We extend the Riemann–Hilbert (RH) method to study the Fokas–Lenells (FL) equation with nonzero boundary conditions at infinity and successfully find its multiple soliton solutions with one high-order pole and N high-order poles. The mathematical structures of the FL equation are constructed, including global conservation laws and local conservation laws. Then, the conditions (analytic, symmetric, and asymptotic properties) needed to construct the RH problem are obtained by analyzing the spectral problem. The reflection coefficient r(z) with two cases appearing in the RH problem is considered, including one high-order pole and N high-order poles. In order to overcome the difficulty of establishing the residue expressions corresponding to high-order poles, we introduce the generalized residue formula. Finally, the expression of exact soliton solutions with reflectionless potential is further derived by a closed algebraic system.
{"title":"Riemann–Hilbert problem for the Fokas–Lenells equation in the presence of high-order discrete spectrum with non-vanishing boundary conditions","authors":"Xiao-fan Zhang, Shou‐Fu Tian","doi":"10.1063/5.0097122","DOIUrl":"https://doi.org/10.1063/5.0097122","url":null,"abstract":"We extend the Riemann–Hilbert (RH) method to study the Fokas–Lenells (FL) equation with nonzero boundary conditions at infinity and successfully find its multiple soliton solutions with one high-order pole and N high-order poles. The mathematical structures of the FL equation are constructed, including global conservation laws and local conservation laws. Then, the conditions (analytic, symmetric, and asymptotic properties) needed to construct the RH problem are obtained by analyzing the spectral problem. The reflection coefficient r(z) with two cases appearing in the RH problem is considered, including one high-order pole and N high-order poles. In order to overcome the difficulty of establishing the residue expressions corresponding to high-order poles, we introduce the generalized residue formula. Finally, the expression of exact soliton solutions with reflectionless potential is further derived by a closed algebraic system.","PeriodicalId":50141,"journal":{"name":"Journal of Mathematical Physics Analysis Geometry","volume":"47 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87344866","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Yang Liu, Byungsoo Moon, Vicentiu D. Rădulescu, Runzhang Xu, Chao Yang
This paper is concerned with the initial boundary value problem for viscoelastic Kirchhoff-like plate equations with rotational inertia, memory, p-Laplacian restoring force, weak damping, strong damping, and nonlinear source terms. We establish the local existence and uniqueness of the solution by linearization and the contraction mapping principle. Then, we obtain the global existence of solutions with subcritical and critical initial energy by applying potential well theory. Then, we prove the asymptotic behavior of the global solution with positive initial energy strictly below the depth of the potential well. Finally, we conduct a comprehensive study on the finite time blow-up of solutions with negative initial energy, null initial energy, and positive initial energy strictly below the depth of the potential well and arbitrary positive initial energy, respectively.
{"title":"Qualitative properties of solution to a viscoelastic Kirchhoff-like plate equation","authors":"Yang Liu, Byungsoo Moon, Vicentiu D. Rădulescu, Runzhang Xu, Chao Yang","doi":"10.1063/5.0149240","DOIUrl":"https://doi.org/10.1063/5.0149240","url":null,"abstract":"This paper is concerned with the initial boundary value problem for viscoelastic Kirchhoff-like plate equations with rotational inertia, memory, p-Laplacian restoring force, weak damping, strong damping, and nonlinear source terms. We establish the local existence and uniqueness of the solution by linearization and the contraction mapping principle. Then, we obtain the global existence of solutions with subcritical and critical initial energy by applying potential well theory. Then, we prove the asymptotic behavior of the global solution with positive initial energy strictly below the depth of the potential well. Finally, we conduct a comprehensive study on the finite time blow-up of solutions with negative initial energy, null initial energy, and positive initial energy strictly below the depth of the potential well and arbitrary positive initial energy, respectively.","PeriodicalId":50141,"journal":{"name":"Journal of Mathematical Physics Analysis Geometry","volume":"19 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82881323","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the existence and Hölder continuity conditions for the higher-order derivative of self-intersection local time for u(t, x), where u(t,x)=u1(t,x),…,ud(t,x) is the solution to a system of non-linear stochastic heat equations driven by a d-dimensional space-time white noise. Moreover, we study the cases of intersection local time and collision local time.
