Studies in Applied Mathematics最新文献

{"title":"Large \u0000 \u0000 x\u0000 $x$\u0000 Asymptotics of the Soliton Gas for the Nonlinear Schrödinger Equation","authors":"Xiaofeng Han,&nbsp;Xiaoen Zhang,&nbsp;Huanhe Dong","doi":"10.1111/sapm.70027","DOIUrl":"https://doi.org/10.1111/sapm.70027","url":null,"abstract":"<div>\u0000 \u0000 <p>In this paper, we construct a Riemann–Hilbert problem of the soliton gas for the nonlinear Schrödinger equation, derived by taking the limit of the <span></span><math>\u0000 <semantics>\u0000 <mi>n</mi>\u0000 <annotation>$n$</annotation>\u0000 </semantics></math> soliton solutions as <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>→</mo>\u0000 <mi>∞</mi>\u0000 </mrow>\u0000 <annotation>$nrightarrow infty$</annotation>\u0000 </semantics></math>. The discrete spectra corresponding to the soliton solutions are located in four disjoint intervals on the imaginary axis, which are symmetric about the real axis. We analyze the large <span></span><math>\u0000 <semantics>\u0000 <mi>x</mi>\u0000 <annotation>$x$</annotation>\u0000 </semantics></math> asymptotics by setting the time variable <span></span><math>\u0000 <semantics>\u0000 <mi>t</mi>\u0000 <annotation>$t$</annotation>\u0000 </semantics></math> to zero. Using the Deift–Zhou nonlinear steepest-descent method, we find that the large <span></span><math>\u0000 <semantics>\u0000 <mi>x</mi>\u0000 <annotation>$x$</annotation>\u0000 </semantics></math> asymptotics at <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 <mo>=</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$t=0$</annotation>\u0000 </semantics></math> behave differently, as <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>x</mi>\u0000 <mo>→</mo>\u0000 <mi>∞</mi>\u0000 </mrow>\u0000 <annotation>$xrightarrow infty$</annotation>\u0000 </semantics></math>, the asymptotics decays to the zero background exponentially, while as <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>x</mi>\u0000 <mo>→</mo>\u0000 <mo>−</mo>\u0000 <mi>∞</mi>\u0000 </mrow>\u0000 <annotation>$xrightarrow -infty$</annotation>\u0000 </semantics></math>, the leading-order term can be expressed with a Riemann-theta function of genus three. In the conclusion, we expand this case to the general <span></span><math>\u0000 <semantics>\u0000 <mi>N</mi>\u0000 <annotation>$N$</annotation>\u0000 </semantics></math> intervals and conjecture on the large <span></span><math>\u0000 <semantics>\u0000 <mi>x</mi>\u0000 <annotation>$x$</annotation>\u0000 </semantics></math> asymptotics.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"154 2","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143466094","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}

