : 2024-09-17DOI: 10.1007/s00332-024-10078-8
Bing Luo, Wei Li
Many flying and swimming creatures have morphing pectoral propulsors (wings or fins) for propulsion, typically with flapping, rowing, and pitching motions; flapping and rowing motions are responsible for the stroke plane angle that is important for a broader performance space of the propulsor, while the stroke plane angle has been less characterized and implemented by artificial propulsors of biomimetic vehicles and thus has lack of stroke plane angle control. In this paper, we consider robotic pectoral propulsors with combined flapping and rowing motions for a stroke plane angle that can be generally specified. We consider two possible rotation axes configurations (i.e., the dependence of the rotation axes for flapping and rowing). For each rotation axes configuration, we propose the kinematic relations between the flapping and rowing motions for a generally specified stroke plane angle and provide the general flapping (or rowing) kinematics as a function of the rowing (or flapping) kinematics, which have not been characterized previously. These results serve as the reference trajectories of the propulsor for specified stroke plane angles and have implications for stroke plane angle control and thus have implications to achieve a broader performance space for biomimetic propulsors.
{"title":"Biologically Inspired Pectoral Propulsors with Flapping and Rowing Control for a Specified Stroke Plane Angle","authors":"Bing Luo, Wei Li","doi":"10.1007/s00332-024-10078-8","DOIUrl":"https://doi.org/10.1007/s00332-024-10078-8","url":null,"abstract":"<p>Many flying and swimming creatures have morphing pectoral propulsors (wings or fins) for propulsion, typically with flapping, rowing, and pitching motions; flapping and rowing motions are responsible for the <i>stroke plane angle</i> that is important for a broader performance space of the propulsor, while the stroke plane angle has been less characterized and implemented by artificial propulsors of biomimetic vehicles and thus has lack of stroke plane angle control. In this paper, we consider robotic pectoral propulsors with combined flapping and rowing motions for a stroke plane angle that can be generally specified. We consider two possible rotation axes configurations (i.e., the dependence of the rotation axes for flapping and rowing). For each rotation axes configuration, we propose the kinematic relations between the flapping and rowing motions for a generally specified stroke plane angle and provide the general flapping (or rowing) kinematics as a function of the rowing (or flapping) kinematics, which have not been characterized previously. These results serve as the reference trajectories of the propulsor for specified stroke plane angles and have implications for stroke plane angle control and thus have implications to achieve a broader performance space for biomimetic propulsors.</p>","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":"77 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269337","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-16DOI: 10.1007/s00332-024-10088-6
Jesús Dueñas, Carmen Núñez, Rafael Obaya
The occurrence of tracking or tipping situations for a transition equation (x'=f(t,x,Gamma (t,x))) with asymptotic limits (x'=f(t,x,Gamma _pm (t,x))) is analyzed. The approaching condition is just (lim _{trightarrow pm infty }(Gamma (t,x)-Gamma _pm (t,x))=0) uniformly on compact real sets, and so there is no restriction to the dependence on time of the asymptotic equations. The hypotheses assume concavity in x either of the maps (xmapsto f(t,x,Gamma _pm (t,x))) or of their derivatives with respect to the state variable (d-concavity), but not of (xmapsto f(t,x,Gamma (t,x))) nor of its derivative. The analysis provides a powerful tool to analyze the occurrence of critical transitions for one-parametric families (x'=f(t,x,Gamma ^c(t,x))). The new approach significatively widens the field of application of the results, since the evolution law of the transition equation can be essentially different from those of the limit equations. Among these applications, some scalar population dynamics models subject to nontrivial predation and migration patterns are analyzed, both theoretically and numerically. Some key points in the proofs are: to understand the transition equation as part of an orbit in its hull which approaches the -limit and -limit sets; to observe that these sets concentrate all the ergodic measures; and to prove that in order to describe the dynamical possibilities of the equation it is sufficient that the concavity or d-concavity conditions hold for a complete measure subset of the equations of the hull.
