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Qualitative Theory of Dynamical Systems最新文献

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Normalized Solutions for Schrödinger Equations with Local Superlinear Nonlinearities 具有局部超线性非线性的薛定谔方程的归一化解
IF 1.4 3区 数学
Qualitative Theory of Dynamical Systems Pub Date : 2024-06-18 DOI: 10.1007/s12346-024-01071-3
Qin Xu, Gui-Dong Li, Shengda Zeng
{"title":"Normalized Solutions for Schrödinger Equations with Local Superlinear Nonlinearities","authors":"Qin Xu, Gui-Dong Li, Shengda Zeng","doi":"10.1007/s12346-024-01071-3","DOIUrl":"https://doi.org/10.1007/s12346-024-01071-3","url":null,"abstract":"<p>In this paper, we consider the following Schrödinger equation: </p><span>$$begin{aligned} {left{ begin{array}{ll} -Delta u=sigma f(u) +lambda u, &amp;{}text {in}quad mathbb {R}^{N}, int _{mathbb {R}^{N}}|u|^{2}~textrm{d}x =a, &amp;{} uin H^1(mathbb {R}^{N}), end{array}right. } end{aligned}$$</span><p>where <span>( N ge 3 )</span>, <span>( a&gt;0 )</span>, <span>(sigma &gt;0)</span>, and <span>( lambda in mathbb {R})</span> appears as a Lagrange multiplier. Assume that the nonlinear term <i>f</i> satisfies conditions only in a neighborhood of zero. For <i>f</i> has a subcritical growth, we prove the existence of the positive normalized solution for the equation with sufficiently small <span>(sigma &gt;0)</span>. For <i>f</i> has a supercritical growth, we derive the existence of the positive normalized solution for the equation with <span>(sigma &gt;0)</span> large enough. In addition, we also obtain infinitely many normalized solutions with sufficiently small <span>(sigma &gt;0)</span> for the subcritical case.</p>","PeriodicalId":48886,"journal":{"name":"Qualitative Theory of Dynamical Systems","volume":"153 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141531344","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}
引用次数: 0
Dynamics and Density Function of a Stochastic SICA Model of a Standard Incidence Rate with Ornstein–Uhlenbeck Process 带有 Ornstein-Uhlenbeck 过程的标准发生率随机 SICA 模型的动态和密度函数
IF 1.4 3区 数学
Qualitative Theory of Dynamical Systems Pub Date : 2024-06-17 DOI: 10.1007/s12346-024-01073-1
Zengchao Wu, Daqing Jiang
{"title":"Dynamics and Density Function of a Stochastic SICA Model of a Standard Incidence Rate with Ornstein–Uhlenbeck Process","authors":"Zengchao Wu, Daqing Jiang","doi":"10.1007/s12346-024-01073-1","DOIUrl":"https://doi.org/10.1007/s12346-024-01073-1","url":null,"abstract":"<p>In this paper, we study an SICA model with a standard incidence rate, where the contact rate <span>(beta )</span> is controlled by the Ornstein–Uhlenbeck process. We first prove the existence and uniqueness of the global positive solution, and by constructing an appropriate Lyapunov function, we demonstrate that when <span>(R_0^s &gt; 1)</span>, the system has a stationary distribution. Furthermore, we obtain a concrete expression of the probability density function near the quasi-positive equilibrium point. By constructing another suitable Lyapunov function, we also derive a threshold value <span>(R_0^e)</span> for disease extinction, and when <span>(R_0^e &lt; 1)</span>, the disease extinguishes at an exponential rate. Finally, our conclusions are verified through numerical simulations.</p>","PeriodicalId":48886,"journal":{"name":"Qualitative Theory of Dynamical Systems","volume":"53 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508859","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}
引用次数: 0
Pontryagin Maximum Principle for Fractional Delay Differential Equations and Controlled Weakly Singular Volterra Delay Integral Equations 分式延迟微分方程和受控弱奇异 Volterra 延迟积分方程的庞特里亚金最大原则
IF 1.4 3区 数学
Qualitative Theory of Dynamical Systems Pub Date : 2024-06-11 DOI: 10.1007/s12346-024-01049-1
Jasarat J. Gasimov, Javad A. Asadzade, Nazim I. Mahmudov