{"title":"On a Bistable Delayed Nonlinear Reaction–Diffusion Equation for a Two-Phase Free Boundary: Semi-Wave and Its Numerical Simulation","authors":"Thanh-Hieu Nguyen","doi":"10.1111/sapm.70024","DOIUrl":"https://doi.org/10.1111/sapm.70024","url":null,"abstract":"<p>In this paper, we investigate the existence and uniqueness of the semi-wave solution to a bistable delayed reaction–diffusion equation with a double free boundary. This equation captures key features of biological systems and has broad applications in mathematical biology, including population dynamics, gene expression, virus propagation, and tumor growth. In particular, we employ the technique that was developed in the previous study by M. Alfaro et al. [<i>Proceedings of the London Mathematical Society</i> (3), 116, no. 4 (2018): 729–759], and use it to prove the existence and uniqueness of semi-wave solutions. Our results demonstrate that the semi-wave solutions depend on both the delay parameter and the free boundary condition. Additionally, we conduct numerical simulations to validate our theoretical results, and explore the effects of various parameters on the semi-wave solution and spreading speed.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"154 2","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.70024","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143438817","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Breather and Rogue Wave Solutions on the Different Periodic Backgrounds in the Focusing Nonlinear Schrödinger Equation","authors":"Fang-Cheng Fan,&nbsp;Wang Tang,&nbsp;Guo-Fu Yu","doi":"10.1111/sapm.70026","DOIUrl":"https://doi.org/10.1111/sapm.70026","url":null,"abstract":"<div>\u0000 \u0000 <p>In this paper, we construct breather and rogue wave solutions on the different periodic backgrounds in the focusing nonlinear Schrödinger equation by using the Darboux transformation. First, we present solutions of the Lax pair related to the periodic seed solutions with trivial and nontrivial phases. In this process, different from the previous approaches of employing the nonlinearization of the Lax pair or the traveling wave transformation, we mainly combine the proper assumption with the method of separation of variables. This strategy is more direct and simpler and can be extended to other nonlinear integrable equations. Second, we construct the Kuznetsov–Ma breather and the spatiotemporally periodic breather on the periodic background. Their asymptotic expressions are obtained, which can be used to show that the related nonlinear waves appear on the periodic background. The corresponding dynamical properties and evolution states are illustrated graphically. Finally, at branch points of breathers, the rogue waves on the periodic background are derived and their characteristics are analyzed. For breather and rogue wave solutions, we both investigate the relationship between parameters and solutions' structures and the limits when the elliptic modulus approach to 0 and 1. All the results in this paper might be helpful for us to understand the dynamics of breathers and rogue waves on the periodic background.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"154 2","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143404719","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}

{"title":"Rarefaction Wave Interaction and Existence of a Global Smooth Solution in the Blood Flow Model With Time-Dependent Body Force","authors":"Rakib Mondal,&nbsp; Minhajul","doi":"10.1111/sapm.70025","DOIUrl":"https://doi.org/10.1111/sapm.70025","url":null,"abstract":"<div>\u0000 \u0000 <p>This paper presents the collision between two rarefaction waves for the one-dimensional blood flow model with non-constant body force. We establish the Riemann solutions using phase plane analysis, which are no longer self-similar. By employing Riemann invariants, we transform the system into a non-reducible diagonal system in Riemann invariant coordinates. This interaction problem then becomes a Goursat boundary value problem (GBVP) within the interaction region. We demonstrate that no vacuum forms within the interaction domain if the boundary data on characteristics is vacuum-free, and a vacuum only appears if the two rarefaction waves fail to penetrate each other within a finite time. Furthermore, we prove the existence and uniqueness of the <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>C</mi>\u0000 <mn>1</mn>\u0000 </msup>\u0000 <annotation>$C^1$</annotation>\u0000 </semantics></math> solution to the GBVP throughout the entire interaction region using <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mtext>a priori</mtext>\u0000 <mspace></mspace>\u0000 <msup>\u0000 <mi>C</mi>\u0000 <mn>1</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$text{a priori} ; C^1$</annotation>\u0000 </semantics></math> bounds. Finally, we present the results of the interaction, showing that either the rarefaction waves completely penetrate each other or form a vacuum in the solution at a sufficiently large time during the process of penetration.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"154 2","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143404662","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}

{"title":"Asymptotic Behavior of a Degenerate Forest Kinematic Model With a Perturbation","authors":"Lu LI,&nbsp;Guillaume Cantin","doi":"10.1111/sapm.70014","DOIUrl":"https://doi.org/10.1111/sapm.70014","url":null,"abstract":"<div>\u0000 \u0000 <p>In this paper, we study the asymptotic behavior of the global solutions to a degenerate forest kinematic model, under the action of a perturbation modeling the impact of climate change. In the case where the main nonlinear term of the model is monotone, we prove that the global solutions converge to a stationary solution, by showing that the Lyapunov function derived from the system satisfies a Łojasiewicz–Simon gradient inequality. We also present an original algorithm, based on the Statistical Model Checking framework, to estimate the probability of convergence toward nonconstant equilibria. Furthermore, under suitable assumptions on the parameters, we prove the continuity of the flow and of the stationary solutions with respect to the perturbation parameter. Then, we succeed in proving the robustness of the weak attractors, by considering a weak topology phase space and establishing the existence of a family of positively invariant regions. At last, we present numerical simulations of the model and explore the behavior of the solutions under the effect of several types of perturbations. We also show that the forest kinematic model can lead to the emergence of chaotic patterns.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"154 2","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143389181","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}

