Studies in Applied Mathematics最新文献

{"title":"Rigid lid limit in shallow water over a flat bottom","authors":"Benjamin Melinand","doi":"10.1111/sapm.12773","DOIUrl":"https://doi.org/10.1111/sapm.12773","url":null,"abstract":"<p>We perform the so-called rigid lid limit on different shallow water models such as the abcd Bousssinesq systems or the Green–Naghdi equations. To do so, we consider an appropriate nondimensionalization of these models where two small parameters are involved: the shallowness parameter <span></span><math>\u0000 <semantics>\u0000 <mi>μ</mi>\u0000 <annotation>$mu$</annotation>\u0000 </semantics></math> and a parameter <span></span><math>\u0000 <semantics>\u0000 <mi>ε</mi>\u0000 <annotation>$epsilon$</annotation>\u0000 </semantics></math> which can be interpreted as a Froude number. When the parameter <span></span><math>\u0000 <semantics>\u0000 <mi>ε</mi>\u0000 <annotation>$epsilon$</annotation>\u0000 </semantics></math> tends to zero, the surface deformation formally goes to the rest state, hence the name rigid lid limit. We carefully study this limit for different topologies. We also provide rates of convergence with respect to <span></span><math>\u0000 <semantics>\u0000 <mi>ε</mi>\u0000 <annotation>$epsilon$</annotation>\u0000 </semantics></math> and careful attention is given to the dependence on the shallowness parameter <span></span><math>\u0000 <semantics>\u0000 <mi>μ</mi>\u0000 <annotation>$mu$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"153 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.12773","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142642371","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":"Higher-order integrable models for oceanic internal wave–current interactions","authors":"David Henry,&nbsp;Rossen I. Ivanov,&nbsp;Zisis N. Sakellaris","doi":"10.1111/sapm.12778","DOIUrl":"https://doi.org/10.1111/sapm.12778","url":null,"abstract":"<p>In this paper, we derive a higher-order Korteweg–de Vries (HKdV) equation as a model to describe the unidirectional propagation of waves on an internal interface separating two fluid layers of varying densities. Our model incorporates underlying currents by permitting a sheared current in both fluid layers, and also accommodates the effect of the Earth's rotation by including Coriolis forces (restricted to the Equatorial region). The resulting governing equations describing the water wave problem in two fluid layers under a “flat-surface” assumption are expressed in a general form as a system of two coupled equations through Dirichlet–Neumann (DN) operators. The DN operators also facilitate a convenient Hamiltonian formulation of the problem. We then derive the HKdV equation from this Hamiltonian formulation, in the long-wave, and small-amplitude, asymptotic regimes. Finally, it is demonstrated that there is an explicit transformation connecting the HKdV we derive with the following integrable equations of a similar type: KdV5, Kaup–Kuperschmidt equation, Sawada–Kotera equation, and Camassa–Holm and Degasperis–Procesi equations.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"153 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.12778","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142642031","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":"Whitham modulation theory and the classification of solutions to the Riemann problem of the Fokas–Lenells equation","authors":"Zhi-Jia Wu,&nbsp;Shou-Fu Tian","doi":"10.1111/sapm.12779","DOIUrl":"https://doi.org/10.1111/sapm.12779","url":null,"abstract":"<p>In this work, we explore the Riemann problem of the Fokas–Lenells (FL) equation given initial data in the form of a step discontinuity by employing the Whitham modulation theory. The periodic wave solutions of the FL equation are characterized by elliptic functions along with the Whitham modulation equations. Moreover, we find that the <span></span><math>\u0000 <semantics>\u0000 <mo>±</mo>\u0000 <annotation>$pm$</annotation>\u0000 </semantics></math> signs for the velocities of the periodic wave solutions remain unchanged during propagation. Thus, when analyzing the propagation behavior of solutions, it is necessary to separately consider the clockwise (negative velocity) and counterclockwise (positive velocity) cases. In this regard, we present the classification of the solutions to the Riemann problem of the FL equation in both clockwise and counterclockwise cases for the first time.