{"title":"Self-intersection local time derivative for systems of non-linear stochastic heat equations","authors":"Qian Yu","doi":"10.1063/5.0117488","DOIUrl":"https://doi.org/10.1063/5.0117488","url":null,"abstract":"We consider the existence and Hölder continuity conditions for the higher-order derivative of self-intersection local time for u(t, x), where u(t,x)=u1(t,x),…,ud(t,x) is the solution to a system of non-linear stochastic heat equations driven by a d-dimensional space-time white noise. Moreover, we study the cases of intersection local time and collision local time.","PeriodicalId":50141,"journal":{"name":"Journal of Mathematical Physics Analysis Geometry","volume":"68 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80597060","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is devoted to a class of singularly perturbed nonlinear Schrödinger systems defined on a smooth bounded domain in RN(N=2,3). We use the Lyapunov–Schmidt reduction method to construct synchronized vector solutions with multiple spikes both on the boundary and in the interior of the domain. For each vector solution that has been constructed, we point out that the interior spikes locate near sphere packing points in the domain, and the boundary spikes locate near the critical points of the mean curvature function related to the boundary of the domain.
{"title":"Multiple mixed interior and boundary peaks synchronized solutions for nonlinear coupled elliptic systems","authors":"Z. Tang, Lushun Wang, Huafei Xie","doi":"10.1063/5.0120617","DOIUrl":"https://doi.org/10.1063/5.0120617","url":null,"abstract":"This paper is devoted to a class of singularly perturbed nonlinear Schrödinger systems defined on a smooth bounded domain in RN(N=2,3). We use the Lyapunov–Schmidt reduction method to construct synchronized vector solutions with multiple spikes both on the boundary and in the interior of the domain. For each vector solution that has been constructed, we point out that the interior spikes locate near sphere packing points in the domain, and the boundary spikes locate near the critical points of the mean curvature function related to the boundary of the domain.","PeriodicalId":50141,"journal":{"name":"Journal of Mathematical Physics Analysis Geometry","volume":"48 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81002476","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article investigates a class of double phase problem with L1 data. Some new criteria to guarantee that the existence and uniqueness of non-negative renormalized solutions for the considered problem are established by using the approximation and energy methods.
研究一类具有L1数据的双相问题。利用逼近法和能量法，建立了该问题非负重整化解存在唯一性的新判据。
{"title":"Renormalized non-negative solutions for the double phase Dirichlet problems with L1 data","authors":"B. Ge, Qing-Hai Cao, Yu Zhang","doi":"10.1063/5.0145741","DOIUrl":"https://doi.org/10.1063/5.0145741","url":null,"abstract":"This article investigates a class of double phase problem with L1 data. Some new criteria to guarantee that the existence and uniqueness of non-negative renormalized solutions for the considered problem are established by using the approximation and energy methods.","PeriodicalId":50141,"journal":{"name":"Journal of Mathematical Physics Analysis Geometry","volume":"116 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87727175","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, we investigate the Cauchy problem for the pressureless Euler–Navier–Stokes system in R3. We first establish the global small solutions of this system with critical regularity and then obtain the optimal time decay rate of the solutions by a suitable energy argument (independent of the spectral analysis). The proof crucially depends on non-standard product estimates and interpolations. In comparison with previous studies about time-decay by Choi and Jung [J. Math. Fluid Mech. 23, 99 (2021); arXiv:2112.14449], the smallness requirement of the low frequencies of initial data could be removed.
{"title":"Optimal well-posedness for the pressureless Euler–Navier–Stokes system","authors":"Xiaoping Zhai, Yiren Chen, Yongsheng Li, Yongye Zhao","doi":"10.1063/5.0136429","DOIUrl":"https://doi.org/10.1063/5.0136429","url":null,"abstract":"In this work, we investigate the Cauchy problem for the pressureless Euler–Navier–Stokes system in R3. We first establish the global small solutions of this system with critical regularity and then obtain the optimal time decay rate of the solutions by a suitable energy argument (independent of the spectral analysis). The proof crucially depends on non-standard product estimates and interpolations. In comparison with previous studies about time-decay by Choi and Jung [J. Math. Fluid Mech. 23, 99 (2021); arXiv:2112.14449], the smallness requirement of the low frequencies of initial data could be removed.","PeriodicalId":50141,"journal":{"name":"Journal of Mathematical Physics Analysis Geometry","volume":"78 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84659736","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this study, we study the solution of a semilinear elliptic equation with nonlocal nonlinearities. By using the mountain pass theorem, Hölder’s inequality, and Sobolev embedding theorem, we obtain the existence result, establish the Liouville type theorem, and consider Morse indices of the equation.
