SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 4084-4103, June 2024. Abstract. We consider Schrödinger operators at a fixed high frequency on simply connected compact Riemannian manifolds with nonpositive sectional curvatures and smooth strictly convex boundaries. We prove that the Dirichlet-to-Neumann map uniquely determines the potential.
{"title":"The Anisotropic Calderón Problem for High Fixed Frequency","authors":"Gunther Uhlmann, Yiran Wang","doi":"10.1137/23m1579029","DOIUrl":"https://doi.org/10.1137/23m1579029","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 4084-4103, June 2024. <br/> Abstract. We consider Schrödinger operators at a fixed high frequency on simply connected compact Riemannian manifolds with nonpositive sectional curvatures and smooth strictly convex boundaries. We prove that the Dirichlet-to-Neumann map uniquely determines the potential.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141259646","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}
Juan Pablo Borthagaray, Wenbo Li, Ricardo H. Nochetto
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 4006-4039, June 2024. Abstract. We prove Besov boundary regularity for solutions of the homogeneous Dirichlet problem for fractional-order quasi-linear operators with variable coefficients on Lipschitz domains [math] of [math]. Our estimates are consistent with the boundary behavior of solutions on smooth domains and apply to fractional [math]-Laplacians and operators with finite horizon. The proof exploits the underlying variational structure and uses a new and flexible local translation operator. We further apply these regularity estimates to derive novel error estimates for finite element approximations of fractional [math]-Laplacians and present several simulations that reveal the boundary behavior of solutions.
{"title":"Quasi-linear Fractional-Order Operators in Lipschitz Domains","authors":"Juan Pablo Borthagaray, Wenbo Li, Ricardo H. Nochetto","doi":"10.1137/23m1575871","DOIUrl":"https://doi.org/10.1137/23m1575871","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 4006-4039, June 2024. <br/> Abstract. We prove Besov boundary regularity for solutions of the homogeneous Dirichlet problem for fractional-order quasi-linear operators with variable coefficients on Lipschitz domains [math] of [math]. Our estimates are consistent with the boundary behavior of solutions on smooth domains and apply to fractional [math]-Laplacians and operators with finite horizon. The proof exploits the underlying variational structure and uses a new and flexible local translation operator. We further apply these regularity estimates to derive novel error estimates for finite element approximations of fractional [math]-Laplacians and present several simulations that reveal the boundary behavior of solutions.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141549361","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}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 4040-4083, June 2024. Abstract. This paper is concerned with the principal spectral theory of time-periodic cooperative systems with nonlocal dispersal and Neumann boundary condition. First we present a sufficient condition for the existence of principal eigenvalues by using the theory of resolvent positive operators with their perturbations. Then we establish the monotonicity of principal eigenvalues with respect to the frequency and investigate the limiting properties of principal eigenvalues as the frequency tends to zero or infinity. We also study the effects of dispersal rates and dispersal ranges on the principal eigenvalues, and the difficulty is that principal eigenvalues of time-periodic cooperative systems with Neumann boundary conditions are not monotone with respect to the domain. Finally, we apply our theory to a man-environment-man epidemic model and consider the impacts of dispersal rates, frequency, and dispersal ranges on the basic reproduction number and positive time-periodic solutions.
{"title":"Principal Spectral Theory of Time-Periodic Nonlocal Dispersal Cooperative Systems and Applications","authors":"Yan-Xia Feng, Wan-Tong Li, Shigui Ruan, Ming-Zhen Xin","doi":"10.1137/22m1543902","DOIUrl":"https://doi.org/10.1137/22m1543902","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 4040-4083, June 2024. <br/> Abstract. This paper is concerned with the principal spectral theory of time-periodic cooperative systems with nonlocal dispersal and Neumann boundary condition. First we present a sufficient condition for the existence of principal eigenvalues by using the theory of resolvent positive operators with their perturbations. Then we establish the monotonicity of principal eigenvalues with respect to the frequency and investigate the limiting properties of principal eigenvalues as the frequency tends to zero or infinity. We also study the effects of dispersal rates and dispersal ranges on the principal eigenvalues, and the difficulty is that principal eigenvalues of time-periodic cooperative systems with Neumann boundary conditions are not monotone with respect to the domain. Finally, we apply our theory to a man-environment-man epidemic model and consider the impacts of dispersal rates, frequency, and dispersal ranges on the basic reproduction number and positive time-periodic solutions.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141259429","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}
Erik Orvehed Hiltunen, Joseph Kraisler, John C. Schotland, Michael I. Weinstein
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3802-3831, June 2024. Abstract. We consider the quantum optics of a single photon interacting with a system of two-level atoms. The wave properties of this interacting system are determined by the spectral properties of a matrix Hamiltonian, involving a nonlocal partial differential operator, acting on photonic and atomic degrees of freedom. We study the spectral problem via a reduction to a spectral problem for a scalar nonlocal operator, which depends nonlinearly on the spectral parameter. We investigate two classes of solutions: Bound states are solutions that decay at infinity, while resonance states have locally finite energy and satisfy a non–self-adjoint outgoing radiation condition at infinity. We have found necessary and sufficient conditions for the existence of bound states, along with an upper bound on the number of such states. We have also considered these problems for atomic models with small, high-contrast inclusions. In this setting, we have derived asymptotic formulas for the resonances. Our results are illustrated with numerical computations.