{"title":"Critical Transitions for Asymptotically Concave or d-Concave Nonautonomous Differential Equations with Applications in Ecology","authors":"Jesús Dueñas, Carmen Núñez, Rafael Obaya","doi":"10.1007/s00332-024-10088-6","DOIUrl":"https://doi.org/10.1007/s00332-024-10088-6","url":null,"abstract":"<p>The occurrence of tracking or tipping situations for a transition equation <span>(x'=f(t,x,Gamma (t,x)))</span> with asymptotic limits <span>(x'=f(t,x,Gamma _pm (t,x)))</span> is analyzed. The approaching condition is just <span>(lim _{trightarrow pm infty }(Gamma (t,x)-Gamma _pm (t,x))=0)</span> uniformly on compact real sets, and so there is no restriction to the dependence on time of the asymptotic equations. The hypotheses assume concavity in <i>x</i> either of the maps <span>(xmapsto f(t,x,Gamma _pm (t,x)))</span> or of their derivatives with respect to the state variable (d-concavity), but not of <span>(xmapsto f(t,x,Gamma (t,x)))</span> nor of its derivative. The analysis provides a powerful tool to analyze the occurrence of critical transitions for one-parametric families <span>(x'=f(t,x,Gamma ^c(t,x)))</span>. The new approach significatively widens the field of application of the results, since the evolution law of the transition equation can be essentially different from those of the limit equations. Among these applications, some scalar population dynamics models subject to nontrivial predation and migration patterns are analyzed, both theoretically and numerically. Some key points in the proofs are: to understand the transition equation as part of an orbit in its hull which approaches the -limit and <img alt=\"\" src=\"//media.springernature.com/lw17/springer-static/image/art%3A10.1007%2Fs00332-024-10088-6/MediaObjects/332_2024_10088_IEq8_HTML.gif\" style=\"width:17px;max-width:none;\"/>-limit sets; to observe that these sets concentrate all the ergodic measures; and to prove that in order to describe the dynamical possibilities of the equation it is sufficient that the concavity or d-concavity conditions hold for a complete measure subset of the equations of the hull.</p>","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":"77 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142257205","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-14DOI: 10.1007/s00332-024-10073-z
R. Ahmad, M. D. Groves, D. Nilsson
We present a Lyapunov centre theorem for an antisymplectically reversible Hamiltonian system exhibiting a nondegenerate 1 : 1 or (1:-1) semisimple resonance as a detuning parameter is varied. The system can be finite- or infinite-dimensional (and quasilinear) and have a non-constant symplectic structure. We allow the origin to be a ‘trivial’ eigenvalue arising from a translational symmetry or, in an infinite-dimensional setting, to lie in the continuous spectrum of the linearised Hamiltonian vector field provided a compatibility condition on its range is satisfied. As an application, we show how Kirchgässner’s spatial dynamics approach can be used to construct doubly periodic travelling waves on the surface of a three-dimensional body of water (of finite or infinite depth) beneath a thin ice sheet (‘hydroelastic waves’). The hydrodynamic problem is formulated as a reversible Hamiltonian system in which an arbitrary horizontal spatial direction is the time-like variable, and the infinite-dimensional phase space consists of wave profiles which are periodic (with fixed period) in a second, different horizontal direction. Applying our Lyapunov centre theorem at a point in parameter space associated with a 1 : 1 or (1:-1) semisimple resonance yields a periodic solution of the spatial Hamiltonian system corresponding to a doubly periodic hydroelastic wave.
