Studies in Applied Mathematics最新文献

{"title":"N\u0000 $N$\u0000 -Soliton Matrix mKdV Solutions: Some Special Solutions Revisited","authors":"Sandra Carillo,&nbsp;Mauro Lo Schiavo,&nbsp;Cornelia Schiebold","doi":"10.1111/sapm.70061","DOIUrl":"https://doi.org/10.1111/sapm.70061","url":null,"abstract":"<p>In this article, a general solution formula is derived for the <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 <mo>×</mo>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 <annotation>${sf d}times {sf d}$</annotation>\u0000 </semantics></math>-matrix modified Korteweg–de Vries equation. Then, a solution class corresponding to special parameter choices is examined in detail. Roughly, this class can be described as <span></span><math>\u0000 <semantics>\u0000 <mi>N</mi>\u0000 <annotation>$N$</annotation>\u0000 </semantics></math>-solitons (in the sense of Goncharenko) with common phase matrix. It turns out that such a solution even takes values in a <i>commutative</i> subalgebra of the <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 <mo>×</mo>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 <annotation>${sf d}times {sf d}$</annotation>\u0000 </semantics></math>-matrices. We arrive at a rich picture of possibilities for generalized 1-solitons and at visual patterns of <span></span><math>\u0000 <semantics>\u0000 <mi>N</mi>\u0000 <annotation>$N$</annotation>\u0000 </semantics></math>-solitons which combine nonlinear with linear features. The impact of the phase matrix is visualized in computer plots.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"154 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.70061","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144206341","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":"Analyticity and Stable Computation of Dirichlet–Neumann Operators for Laplace's Equation Under Quasiperiodic Boundary Conditions in Two and Three Dimensions","authors":"David P. Nicholls,&nbsp;Jon Wilkening,&nbsp;Xinyu Zhao","doi":"10.1111/sapm.70059","DOIUrl":"https://doi.org/10.1111/sapm.70059","url":null,"abstract":"<p>Dirichlet–Neumann operators (DNOs) are important to the formulation, analysis, and simulation of many crucial models found in engineering and the sciences. For instance, these operators permit moving-boundary problems, such as the classical water wave problem (free-surface ideal fluid flow under the influence of gravity and capillarity), to be restated in terms of interfacial quantities, which not only eliminates the boundary tracking problem, but also reduces the problem's dimension. While these DNOs have been the object of much recent study regarding their numerical simulation and rigorous analysis, they have yet to be examined in the setting of laterally quasiperiodic boundary conditions. The purpose of this contribution is to begin this investigation with a particular focus on the more realistic simulation of two- and three-dimensional free-surface water waves. Here, we not only carefully define the DNO with respect to these boundary conditions for Laplace's equation, but we also show the rigorous analyticity of these operators with respect to sufficiently smooth boundary perturbations. These theoretical developments suggest a novel algorithm for the stable and high-order simulation of the DNO, which we implement and extensively test.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"154 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.70059","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144148446","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":"Zero-Mach Limit of the Viscous Compressible Magnetohydrodynamic Flows Inside a Perfectly Conducting Wall for All Time","authors":"Qiangchang Ju,&nbsp;Zilai Li,&nbsp;Jianjun Xu","doi":"10.1111/sapm.70066","DOIUrl":"https://doi.org/10.1111/sapm.70066","url":null,"abstract":"<div>\u0000 \u0000 <p>We consider the two-dimensional viscous compressible magnetohydrodynamic flows inside a perfectly conducting wall with a nonslip boundary condition. By virtue of the exponential decay of strong solutions to the incompressible system, we establish all-time existence of strong solutions to the initial-boundary value problem for slightly compressible magnetohydrodynamic flows without smallness restrictions on the initial velocity and magnetic field. Furthermore, we prove that solutions of the compressible system uniformly converge to that of the incompressible system for all time as the Mach number approaches zero.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"154 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144140756","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":"A Theory of Generalized Coordinates for Stochastic Differential Equations","authors":"Lancelot Da Costa,&nbsp;Nathaël Da Costa,&nbsp;Conor Heins,&nbsp;Johan Medrano,&nbsp;Grigorios A. Pavliotis,&nbsp;Thomas Parr,&nbsp;Ajith Anil Meera,&nbsp;Karl Friston","doi":"10.1111/sapm.70062","DOIUrl":"https://doi.org/10.1111/sapm.70062","url":null,"abstract":"<p>Stochastic differential equations are ubiquitous modeling tools in applied mathematics and the sciences. In most modeling scenarios, random fluctuations driving dynamics or motion have some nontrivial temporal correlation structure, which renders the SDE non-Markovian; a phenomenon commonly known as ‘colored’’ noise. Thus, an important objective is to develop effective tools for mathematically and numerically studying (possibly non-Markovian) SDEs. In this paper, we formalize a mathematical theory for analyzing and numerically studying SDEs based on so-called “generalized coordinates of motion.” Like the theory of rough paths, we analyze SDEs pathwise for any given realization of the noise, not solely probabilistically. Like the established theory of Markovian realization, we realize non-Markovian SDEs as a Markov process in an extended space. Unlike the established theory of Markovian realization however, the Markovian realizations here are accurate on short timescales and may be exact globally in time, when flows and fluctuations are analytic. This theory is exact for SDEs with analytic flows and fluctuations, and is approximate when flows and fluctuations are differentiable. It provides useful analysis tools, which we employ to solve linear SDEs with analytic fluctuations. It may also be useful for studying rougher SDEs, as these may be identified as the limit of smoother ones. This theory supplies effective, computationally straightforward methods for simulation, filtering and control of SDEs; among others, we rederive generalized Bayesian filtering, a state-of-the-art method for time-series analysis. Looking forward, this paper suggests that generalized coordinates have far-reaching applications throughout stochastic differential equations.