{"title":"Nonlocal Partial Differential Equations and Quantum Optics: Bound States and Resonances","authors":"Erik Orvehed Hiltunen, Joseph Kraisler, John C. Schotland, Michael I. Weinstein","doi":"10.1137/23m158142x","DOIUrl":"https://doi.org/10.1137/23m158142x","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3802-3831, June 2024. <br/> Abstract. We consider the quantum optics of a single photon interacting with a system of two-level atoms. The wave properties of this interacting system are determined by the spectral properties of a matrix Hamiltonian, involving a nonlocal partial differential operator, acting on photonic and atomic degrees of freedom. We study the spectral problem via a reduction to a spectral problem for a scalar nonlocal operator, which depends nonlinearly on the spectral parameter. We investigate two classes of solutions: Bound states are solutions that decay at infinity, while resonance states have locally finite energy and satisfy a non–self-adjoint outgoing radiation condition at infinity. We have found necessary and sufficient conditions for the existence of bound states, along with an upper bound on the number of such states. We have also considered these problems for atomic models with small, high-contrast inclusions. In this setting, we have derived asymptotic formulas for the resonances. Our results are illustrated with numerical computations.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141259527","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}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3861-3885, June 2024. Abstract. In this paper, we study the following first order Hamiltonian systems: [math] where [math], [math], [math] arises as the Lagrange multiplier, and [math] are [math] real matrices with [math]. Using the multiplicity theorem of Ljusternik–Schnirelmann together with variational methods, we show the existence of multiple normalized homoclinic solutions for this problem. We deal with not only the case det[math] for all [math] in a set of nonzero measure, but also the case det[math] for all [math]. In particular, we also obtain bifurcation results of this problem.
{"title":"Multiple Normalized Solutions for First Order Hamiltonian Systems","authors":"Yuxia Guo, Yuanyang Yu","doi":"10.1137/23m1584575","DOIUrl":"https://doi.org/10.1137/23m1584575","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3861-3885, June 2024. <br/> Abstract. In this paper, we study the following first order Hamiltonian systems: [math] where [math], [math], [math] arises as the Lagrange multiplier, and [math] are [math] real matrices with [math]. Using the multiplicity theorem of Ljusternik–Schnirelmann together with variational methods, we show the existence of multiple normalized homoclinic solutions for this problem. We deal with not only the case det[math] for all [math] in a set of nonzero measure, but also the case det[math] for all [math]. In particular, we also obtain bifurcation results of this problem.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141259419","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}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3832-3860, June 2024. Abstract. We are concerned with the critical Choquard equation [math] where [math], [math] is the Riesz potential with order [math], and the exponent [math] is critical with respect to the Hardy–Littlewood–Sobolev inequality. By combining the variational gluing method and a penalization technique, for every [math], we prove the existence of infinitely many [math]-bump positive solutions for this nonlocal equation exhibiting a polynomial decay at infinity if the potential [math] is periodic in one of its variables and permits a global maxima with a fast decay rate near the maximum point. Our results demonstrate the nonlocal features of the Choquard equation and do not depend on the uniqueness or nondegeneracy property of positive solutions, which is in contrast to the results of the local Yamabe equation.
{"title":"Multibump Solutions for Critical Choquard Equation","authors":"Jiankang Xia, Xu Zhang","doi":"10.1137/23m1581820","DOIUrl":"https://doi.org/10.1137/23m1581820","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3832-3860, June 2024. <br/> Abstract. We are concerned with the critical Choquard equation [math] where [math], [math] is the Riesz potential with order [math], and the exponent [math] is critical with respect to the Hardy–Littlewood–Sobolev inequality. By combining the variational gluing method and a penalization technique, for every [math], we prove the existence of infinitely many [math]-bump positive solutions for this nonlocal equation exhibiting a polynomial decay at infinity if the potential [math] is periodic in one of its variables and permits a global maxima with a fast decay rate near the maximum point. Our results demonstrate the nonlocal features of the Choquard equation and do not depend on the uniqueness or nondegeneracy property of positive solutions, which is in contrast to the results of the local Yamabe equation.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141259451","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}
Elvise Berchio, Denis Bonheure, Giovanni P. Galdi, Filippo Gazzola, Simona Perotto
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3759-3801, June 2024. Abstract. We study the equilibrium configurations for several fluid-structure interaction problems. The fluid is confined in a 2D unbounded channel that contains a body, free to move inside the channel with rigid motions (transversal translations and rotations). The motion of the fluid is generated by a Poiseuille inflow/outflow at infinity and governed by the stationary Navier–Stokes equations. For a model where the fluid is the air and the body represents the cross-section of a suspension bridge, therefore also subject to restoring elastic forces, we prove that for small inflows there exists a unique equilibrium position, while for large inflows we numerically show the appearance of additional equilibria. A similar uniqueness result is also obtained for a discretized 3D bridge, consisting in a finite number of cross-sections interacting with the adjacent ones. The very same model, but without restoring forces, is used to describe the mechanism of the Leonardo da Vinci ferry, which is able to cross a river without engines. We numerically determine the optimal orientation of the ferry that allows it to cross the river in minimal time.