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"153 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142642032","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":"Hydrodynamics of a discrete conservation law","authors":"Patrick Sprenger,&nbsp;Christopher Chong,&nbsp;Emmanuel Okyere,&nbsp;Michael Herrmann,&nbsp;P. G. Kevrekidis,&nbsp;Mark A. Hoefer","doi":"10.1111/sapm.12767","DOIUrl":"https://doi.org/10.1111/sapm.12767","url":null,"abstract":"<p>The Riemann problem for the discrete conservation law <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <msub>\u0000 <mover>\u0000 <mi>u</mi>\u0000 <mo>̇</mo>\u0000 </mover>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mo>+</mo>\u0000 <msubsup>\u0000 <mi>u</mi>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <mn>2</mn>\u0000 </msubsup>\u0000 <mo>−</mo>\u0000 <msubsup>\u0000 <mi>u</mi>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <mn>2</mn>\u0000 </msubsup>\u0000 <mo>=</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$2 dot{u}_n + u^2_{n+1} - u^2_{n-1} = 0$</annotation>\u0000 </semantics></math> is classified using Whitham modulation theory, a quasi-continuum approximation, and numerical simulations. A surprisingly elaborate set of solutions to this simple discrete regularization of the inviscid Burgers' equation is obtained. In addition to discrete analogs of well-known dispersive hydrodynamic solutions—rarefaction waves (RWs) and dispersive shock waves (DSWs)—additional unsteady solution families and finite-time blowup are observed. Two solution types exhibit no known conservative continuum correlates: (i) a counterpropagating DSW and RW solution separated by a symmetric, stationary shock and (ii) an unsteady shock emitting two counterpropagating periodic wavetrains with the same frequency connected to a partial DSW or an RW. Another class of solutions called traveling DSWs, (iii), consists of a partial DSW connected to a traveling wave comprised of a periodic wavetrain with a rapid transition to a constant. Portions of solutions (ii) and (iii) are interpreted as shock solutions of the Whitham modulation equations.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"153 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142642049","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":"Stability of bound states for regularized nonlinear Schrödinger equations","authors":"John Albert,&nbsp;Jack Arbunich","doi":"10.1111/sapm.12780","DOIUrl":"https://doi.org/10.1111/sapm.12780","url":null,"abstract":"<p>We consider the stability of bound-state solutions of a family of regularized nonlinear Schrödinger equations which were introduced by Dumas et al. as models for the propagation of laser beams. Among these bound-state solutions are ground states, which are defined as solutions of a variational problem. We give a sufficient condition for existence and orbital stability of ground states, and use it to verify that ground states exist and are stable over a wider range of nonlinearities than for the nonregularized nonlinear Schrödinger equation. We also give another sufficient and almost necessary condition for stability of general bound states, and show that some stable bound states exist which are not ground states.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"153 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142642033","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":"Weak compactons of nonlinearly dispersive KdV and KP equations","authors":"S. C. Anco,&nbsp;M. L. Gandarias","doi":"10.1111/sapm.12777","DOIUrl":"https://doi.org/10.1111/sapm.12777","url":null,"abstract":"<p>A weak formulation is devised for the <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>K</mi>\u0000 <mo>(</mo>\u0000 <mi>m</mi>\u0000 <mo>,</mo>\u0000 <mi>n</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$K(m,n)$</annotation>\u0000 </semantics></math> equation, which is a nonlinearly dispersive generalization of the gKdV equation  having compacton solutions. With this formulation, explicit weak compacton solutions are derived, including ones that do not exist as classical (strong) solutions. Similar results are obtained for a nonlinearly dispersive generalization of the gKP equation in two dimensions, which possesses line compacton solutions.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"153 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.12777","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142642022","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":"Singularity formation for the relativistic Euler equations of Chaplygin gases in Schwarzschild spacetime","authors":"Yanbo Hu,&nbsp;Houbin Guo","doi":"10.1111/sapm.12775","DOIUrl":"https://doi.org/10.1111/sapm.12775","url":null,"abstract":"<p>We study the formation of singularities of smooth solutions to the relativistic Euler equations of Chaplygin gases in Schwarzschild spacetime. The system is in the spherically symmetric form, and its coefficients and nonhomogeneous terms contain a parameter reflecting the mass of the black hole, which makes it highly nonlinear and complicated. To overcome the influence of the mass parameter of black hole, we introduce a pair of suitable auxiliary variables related to it and derive their characteristic decompositions to establish the estimates of the smooth solution. We show that, for a kind of initial data, the smooth solution develops singularity in finite time and the mass-energy density itself approaches infinity at the blowup point.