{"title":"Detecting (the Absence of) Species Interactions in Microbial Ecological Systems","authors":"Thomas Beardsley,&nbsp;Megan Behringer,&nbsp;Natalia L. Komarova","doi":"10.1111/sapm.70009","DOIUrl":"https://doi.org/10.1111/sapm.70009","url":null,"abstract":"<div>\u0000 \u0000 <p>Microbial communities are complex ecological systems of organisms that evolve in time, with new variants created, while others disappear. Understanding how species interact within communities can help us shed light into the mechanisms that drive ecosystem processes. We studied systems with serial propagation, where the community is kept alive by taking a subsample at regular intervals and replating it in fresh medium. The data that are usually collected consist of the % of the population for each of the species, at several time points. In order to utilize this type of data, we formulated a system of equations (based on the generalized Lotka–Volterra model) and derived conditions of species noninteraction. This was possible to achieve by reformulating the problem as a problem of finding feasibility domains, which can be solved by a number of efficient algorithms. This methodology provides a cost-effective way to investigate interactions in microbial communities.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"154 2","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143380161","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}

{"title":"Smoothing of the Higher-Order Stokes Phenomenon","authors":"Chris J. Howls,&nbsp;John R. King,&nbsp;Gergő Nemes,&nbsp;Adri B. Olde Daalhuis","doi":"10.1111/sapm.70008","DOIUrl":"https://doi.org/10.1111/sapm.70008","url":null,"abstract":"<p>For over a century, the Stokes phenomenon had been perceived as a discontinuous change in the asymptotic representation of a function. In 1989, Berry demonstrated it is possible to smooth this discontinuity in broad classes of problems with the prefactor for the exponentially small contribution switching on/off taking a universal error function form. Following pioneering work of Berk, Nevins, and Roberts and the Japanese school of formally exact asymptotics, the concept of the higher-order Stokes phenomenon was introduced, whereby the ability for the exponentially small terms to cause a Stokes phenomenon may change, depending on the values of parameters in the problem, corresponding to the associated Borel-plane singularities transitioning between Riemann sheets. Until now, the higher-order Stokes phenomenon has also been treated as a discontinuous event. In this paper, we show how the higher-order Stokes phenomenon is also smooth and occurs universally with a prefactor that takes the form of a new special function, based on a Gaussian convolution of an error function. We provide a rigorous derivation of the result, with examples spanning the gamma function, a second-order nonlinear ODE, and the telegraph equation, giving rise to a ghost-like smooth contribution present in the vicinity of a Stokes line, but which rapidly tends to zero on either side. We also include a rigorous derivation of the effect of the smoothed higher-order Stokes phenomenon on the individual terms in the asymptotic series, where the additional contributions appear prefactored by an error function.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"154 2","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.70008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143380162","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Evolution of Dispersal in a Stream With Better Resources at Downstream Locations","authors":"Kuiyue Liu,&nbsp; De Tang,&nbsp;Shanshan Chen","doi":"10.1111/sapm.70017","DOIUrl":"https://doi.org/10.1111/sapm.70017","url":null,"abstract":"<div>\u0000 \u0000 <p>This paper is concerned with a two-species Lotka–Volterra competition patch model over a stream with better resources at downstream locations. Treating one species as the resident species and the other one as a mutant species, we first show that there exist two quantities <span></span><math>\u0000 <semantics>\u0000 <mover>\u0000 <mi>d</mi>\u0000 <mo>¯</mo>\u0000 </mover>\u0000 <annotation>$overline{d}$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <munder>\u0000 <mi>d</mi>\u0000 <mo>̲</mo>\u0000 </munder>\u0000 <annotation>$underline{d}$</annotation>\u0000 </semantics></math> depending on the drift rate: if the dispersal rate of the resident species is smaller (respectively, larger) than <span></span><math>\u0000 <semantics>\u0000 <munder>\u0000 <mi>d</mi>\u0000 <mo>̲</mo>\u0000 </munder>\u0000 <annotation>$underline{d}$</annotation>\u0000 </semantics></math> (respectively, <span></span><math>\u0000 <semantics>\u0000 <mover>\u0000 <mi>d</mi>\u0000 <mo>¯</mo>\u0000 </mover>\u0000 <annotation>$overline{d}$</annotation>\u0000 </semantics></math>), then a rare mutant species can invade only when its dispersal rate is faster (respectively, slower) than the resident species. Then, we show that there exists some intermediate dispersal rate, which is the unique evolutionarily stable strategy for the resident species under certain conditions. Moreover, the global dynamics of the model is obtained, and both competition exclusion and coexistence can occur. Our method for the patch model can be used for the corresponding reaction–diffusion model, and some existing results are improved.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"154 2","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143362444","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}