{"title":"A Resonant Lyapunov Centre Theorem with an Application to Doubly Periodic Travelling Hydroelastic Waves","authors":"R. Ahmad, M. D. Groves, D. Nilsson","doi":"10.1007/s00332-024-10073-z","DOIUrl":"https://doi.org/10.1007/s00332-024-10073-z","url":null,"abstract":"<p>We present a Lyapunov centre theorem for an antisymplectically reversible Hamiltonian system exhibiting a nondegenerate 1 : 1 or <span>(1:-1)</span> semisimple resonance as a detuning parameter is varied. The system can be finite- or infinite-dimensional (and quasilinear) and have a non-constant symplectic structure. We allow the origin to be a ‘trivial’ eigenvalue arising from a translational symmetry or, in an infinite-dimensional setting, to lie in the continuous spectrum of the linearised Hamiltonian vector field provided a compatibility condition on its range is satisfied. As an application, we show how Kirchgässner’s spatial dynamics approach can be used to construct doubly periodic travelling waves on the surface of a three-dimensional body of water (of finite or infinite depth) beneath a thin ice sheet (‘hydroelastic waves’). The hydrodynamic problem is formulated as a reversible Hamiltonian system in which an arbitrary horizontal spatial direction is the time-like variable, and the infinite-dimensional phase space consists of wave profiles which are periodic (with fixed period) in a second, different horizontal direction. Applying our Lyapunov centre theorem at a point in parameter space associated with a 1 : 1 or <span>(1:-1)</span> semisimple resonance yields a periodic solution of the spatial Hamiltonian system corresponding to a doubly periodic hydroelastic wave.</p>","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":"18 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142257206","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-11DOI: 10.1007/s00332-024-10086-8
Angel Durán, Nuria Reguera
The present paper is the first part of a project devoted to the fractional nonlinear Schrödinger (fNLS) equation. It is concerned with the existence and numerical generation of the solitary-wave solutions. For the first point, some conserved quantities of the problem are used to search for solitary-wave solutions from a constrained critical point problem and the application of the concentration-compactness theory. Several properties of the waves, such as the regularity and the asymptotic decay in some cases, are derived from the existence result. Some other properties, such as the monotone behavior and the speed-amplitude relation, will be explored computationally. To this end, a numerical procedure for the generation of the profiles is proposed. The method is based on a Fourier pseudospectral approximation of the differential system for the profiles and the use of Petviashvili’s iteration with extrapolation.
{"title":"Solitary-Wave Solutions of the Fractional Nonlinear Schrödinger Equation: I—Existence and Numerical Generation","authors":"Angel Durán, Nuria Reguera","doi":"10.1007/s00332-024-10086-8","DOIUrl":"https://doi.org/10.1007/s00332-024-10086-8","url":null,"abstract":"<p>The present paper is the first part of a project devoted to the fractional nonlinear Schrödinger (fNLS) equation. It is concerned with the existence and numerical generation of the solitary-wave solutions. For the first point, some conserved quantities of the problem are used to search for solitary-wave solutions from a constrained critical point problem and the application of the concentration-compactness theory. Several properties of the waves, such as the regularity and the asymptotic decay in some cases, are derived from the existence result. Some other properties, such as the monotone behavior and the speed-amplitude relation, will be explored computationally. To this end, a numerical procedure for the generation of the profiles is proposed. The method is based on a Fourier pseudospectral approximation of the differential system for the profiles and the use of Petviashvili’s iteration with extrapolation.\u0000</p>","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":"10 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206260","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-11DOI: 10.1007/s00332-024-10079-7
Ernest Fontich, Antonio Garijo, Xavier Jarque
In this paper, we give analytic proofs of the existence of transversal homoclinic points for a family of non-globally smooth diffeomorphisms having the origin as a fixed point which come out as a truncated map governing the local dynamics near a critical period three-cycle associated with the Secant map. Using Moser’s version of Birkhoff–Smale’s theorem, we prove that the boundary of the basin of attraction of the origin contains a Cantor-like invariant subset such that the restricted dynamics to it is conjugate to the full shift of N-symbols for any integer (Nge 2) or infinity.