{"title":"Pontryagin Maximum Principle for Fractional Delay Differential Equations and Controlled Weakly Singular Volterra Delay Integral Equations","authors":"Jasarat J. Gasimov, Javad A. Asadzade, Nazim I. Mahmudov","doi":"10.1007/s12346-024-01049-1","DOIUrl":"https://doi.org/10.1007/s12346-024-01049-1","url":null,"abstract":"<p>This article explores two distinct issues. To begin with, we analyze the Pontriagin maximum principle concerning fractional delay differential equations. Furthermore, we investigate the most effective method for resolving the control problem associated with Eq. (1.1) and its corresponding payoff function (1.2). Subsequently, we explore the Pontryagin Maximum principle within the framework of Volterra delay integral equations (1.3). We enhance the outcomes of our investigation by presenting illustrative examples towards the conclusion of the article.\u0000</p>","PeriodicalId":48886,"journal":{"name":"Qualitative Theory of Dynamical Systems","volume":"8 5 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141519481","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}
引用次数: 0
Complete Description of Local Conservation Laws for Generalized Dissipative Westervelt Equation 广义耗散韦斯特韦尔特方程局部守恒定律的完整描述
IF 1.4 3区 数学
Qualitative Theory of Dynamical Systems Pub Date : 2024-06-04 DOI: 10.1007/s12346-024-01066-0
Artur Sergyeyev
{"title":"Complete Description of Local Conservation Laws for Generalized Dissipative Westervelt Equation","authors":"Artur Sergyeyev","doi":"10.1007/s12346-024-01066-0","DOIUrl":"https://doi.org/10.1007/s12346-024-01066-0","url":null,"abstract":"<p>We give a complete description of inequivalent nontrivial local conservation laws of all orders for a natural generalization of the dissipative Westervelt equation and, in particular, show that the equation under study admits an infinite number of inequivalent nontrivial local conservation laws for the case of more than two independent variables.</p>","PeriodicalId":48886,"journal":{"name":"Qualitative Theory of Dynamical Systems","volume":"13 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141259640","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}
引用次数: 0
Initial Value and Terminal Value Problems for Distributed Order Fractional Diffusion Equations 分布阶分数扩散方程的初值和终值问题
IF 1.4 3区 数学
Qualitative Theory of Dynamical Systems Pub Date : 2024-06-03 DOI: 10.1007/s12346-024-01061-5
Li Peng, Yong Zhou
{"title":"Initial Value and Terminal Value Problems for Distributed Order Fractional Diffusion Equations","authors":"Li Peng, Yong Zhou","doi":"10.1007/s12346-024-01061-5","DOIUrl":"https://doi.org/10.1007/s12346-024-01061-5","url":null,"abstract":"<p>In this work, we introduce and study two problems for diffusion equations with the distributed order fractional derivatives including the initial value problem and the terminal value problem. For the initial value problem, we establish some existence results and Hölder regularity for the mild solution. On the other hand, we also show the existence results and a decay estimate of the mild solution for the terminal value problems. Especially, the polynomial decay of the solutions to the terminal value problems is firstly included when the source function is equal to zero.</p>","PeriodicalId":48886,"journal":{"name":"Qualitative Theory of Dynamical Systems","volume":"37 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141257375","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}
引用次数: 0
Auto-Bäcklund Transformation and Exact Solutions for a New Integrable (3+1)-dimensional KdV-CBS Equation 一个新的可积分 (3+1)-dimensional KdV-CBS 方程的自贝克伦变换和精确解
IF 1.4 3区 数学
Qualitative Theory of Dynamical Systems Pub Date : 2024-05-30 DOI: 10.1007/s12346-024-01062-4
Xinyue Guo, Lianzhong Li