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"154 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.70062","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144135723","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":"Mechanical Hamiltonization of Unreduced \u0000 \u0000 ϕ\u0000 $phi$\u0000 -Simple Chaplygin Systems","authors":"Alexandre Anahory Simoes,&nbsp;Juan Carlos Marrero,&nbsp;David Martín de Diego","doi":"10.1111/sapm.70065","DOIUrl":"https://doi.org/10.1111/sapm.70065","url":null,"abstract":"<p>In this paper, we prove that the trajectories of unreduced <span></span><math>\u0000 <semantics>\u0000 <mi>ϕ</mi>\u0000 <annotation>$phi$</annotation>\u0000 </semantics></math>-simple Chaplygin kinetic systems are reparameterizations of horizontal geodesics with respect to a modified Riemannian metric. Furthermore, our proof is constructive and these Riemannian metrics, which are not unique, are obtained explicitly in interesting examples. We also extend these results to <span></span><math>\u0000 <semantics>\u0000 <mi>ϕ</mi>\u0000 <annotation>$phi$</annotation>\u0000 </semantics></math>-simple Chaplygin mechanical systems (not necessarily kinetic).</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"154 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.70065","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144126026","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":"Long-Wavelength Limit of a Quantum Euler–Poisson System in the (3+1) Dimensions for a Dense Magnetoplasma","authors":"Rong Rong,&nbsp;Xueke Pu","doi":"10.1111/sapm.70064","DOIUrl":"https://doi.org/10.1111/sapm.70064","url":null,"abstract":"<div>\u0000 \u0000 <p>This paper presents the derivation of a (3+1)-dimensional quantum Zakharov–Kuznetsov (QZK) equation for ion acoustic waves. Using a singular perturbation method within the long-wavelength limit of the (3+1)-dimensional quantum Euler–Poisson system, we demonstrate that the QZK equation can be systematically derived through the Gardner–Morikawa transformation. The derived equation is valid over a time interval of order <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>O</mi>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mi>ε</mi>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mn>3</mn>\u0000 <mo>/</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$O(varepsilon ^{-3/2})$</annotation>\u0000 </semantics></math>.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"154 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144100674","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 Behaviors of Chandrasekhar Variational Problem for Neutron Stars With Slater-Type Modification","authors":"Deke Li,&nbsp;Qingxuan Wang","doi":"10.1111/sapm.70058","DOIUrl":"https://doi.org/10.1111/sapm.70058","url":null,"abstract":"&lt;div&gt;\u0000 \u0000 &lt;p&gt;In this paper, we consider the Chandrasekhar variational model for neutron stars with defocusing Slater-type modifications. First, we show the existence and nonexistence of the ground state &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ρ&lt;/mi&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$rho _{varepsilon }$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; by concentration–compactness method, and particularly use two auxiliary functions to prove the strongly binding inequality. Second, we characterize perturbation limit behaviors of ground states &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ρ&lt;/mi&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$rho _{varepsilon }$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; as &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;mo&gt;→&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$varepsilon rightarrow 0^+$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and obtain two different blow-up profiles corresponding to two limit equations for &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;mo&gt;∗&lt;/mo&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$N= N_*$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;mo&gt;&gt;&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;mo&gt;∗&lt;/mo&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$N&gt; N_*$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, where &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;annotation&gt;$varepsilon$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a parameter corresponding to Slater-type modifications, and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;mo&gt;∗&lt;/mo&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$N_*$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a threshold value related to the &lt;i&gt;Chandrasekhar limit&lt;/i&gt;. Finally, we study the limit behaviors for &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;mo&gt;≥&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;mo&gt;∗&lt;/mo&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$Nge N_*$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; as &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;m","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"154 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143950103","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 