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"153 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142642149","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 direct approach for solving the cubic Szegö equation","authors":"Yoshimasa Matsuno","doi":"10.1111/sapm.12770","DOIUrl":"https://doi.org/10.1111/sapm.12770","url":null,"abstract":"<p>We study the cubic Szegö equation which is an integrable nonlinear nondispersive and nonlocal evolution equation. In particular, we present a direct approach for obtaining the multiphase and multisoliton solutions as well as a special class of periodic solutions. Our method is substantially different from the existing one which relies mainly on the spectral analysis of the Hankel operator. We show that the cubic Szegö equation can be bilinearized through appropriate dependent variable transformations and then the solutions satisfy a set of bilinear equations. The proof is carried out within the framework of an elementary theory of determinants. Furthermore, we demonstrate that the eigenfunctions associated with the multiphase solutions satisfy the Lax pair for the cubic Szegö equation, providing an alternative proof of the solutions. Last, the eigenvalue problem for a periodic solution is solved exactly to obtain the analytical expressions of the eigenvalues.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"153 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142642110","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 posteriori error estimation for Runge–Kutta discontinuous Galerkin methods for linear hyperbolic problems","authors":"Emmanuil H. Georgoulis,&nbsp;Edward J. C. Hall,&nbsp;Charalambos G. Makridakis","doi":"10.1111/sapm.12772","DOIUrl":"https://doi.org/10.1111/sapm.12772","url":null,"abstract":"<p>A posteriori bounds for the error measured in various norms for a standard second-order explicit-in-time Runge–Kutta discontinuous Galerkin (RKDG) discretization of a one-dimensional (in space) linear transport problem are derived. The proof is based on a novel space-time polynomial reconstruction, hinging on high-order temporal reconstructions for continuous and discontinuous Galerkin time-stepping methods. Of particular interest is the question of error estimation under dynamic mesh modification. The theoretical findings are tested by numerical experiments.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"153 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.12772","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142641404","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":"The hysteretic Aw–Rascle–Zhang model","authors":"Andrea Corli,&nbsp;Haitao Fan","doi":"10.1111/sapm.12769","DOIUrl":"https://doi.org/10.1111/sapm.12769","url":null,"abstract":"<p>A novel hyperbolic system of partial differential equations is introduced to model traffic flows. This system comprises three equations, with two being linearly degenerate; its main feature is the inclusion of a hysteretic term in a generalized Aw–Rascle–Zhang (ARZ) model. First, a maximum principle for the diffusive version of the model is proven. Then, it is demonstrated that a solution to the Riemann problem exists, which is unique among solutions that are monotone in velocity; all waves exploited in the construction have suitable viscous profiles. Through several examples it is shown how, as a consequence of different driving habits, the system can model the decay, emergence, or persistence of stop-and-go waves (a feature that is missing in the ARZ model), and such behavior is characterized by a simple geometric condition. Furthermore, the model allows the study of traffic flows with a mixture of drivers whose hysteresis loops are either clockwise or counterclockwise. In particular, the presence of sufficiently many of the former dampens speed oscillations.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"153 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.12769","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142641668","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}