{"title":"Auto-Bäcklund Transformation and Exact Solutions for a New Integrable (3+1)-dimensional KdV-CBS Equation","authors":"Xinyue Guo, Lianzhong Li","doi":"10.1007/s12346-024-01062-4","DOIUrl":"https://doi.org/10.1007/s12346-024-01062-4","url":null,"abstract":"<p>The Korteweg-de Vries–Calogero–Bogoyavlenskii–Schiff (KdV-CBS) equation is often used in dealing with long-wave propagation interactions, and is widely used in mathematics, physics, and engineering. This paper proposes a new extended (3+1)-dimensional KdV-CBS equation, and it’s never been studied. Additionally, we verify the integrability of the equation based on the Painlevé test. By employing Hirota’s method, a bilinear auto-Bäcklund transformation, the multiple-soliton solutions, and the soliton molecules of the equation are derived. New exact solutions of the equation are constructed utilizing the power series expansion method and <span>((G'/G))</span>-expansion method. These exact solutions are also presented graphically. Finally, the conservation laws of the equation are obtained. Our results are helpful for understanding nonlinear wave phenomena.\u0000</p>","PeriodicalId":48886,"journal":{"name":"Qualitative Theory of Dynamical Systems","volume":"56 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141193548","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}
引用次数: 0
Existence and Asymptotical Behavior of $$L^2$$ -Normalized Standing Wave Solutions to HLS Lower Critical Choquard Equation with a Nonlocal Perturbation 具有非局部扰动的 HLS 下临界 Choquard 方程的 $$L^2$$ 归一化驻波解的存在性和渐近行为
IF 1.4 3区 数学
Qualitative Theory of Dynamical Systems Pub Date : 2024-05-29 DOI: 10.1007/s12346-024-01060-6
Zi-Heng Zhang, Jian-Lun Liu, Hong-Rui Sun
{"title":"Existence and Asymptotical Behavior of $$L^2$$ -Normalized Standing Wave Solutions to HLS Lower Critical Choquard Equation with a Nonlocal Perturbation","authors":"Zi-Heng Zhang, Jian-Lun Liu, Hong-Rui Sun","doi":"10.1007/s12346-024-01060-6","DOIUrl":"https://doi.org/10.1007/s12346-024-01060-6","url":null,"abstract":"<p>This paper is concerned with the following HLS lower critical Choquard equation with a nonlocal perturbation </p><span>$$begin{aligned} {left{ begin{array}{ll} -{Delta }u-bigg (I_alpha *bigg [h|u|^frac{N+alpha }{N}bigg ]bigg )h|u|^{frac{N+alpha }{N}-2}u-mu (I_alpha *|u|^q)|u|^{q-2}u=lambda u text{ in } {mathbb {R}}^N, int _{{mathbb {R}}^N} u^2 dx = c, end{array}right. } end{aligned}$$</span><p>where <span>(alpha in (0,N))</span>, <span>(N ge 3)</span>, <span>(mu , c&gt;0)</span>, <span>(frac{N+alpha }{N}&lt;q&lt;frac{N+alpha +2}{N})</span>, <span>(lambda in {mathbb {R}})</span> is an unknown Lagrange multiplier and <span>(h:{mathbb {R}}^Nrightarrow (0,infty ))</span> is a continuous function. The novelty of this paper is that we not only investigate autonomous case but also handle nonautonomous situation for the above problem. For both cases, we prove the existence and discuss asymptotic behavior of ground state normalized solutions. Compared with the existing references, we extend the recent results obtained by Ye et al. (J Geom Anal 32:242, 2022) to the HLS lower critical case.</p>","PeriodicalId":48886,"journal":{"name":"Qualitative Theory of Dynamical Systems","volume":"18 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141166603","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}
引用次数: 0
A Complement to the Uniqueness of the Limit Cycle of a Family of Systems with Homogeneous Components 同质成分系统族极限循环唯一性的补充
IF 1.4 3区 数学
Qualitative Theory of Dynamical Systems Pub Date : 2024-05-28 DOI: 10.1007/s12346-024-00991-4
Ziwei Zhuang, Changjian Liu
{"title":"A Complement to the Uniqueness of the Limit Cycle of a Family of Systems with Homogeneous Components","authors":"Ziwei Zhuang, Changjian Liu","doi":"10.1007/s12346-024-00991-4","DOIUrl":"https://doi.org/10.1007/s12346-024-00991-4","url":null,"abstract":"<p>Consider the number of limit cycles of a family of systems with homogeneous components: <span>( {dot{x}}=y, {dot{y}}=-x^3+alpha x^2y+y^3. )</span> We show that there is an <span>(alpha ^*&lt;0)</span> such that the system has exactly one limit cycle for <span>(alpha in (alpha ^*,0),)</span> while no limit cycle for the else region. This completes a previous result and also gives a positive answer to the second part of Gasull’s 3rd problem listed in the paper (SeMA J 78(3):233–269, 2021). To obtain this result, we mainly analyse the behavior of the heteroclinic separatrices at infinity.</p>","PeriodicalId":48886,"journal":{"name":"Qualitative Theory of Dynamical Systems","volume":"36 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141173466","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}