the Coupled Maxwell–Bloch System of Equations With Nondecaying Fields at Infinity","authors":"Sitai Li,&nbsp;Gino Biondini,&nbsp;Gregor Kovačič","doi":"10.1111/sapm.70055","DOIUrl":"https://doi.org/10.1111/sapm.70055","url":null,"abstract":"<div>\u0000 \u0000 <p>We study an initial-boundary-value problem (IBVP) for a system of coupled Maxwell–Bloch equations (CMBE) that model two colors or polarizations of light resonantly interacting with a degenerate, two-level, active optical medium with an excited state and a pair of degenerate ground states. We assume that the electromagnetic field approaches nonvanishing plane waves in the far past and future. This type of interaction has been found to underlie nonlinear optical phenomena including electromagnetically induced transparency, slow light, stopped light, and quantum memory. Under the assumptions of unidirectional, lossless propagation of slowly modulated plane waves, the resulting CMBE become completely integrable in the sense of possessing a Lax pair. In this paper, we formulate an inverse scattering transform (IST) corresponding to these CMBE and their Lax pair, allowing for the spectral line of the atomic transitions in the active medium to have a finite width. The scattering problem for this Lax pair is the same as for the Manakov system. The main advancement in this IST for CMBE is calculating the nontrivial spatial propagation of the spectral data and determining the state of the optical medium in the distant future from that in the distant past, which is needed for the complete formulation of the IBVP. The Riemann–Hilbert problem is used to extract the spatio-temporal dependence of the solution from the evolving spectral data. We further derive and analyze several types of solitons and determine their velocity and stability, as well as find dark states of the medium, which fail to interact with a given pulse.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"154 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143950102","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":"Derivation of the Bacterial Run-and-Tumble Kinetic Model: Quantitative and Strong Convergence Results","authors":"Alain Blaustein","doi":"10.1111/sapm.70060","DOIUrl":"https://doi.org/10.1111/sapm.70060","url":null,"abstract":"<p>During the past century, biologists and mathematicians investigated two mechanisms underlying bacteria motion: the run phase during which bacteria move in straight lines and the tumble phase in which they change their orientation. When surrounded by a chemical attractant, experiments show that bacteria increase their run time as moving up concentration gradients, leading to a biased random walk toward favorable regions. This observation raises the following question, which has drawn intense interest from both biological and mathematical communities: what cellular mechanisms enable bacteria to feel concentration gradients? In this article, we investigate an asymptotic regime that was proposed to explain this ability thanks to internal mechanisms. More precisely, we derive the run-and-tumble kinetic equation with concentration's gradient-dependent tumbling rate from a more comprehensive model, which incorporates internal cellular mechanisms. Our result improves on previous investigations, as we obtain strong convergence toward the gradient-dependent kinetic model with quantitative and formally optimal convergence rates. The main ingredient consists in identifying a set of coordinates for the internal cellular dynamics in which concentration gradients arise explicitly. Then, we use relative entropy methods in order to capture quantitative measurement of the distance between the model incorporating cellular mechanisms and the one with concentration-gradient-dependent tumbling rate.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"154 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.70060","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143944717","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":"Separation of the Initial Conditions in the Inverse Problem for One-Dimensional Nonlinear Tsunami Wave Run-Up Theory","authors":"Alexei Rybkin,&nbsp;Oleksandr Bobrovnikov,&nbsp;Noah Palmer,&nbsp;Daniel Abramowicz,&nbsp;Efim Pelinovsky","doi":"10.1111/sapm.70054","DOIUrl":"https://doi.org/10.1111/sapm.70054","url":null,"abstract":"&lt;div&gt;\u0000 \u0000 &lt;p&gt;We investigate the inverse tsunami wave problem within the framework of the one-dimensional (1D) nonlinear shallow water equations (SWE). Specifically, we show that the initial displacement &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;η&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$eta _0(x)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and velocity &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$u_0(x)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of the wave can be recovered, given the known motion of the shoreline &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$R(t)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; (the wet/dry free boundary), in terms of the Abel transform. We demonstrate that for power-shaped inclined bathymetries, this problem admits a complete solution for any &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;η&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$eta _0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$u_0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, provided the wave does not break.&lt;/p&gt;\u0000 &lt;p&gt;It is important to note that, in contrast to the direct problem (also known as the tsunami wave run-up problem), where &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$R(t)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; can be computed exactly only for &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"154 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143919783","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}