{"title":"Chaotic Dynamics at the Boundary of a Basin of Attraction via Non-transversal Intersections for a Non-global Smooth Diffeomorphism","authors":"Ernest Fontich, Antonio Garijo, Xavier Jarque","doi":"10.1007/s00332-024-10079-7","DOIUrl":"https://doi.org/10.1007/s00332-024-10079-7","url":null,"abstract":"<p>In this paper, we give analytic proofs of the existence of transversal homoclinic points for a family of non-globally smooth diffeomorphisms having the origin as a fixed point which come out as a truncated map governing the local dynamics near a critical period three-cycle associated with the Secant map. Using Moser’s version of Birkhoff–Smale’s theorem, we prove that the boundary of the basin of attraction of the origin contains a Cantor-like invariant subset such that the restricted dynamics to it is conjugate to the full shift of <i>N</i>-symbols for any integer <span>(Nge 2)</span> or infinity.</p>","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":"13 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206261","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-04DOI: 10.1007/s00332-024-10081-z
Gaoyang She, Fengqi Yi
In this paper, we are concerned with the dynamics of a reaction–diffusion SIRS epidemic model with the general saturated nonlinear incidence rates. Firstly, we show the global existence and boundedness of the in-time solutions for the parabolic system. Secondly, for the ODEs system, we analyze the existence and stability of the disease-free equilibrium solution, the endemic equilibrium solutions as well as the bifurcating periodic solution. In particular, in the language of the basic reproduction number, we are able to address the existence of the saddle-node-like bifurcation and the secondary bifurcation (Hopf bifurcation). Our results also suggest that the ODEs system has a Allee effect, i.e., one can expect either the coexistence of a stable disease-free equilibrium and a stable endemic equilibrium solution, or the coexistence of a stable disease-free equilibrium solution and a stable periodic solution. Finally, for the PDEs system, we are capable of deriving the Turing instability criteria in terms of the diffusion rates for both the endemic equilibrium solutions and the Hopf bifurcating periodic solution. The onset of Turing instability can bring out multi-level bifurcations and manifest itself as the appearance of new spatiotemporal patterns. It seems also interesting to note that p and k, appearing in the saturated incidence rate (kSI^p/(1+alpha I^p)), tend to play far reaching roles in the spatiotemporal pattern formations.
本文关注的是具有一般饱和非线性发病率的反应扩散 SIRS 流行病模型的动力学。首先,我们证明了抛物线系统的全局存在性和实时解的有界性。其次,对于 ODEs 系统,我们分析了无病平衡解、流行平衡解以及分岔周期解的存在性和稳定性。特别是,在基本繁殖数的语言中,我们能够解决鞍结状分岔和二次分岔(霍普夫分岔)的存在问题。我们的结果还表明,ODEs 系统具有阿利效应,即可以预期稳定的无病平衡解和稳定的地方病平衡解共存,或者稳定的无病平衡解和稳定的周期解共存。最后,对于 PDEs 系统,我们能够根据地方性平衡解和霍普夫分岔周期解的扩散率推导出图灵不稳定性标准。图灵不稳定性的出现会带来多级分岔,并表现为新的时空模式的出现。值得注意的是,出现在饱和发生率 (kSI^p/(1+alpha I^p)) 中的 p 和 k 往往在时空模式形成中发挥深远的作用。
{"title":"Stability and Bifurcation Analysis of a Reaction–Diffusion SIRS Epidemic Model with the General Saturated Incidence Rate","authors":"Gaoyang She, Fengqi Yi","doi":"10.1007/s00332-024-10081-z","DOIUrl":"https://doi.org/10.1007/s00332-024-10081-z","url":null,"abstract":"<p>In this paper, we are concerned with the dynamics of a reaction–diffusion SIRS epidemic model with the general saturated nonlinear incidence rates. Firstly, we show the global existence and boundedness of the in-time solutions for the parabolic system. Secondly, for the ODEs system, we analyze the existence and stability of the disease-free equilibrium solution, the endemic equilibrium solutions as well as the bifurcating periodic solution. In particular, in the language of the basic reproduction number, we are able to address the existence of the saddle-node-like bifurcation and the secondary bifurcation (Hopf bifurcation). Our results also suggest that the ODEs system has a Allee effect, i.e., one can expect either the coexistence of a stable disease-free equilibrium and a stable endemic equilibrium solution, or the coexistence of a stable disease-free equilibrium solution and a stable periodic solution. Finally, for the PDEs system, we are capable of deriving the Turing instability criteria in terms of the diffusion rates for both the endemic equilibrium solutions and the Hopf bifurcating periodic solution. The onset of Turing instability can bring out multi-level bifurcations and manifest itself as the appearance of new spatiotemporal patterns. It seems also interesting to note that <i>p</i> and <i>k</i>, appearing in the saturated incidence rate <span>(kSI^p/(1+alpha I^p))</span>, tend to play far reaching roles in the spatiotemporal pattern formations.</p>","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":"185 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206262","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-03DOI: 10.1007/s00332-024-10076-w
Stefan Krömer, Martin Kružík, Marco Morandotti, Elvira Zappale
Measure-valued structured deformations are introduced to present a unified theory of deformations of continua. The energy associated with a measure-valued structured deformation is defined via relaxation departing either from energies associated with classical deformations or from energies associated with structured deformations. A concise integral representation of the energy functional is provided both in the unconstrained case and under Dirichlet conditions on a part of the boundary.