{"title":"Liouville type theorem and Morse indices of a semilinear elliptic equation with nonlocal nonlinearities","authors":"Xiaowei An, Huixia He, Xianfa Song","doi":"10.1063/5.0091269","DOIUrl":"https://doi.org/10.1063/5.0091269","url":null,"abstract":"In this study, we study the solution of a semilinear elliptic equation with nonlocal nonlinearities. By using the mountain pass theorem, Hölder’s inequality, and Sobolev embedding theorem, we obtain the existence result, establish the Liouville type theorem, and consider Morse indices of the equation.","PeriodicalId":50141,"journal":{"name":"Journal of Mathematical Physics Analysis Geometry","volume":"105 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90967691","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is concerned with the pathwise dynamics of the stochastic evolution equation: du + Audt = F(u)dt + G(u)dW(t) on a separable Hilbert space H with the Lipschitz continuous drift term F(u) as well as the Lipschitz continuous diffusion term G(u). We first introduce the notion of generalized random dynamical systems (GRDSs) and show that the equation can generate a GRDS. We then construct a pathwise unstable manifold for the GRDS provided that the Lipschitz constants of the drift term and the diffusion term satisfy a spectral gap condition. At last, we present a pathwise unstable manifold reduction for the GRDS.
{"title":"Pathwise unstable invariant manifolds reduction for stochastic evolution equations driven by nonlinear noise","authors":"Xuewei Ju","doi":"10.1063/5.0101516","DOIUrl":"https://doi.org/10.1063/5.0101516","url":null,"abstract":"This paper is concerned with the pathwise dynamics of the stochastic evolution equation: du + Audt = F(u)dt + G(u)dW(t) on a separable Hilbert space H with the Lipschitz continuous drift term F(u) as well as the Lipschitz continuous diffusion term G(u). We first introduce the notion of generalized random dynamical systems (GRDSs) and show that the equation can generate a GRDS. We then construct a pathwise unstable manifold for the GRDS provided that the Lipschitz constants of the drift term and the diffusion term satisfy a spectral gap condition. At last, we present a pathwise unstable manifold reduction for the GRDS.","PeriodicalId":50141,"journal":{"name":"Journal of Mathematical Physics Analysis Geometry","volume":"45 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86526386","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we will show that the Dubrovin–Frobenius prepotentials on the orbit space of the Coxeter group Bn constructed by Arsie et al. [Sel. Math. New Ser. 29, 1 (2023)] coincide with the solutions of Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations associated with the constrained KP hierarchy.
{"title":"Dubrovin–Frobenius manifolds associated with Bn and the constrained KP hierarchy","authors":"Shilin Ma, Dafeng Zuo","doi":"10.1063/5.0142578","DOIUrl":"https://doi.org/10.1063/5.0142578","url":null,"abstract":"In this paper, we will show that the Dubrovin–Frobenius prepotentials on the orbit space of the Coxeter group Bn constructed by Arsie et al. [Sel. Math. New Ser. 29, 1 (2023)] coincide with the solutions of Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations associated with the constrained KP hierarchy.","PeriodicalId":50141,"journal":{"name":"Journal of Mathematical Physics Analysis Geometry","volume":"38 9-10","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72608353","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study the boundedness of atmospheric Ekman flows with classical boundary conditions. We consider the system with a two-layer eddy viscosity, consisting of a constant eddy viscosity in the upper layer and a continuous eddy viscosity in the lower layer. We analyze the boundedness of the solution by using the logarithmic matrix norm.
{"title":"Boundedness of atmospheric Ekman flows with two-layer eddy viscosity","authors":"Y. Guan","doi":"10.1063/5.0142172","DOIUrl":"https://doi.org/10.1063/5.0142172","url":null,"abstract":"In this paper, we study the boundedness of atmospheric Ekman flows with classical boundary conditions. We consider the system with a two-layer eddy viscosity, consisting of a constant eddy viscosity in the upper layer and a continuous eddy viscosity in the lower layer. We analyze the boundedness of the solution by using the logarithmic matrix norm.","PeriodicalId":50141,"journal":{"name":"Journal of Mathematical Physics Analysis Geometry","volume":"3 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74346333","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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