{"title":"Equilibrium Configurations of a Symmetric Body Immersed in a Stationary Navier–Stokes Flow in a Planar Channel","authors":"Elvise Berchio, Denis Bonheure, Giovanni P. Galdi, Filippo Gazzola, Simona Perotto","doi":"10.1137/23m1568752","DOIUrl":"https://doi.org/10.1137/23m1568752","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3759-3801, June 2024. <br/> Abstract. We study the equilibrium configurations for several fluid-structure interaction problems. The fluid is confined in a 2D unbounded channel that contains a body, free to move inside the channel with rigid motions (transversal translations and rotations). The motion of the fluid is generated by a Poiseuille inflow/outflow at infinity and governed by the stationary Navier–Stokes equations. For a model where the fluid is the air and the body represents the cross-section of a suspension bridge, therefore also subject to restoring elastic forces, we prove that for small inflows there exists a unique equilibrium position, while for large inflows we numerically show the appearance of additional equilibria. A similar uniqueness result is also obtained for a discretized 3D bridge, consisting in a finite number of cross-sections interacting with the adjacent ones. The very same model, but without restoring forces, is used to describe the mechanism of the Leonardo da Vinci ferry, which is able to cross a river without engines. We numerically determine the optimal orientation of the ferry that allows it to cross the river in minimal time.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141259338","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}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3886-3923, June 2024. Abstract. We introduce a stochastic version of the Proudman–Taylor model, a 2D-3C fluid approximation of the 3D Navier–Stokes equations, with the small-scale turbulence modeled by a transport-stretching noise. For this model we may rigorously take a scaling limit leading to a deterministic model with additional viscosity on large scales. In certain choice of noises without mirror symmetry, we identify an anisotropic kinetic alpha (AKA) effect. This is the first example with a 3D structure and a stretching noise term.
{"title":"On the Boussinesq Hypothesis for a Stochastic Proudman–Taylor Model","authors":"Franco Flandoli, Dejun Luo","doi":"10.1137/23m1587944","DOIUrl":"https://doi.org/10.1137/23m1587944","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3886-3923, June 2024. <br/> Abstract. We introduce a stochastic version of the Proudman–Taylor model, a 2D-3C fluid approximation of the 3D Navier–Stokes equations, with the small-scale turbulence modeled by a transport-stretching noise. For this model we may rigorously take a scaling limit leading to a deterministic model with additional viscosity on large scales. In certain choice of noises without mirror symmetry, we identify an anisotropic kinetic alpha (AKA) effect. This is the first example with a 3D structure and a stretching noise term.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141259435","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}
Bogdan Raiţă, Angkana Rüland, Camillo Tissot, Antonio Tribuzio
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3720-3758, June 2024. Abstract. We study the scaling behavior of a class of compatible two-well problems for higher order, homogeneous linear differential operators. To this end, we first deduce general lower scaling bounds which are determined by the vanishing order of the symbol of the operator on the unit sphere in the direction of the associated element in the wave cone. We complement the lower bound estimates by a detailed analysis of the two-well problem for generalized (tensor-valued) symmetrized derivatives with the help of the (tensor-valued) Saint-Venant compatibility conditions. In two spatial dimensions for highly symmetric boundary data (but arbitrary tensor order [math]) we provide upper bound constructions matching the lower bound estimates. This illustrates that for the two-well problem for higher order operators new scaling laws emerge which are determined by the Fourier symbol in the direction of the wave cone. The scaling for the symmetrized gradient from [A. Chan and S. Conti, Math. Models Methods Appl. Sci., 25 (2015), pp. 1091–1124] which was also discussed in [B. Raiță, A. Rüland, and C. Tissot, Acta Appl. Math., 184 (2023), 5] provides an example of this family of new scaling laws.