{"title":"Measure-Valued Structured Deformations","authors":"Stefan Krömer, Martin Kružík, Marco Morandotti, Elvira Zappale","doi":"10.1007/s00332-024-10076-w","DOIUrl":"https://doi.org/10.1007/s00332-024-10076-w","url":null,"abstract":"<p>Measure-valued structured deformations are introduced to present a unified theory of deformations of continua. The energy associated with a measure-valued structured deformation is defined via relaxation departing either from energies associated with classical deformations or from energies associated with structured deformations. A concise integral representation of the energy functional is provided both in the unconstrained case and under Dirichlet conditions on a part of the boundary.\u0000</p>","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":"20 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206280","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-02DOI: 10.1007/s00332-024-10085-9
Guowei Dai, Fei Xu, Yong Zhang
In this paper, we consider two-dimensional periodic capillary-gravity waves traveling under the influence of a vertical electric field. The full system is a nonlinear, two-layered, free boundary problem. The interface dynamics are derived by coupling Euler equations for the velocity field of the fluid with voltage potential equations governing the electric field. We first introduce the naive flattening technique to transform the free boundary problem into a fixed boundary problem. We then prove the existence of small-amplitude electrohydrodynamic waves with constant vorticity using local bifurcation theory. Moreover, we show that these electrohydrodynamic waves are formally stable in the linearized sense. Furthermore, we obtain a secondary bifurcation curve that emerges from the primary branch, consisting of ripple solutions on the interface. As far as we know, such solutions in electrohydrodynamics are established for the first time. It is worth noting that the electric field (E_0) plays a key role in controlling the shapes and types of waves on the interface.
{"title":"The Dynamics of Periodic Traveling Interfacial Electrohydrodynamic Waves: Bifurcation and Secondary Bifurcation","authors":"Guowei Dai, Fei Xu, Yong Zhang","doi":"10.1007/s00332-024-10085-9","DOIUrl":"https://doi.org/10.1007/s00332-024-10085-9","url":null,"abstract":"<p>In this paper, we consider two-dimensional periodic capillary-gravity waves traveling under the influence of a vertical electric field. The full system is a nonlinear, two-layered, free boundary problem. The interface dynamics are derived by coupling Euler equations for the velocity field of the fluid with voltage potential equations governing the electric field. We first introduce the naive flattening technique to transform the free boundary problem into a fixed boundary problem. We then prove the existence of small-amplitude electrohydrodynamic waves with constant vorticity using local bifurcation theory. Moreover, we show that these electrohydrodynamic waves are formally stable in the linearized sense. Furthermore, we obtain a secondary bifurcation curve that emerges from the primary branch, consisting of ripple solutions on the interface. As far as we know, such solutions in electrohydrodynamics are established for the first time. It is worth noting that the electric field <span>(E_0)</span> plays a key role in controlling the shapes and types of waves on the interface.</p>","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":"8 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206263","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
By introducing bilinear operators of trigonometric type, we propose several novel integrable variants of the famous Toda lattice, two of which can be regarded as integrable discretizations of the Kadomtsev–Petviashvili equation—a universal model describing weakly nonlinear waves in media with dispersion of velocity. We also demonstrate that two one-dimensional reductions of these variants can approximate the nonlinear Schrödinger equation and a generalized nonlinear Schrödinger equation well. It turns out that these equations admit meaningful solutions including solitons, breathers, lumps and rogue waves, which are expressed in terms of explicit and closed forms. In particular, it seems to be the first time that rogue wave solutions have been obtained for Toda-type equations. Furthermore, g-periodic wave solutions are also produced in terms of Riemann theta function. An approximation solution of the three-periodic wave is successfully carried out by using a deep neural network. The introduction of trigonometric-type bilinear operators is also efficient in generating new variants together with rich properties for some other integrable equations.