{"title":"Integrable Hierarchy for Homogeneous Realization of the Toroidal Lie Algebra \u0000 \u0000 \u0000 \u0000 L\u0000 \u0000 r\u0000 +\u0000 1\u0000 \u0000 tor\u0000 \u0000 \u0000 (\u0000 \u0000 sl\u0000 ℓ\u0000 \u0000 )\u0000 \u0000 \u0000 $mathcal {L}^{mathrm{tor}}_{r+1}(mathfrak {sl}_ell)$","authors":"Chao-Zhong Wu,&nbsp;Yi Yang","doi":"10.1111/sapm.70021","DOIUrl":"https://doi.org/10.1111/sapm.70021","url":null,"abstract":"<div>\u0000 \u0000 <p>Starting from a fairly explicit homogeneous realization of the toroidal Lie algebra <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>L</mi>\u0000 <mrow>\u0000 <mi>r</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <mi>tor</mi>\u0000 </msubsup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>sl</mi>\u0000 <mi>ℓ</mi>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$mathcal {L}^{mathrm{tor}}_{r+1}(mathfrak {sl}_ell)$</annotation>\u0000 </semantics></math> via a lattice vertex algebra, we derive an integrable hierarchy of Hirota bilinear equations. Moreover, we represent this hierarchy in the form of Lax equations, and show that it is an extension of a certain reduction of the <span></span><math>\u0000 <semantics>\u0000 <mi>ℓ</mi>\u0000 <annotation>$ell$</annotation>\u0000 </semantics></math>-component KP hierarchy.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"154 2","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143362443","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}

{"title":"Complex Band Structure for Subwavelength Evanescent Waves","authors":"Yannick De Bruijn,&nbsp;Erik Orvehed Hiltunen","doi":"10.1111/sapm.70022","DOIUrl":"https://doi.org/10.1111/sapm.70022","url":null,"abstract":"<div>\u0000 \u0000 <p>We present the mathematical and numerical theory for evanescent waves in subwavelength bandgap materials. We begin in the one-dimensional case, whereby fully explicit formulas for the complex band structure, in terms of the capacitance matrix, are available. As an example, we show that the gap functions can be used to accurately predict the decay rate of the interface mode of a photonic analogue of the Su–Schrieffer–Heeger model. In two dimensions, we derive the bandgap Green's function and characterize the subwavelength gap functions via layer potential techniques. By generalizing existing lattice-summation techniques, we illustrate our results numerically by computing the complex band structure in a variety of settings.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"154 2","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143248817","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}