SIAM 数学分析期刊》，第 56 卷第 3 期，第 3720-3758 页，2024 年 6 月。 摘要。我们研究了一类高阶同质线性微分算子的兼容两阱问题的缩放行为。为此，我们首先推导出一般的缩放下界，这些下界由单位球上的算子符号在波锥中相关元素方向上的消失阶决定。我们借助（张量值）Saint-Venant 相容性条件，对广义（张量值）对称导数的两井问题进行了详细分析，从而补充了下限估计。在高度对称边界数据的两个空间维度上（但是任意张量阶[math]），我们提供了与下界估计相匹配的上界构造。这说明，对于高阶算子的双井问题，出现了新的缩放规律，它是由波锥方向上的傅里叶符号决定的。对称梯度的缩放规律来自 [A. Chan and S. Conti, M. Mathematics, 2000]。Chan and S. Conti, Math.Models Methods Appl. Sci., 25 (2015), pp.
{"title":"On Scaling Properties for a Class of Two-Well Problems for Higher Order Homogeneous Linear Differential Operators","authors":"Bogdan Raiţă, Angkana Rüland, Camillo Tissot, Antonio Tribuzio","doi":"10.1137/23m1588287","DOIUrl":"https://doi.org/10.1137/23m1588287","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3720-3758, June 2024. <br/> Abstract. We study the scaling behavior of a class of compatible two-well problems for higher order, homogeneous linear differential operators. To this end, we first deduce general lower scaling bounds which are determined by the vanishing order of the symbol of the operator on the unit sphere in the direction of the associated element in the wave cone. We complement the lower bound estimates by a detailed analysis of the two-well problem for generalized (tensor-valued) symmetrized derivatives with the help of the (tensor-valued) Saint-Venant compatibility conditions. In two spatial dimensions for highly symmetric boundary data (but arbitrary tensor order [math]) we provide upper bound constructions matching the lower bound estimates. This illustrates that for the two-well problem for higher order operators new scaling laws emerge which are determined by the Fourier symbol in the direction of the wave cone. The scaling for the symmetrized gradient from [A. Chan and S. Conti, Math. Models Methods Appl. Sci., 25 (2015), pp. 1091–1124] which was also discussed in [B. Raiță, A. Rüland, and C. Tissot, Acta Appl. Math., 184 (2023), 5] provides an example of this family of new scaling laws.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141190271","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}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3679-3702, June 2024. Abstract. We consider a variant of the Lugiato–Lefever equation (LLE), which is a nonlinear Schrödinger equation on a one-dimensional torus with forcing and damping, to which we add a first-order derivative term with a potential [math]. The potential breaks the translation invariance of LLE. Depending on the existence of zeroes of the effective potential [math], which is a suitably weighted and integrated version of [math], we show that stationary solutions from [math] can be continued locally into the range [math]. Moreover, the extremal points of the [math]-continued solutions are located near zeros of [math]. We therefore call this phenomenon pinning of stationary solutions. If we assume additionally that the starting stationary solution at [math] is spectrally stable with the simple zero eigenvalue due to translation invariance being the only eigenvalue on the imaginary axis, we can prove asymptotic stability or instability of its [math]-continuation depending on the sign of [math] at the zero of [math] and the sign of [math]. The variant of the LLE arises in the description of optical frequency combs in a Kerr nonlinear ring-shaped microresonator which is pumped by two different continuous monochromatic light sources of different frequencies and different powers. Our analytical findings are illustrated by numerical simulations.
{"title":"Pinning in the Extended Lugiato–Lefever Equation","authors":"Lukas Bengel, Dmitry Pelinovsky, Wolfgang Reichel","doi":"10.1137/23m1550700","DOIUrl":"https://doi.org/10.1137/23m1550700","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3679-3702, June 2024. <br/> Abstract. We consider a variant of the Lugiato–Lefever equation (LLE), which is a nonlinear Schrödinger equation on a one-dimensional torus with forcing and damping, to which we add a first-order derivative term with a potential [math]. The potential breaks the translation invariance of LLE. Depending on the existence of zeroes of the effective potential [math], which is a suitably weighted and integrated version of [math], we show that stationary solutions from [math] can be continued locally into the range [math]. Moreover, the extremal points of the [math]-continued solutions are located near zeros of [math]. We therefore call this phenomenon pinning of stationary solutions. If we assume additionally that the starting stationary solution at [math] is spectrally stable with the simple zero eigenvalue due to translation invariance being the only eigenvalue on the imaginary axis, we can prove asymptotic stability or instability of its [math]-continuation depending on the sign of [math] at the zero of [math] and the sign of [math]. The variant of the LLE arises in the description of optical frequency combs in a Kerr nonlinear ring-shaped microresonator which is pumped by two different continuous monochromatic light sources of different frequencies and different powers. Our analytical findings are illustrated by numerical simulations.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141190270","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}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号