{"title":"Integrable Variants of the Toda Lattice","authors":"Ya-Jie Liu, Hui Alan Wang, Xiang-Ke Chang, Xing-Biao Hu, Ying-Nan Zhang","doi":"10.1007/s00332-024-10072-0","DOIUrl":"https://doi.org/10.1007/s00332-024-10072-0","url":null,"abstract":"<p>By introducing bilinear operators of trigonometric type, we propose several novel integrable variants of the famous Toda lattice, two of which can be regarded as integrable discretizations of the Kadomtsev–Petviashvili equation—a universal model describing weakly nonlinear waves in media with dispersion of velocity. We also demonstrate that two one-dimensional reductions of these variants can approximate the nonlinear Schrödinger equation and a generalized nonlinear Schrödinger equation well. It turns out that these equations admit meaningful solutions including solitons, breathers, lumps and rogue waves, which are expressed in terms of explicit and closed forms. In particular, it seems to be the first time that rogue wave solutions have been obtained for Toda-type equations. Furthermore, <i>g</i>-periodic wave solutions are also produced in terms of Riemann theta function. An approximation solution of the three-periodic wave is successfully carried out by using a deep neural network. The introduction of trigonometric-type bilinear operators is also efficient in generating new variants together with rich properties for some other integrable equations.</p>","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":"71 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206281","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-28DOI: 10.1007/s00332-024-10077-9
Andrew R. Axelsen, Courtney R. Quinn, Andrew P. Bassom
We consider a coupling of the Stommel box model and the Lorenz model, with the goal of investigating the so-called crises that are known to occur given sufficient forcing. In this context, a crisis is characterized as the destruction of a chaotic attractor under a critical forcing strength. We document the variety of chaotic attractors and crises possible in our model, focusing on the parameter region where the Lorenz model is always chaotic and where bistability exists in the Stommel box model. The chaotic saddle collisions that occur in a boundary crisis are visualized, with the chaotic saddle computed using the Saddle-Straddle Algorithm. We identify a novel sub-type of boundary crisis, namely a vanishing basin crisis. For forcing strength beyond the crisis, we demonstrate the possibility of a merging between the persisting chaotic attractor and either a chaotic transient or a ghost attractor depending on the type of boundary crisis. An investigation of the finite-time Lyapunov exponents around crisis levels of forcing reveals a convergence between two near-neutral exponents, particularly at points of a trajectory most sensitive to divergence. This points to loss of hyperbolicity associated with crisis occurrence. Finally, we generalize our findings by coupling the Stommel box model to other strange attractors and thereby show that the behaviors are quite generic and robust.
{"title":"Finite-Time Analysis of Crises in a Chaotically Forced Ocean Model","authors":"Andrew R. Axelsen, Courtney R. Quinn, Andrew P. Bassom","doi":"10.1007/s00332-024-10077-9","DOIUrl":"https://doi.org/10.1007/s00332-024-10077-9","url":null,"abstract":"<p>We consider a coupling of the Stommel box model and the Lorenz model, with the goal of investigating the so-called crises that are known to occur given sufficient forcing. In this context, a crisis is characterized as the destruction of a chaotic attractor under a critical forcing strength. We document the variety of chaotic attractors and crises possible in our model, focusing on the parameter region where the Lorenz model is always chaotic and where bistability exists in the Stommel box model. The chaotic saddle collisions that occur in a boundary crisis are visualized, with the chaotic saddle computed using the Saddle-Straddle Algorithm. We identify a novel sub-type of boundary crisis, namely a vanishing basin crisis. For forcing strength beyond the crisis, we demonstrate the possibility of a merging between the persisting chaotic attractor and either a chaotic transient or a ghost attractor depending on the type of boundary crisis. An investigation of the finite-time Lyapunov exponents around crisis levels of forcing reveals a convergence between two near-neutral exponents, particularly at points of a trajectory most sensitive to divergence. This points to loss of hyperbolicity associated with crisis occurrence. Finally, we generalize our findings by coupling the Stommel box model to other strange attractors and thereby show that the behaviors are quite generic and robust.</p>","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":"185 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226303","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}