引用次数: 0
Extended Lie Method for Mixed Fractional Derivatives, Unconventional Invariants and Reduction, Conservation Laws and Acoustic Waves Propagated via Nonlinear Dispersive Equation 混合分数衍生物的扩展李法、非常规不变式和还原、守恒定律以及通过非线性分散方程传播的声波
IF 1.4 3区 数学
Qualitative Theory of Dynamical Systems Pub Date : 2024-05-27 DOI: 10.1007/s12346-024-01064-2
Rajesh Kumar Gupta, Poonam Yadav
{"title":"Extended Lie Method for Mixed Fractional Derivatives, Unconventional Invariants and Reduction, Conservation Laws and Acoustic Waves Propagated via Nonlinear Dispersive Equation","authors":"Rajesh Kumar Gupta, Poonam Yadav","doi":"10.1007/s12346-024-01064-2","DOIUrl":"https://doi.org/10.1007/s12346-024-01064-2","url":null,"abstract":"<p>This study primarily aims to investigate the application of the Lie symmetry method and conservation law theories in the analysis of mixed fractional partial differential equations where both Riemann–Liouville time-fractional and integer-order <i>x</i>-derivatives are present simultaneously. Specifically, the focus is on the (2+1) dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation. The fractionally modified equation is subjected to invariant analysis using the prolongation formula for mixed derivatives <span>(partial _{t}^{alpha }(u_{x}))</span> and <span>(partial _{t}^{alpha }(u_{xxx}))</span> for the first time. Through the introduction of a novel reduction method, we utilize the Lie symmetry technique to convert the (2+1) dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation into a fractional ordinary differential equation. It’s worth noting that this transformation is carried out without employing the Erdélyi–Kober fractional differential operator. Following this, we introduce a comprehensive expression for deriving conservation laws, involving the notion of nonlinear self-adjointness. Further, two different versatile techniques, the extended Kudryashov method and the Sardar subequation method have been used to extract a wide array of fresh sets of solitary wave solutions encompassing variations like kink, bright, singular kink, and periodic soliton solutions. To provide an intuitive grasp and investigate the ramifications of the fractional derivative parameter on these solitary wave solutions, we conduct a visual exploration employing both 3D and 2D plots.</p>","PeriodicalId":48886,"journal":{"name":"Qualitative Theory of Dynamical Systems","volume":"62 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141166605","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}
引用次数: 0
Well-Posedness of a Class of Fractional Langevin Equations 一类分式朗文方程的好求解性
IF 1.4 3区 数学
Qualitative Theory of Dynamical Systems Pub Date : 2024-05-27 DOI: 10.1007/s12346-024-00956-7
Mi Zhou, Lu Zhang
{"title":"Well-Posedness of a Class of Fractional Langevin Equations","authors":"Mi Zhou, Lu Zhang","doi":"10.1007/s12346-024-00956-7","DOIUrl":"https://doi.org/10.1007/s12346-024-00956-7","url":null,"abstract":"<p>In this work, we deal with a more general form of fractional Langevin equation. The equation’s nonlinearity term <i>f</i> is relevant to fractional integral and fractional derivative. By using the fixed point theorems, we study the existence and uniqueness of solutions of initial value problem for the nonlinear fractional Langevin equation and obtain some new results. Further, by using the technique of nonlinear functional analysis, we study the stability of Ulam-Hyers, Ulam-Hyers-Rassias and semi-Ulam-Hyers-Rassias for the initial value problem of nonlinear Langevin equation. Finally, some examples are given to show the effectiveness of theoretical results.\u0000</p>","PeriodicalId":48886,"journal":{"name":"Qualitative Theory of Dynamical Systems","volume":"49 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141166724","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}
引用次数: 0
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