The Experts below are selected from a list of 2715 Experts worldwide ranked by ideXlab platform
Wolfgang Reichel - One of the best experts on this subject based on the ideXlab platform.
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Ground states of a nonlinear curl-curl problem in cylindrically symmetric media
Nonlinear Differential Equations and Applications NoDEA, 2016Co-Authors: Thomas Bartsch, Tomáš Dohnal, Michael Plum, Wolfgang ReichelAbstract:We consider the nonlinear curl-curl problem \({\Nabla\times\Nabla\times U + V(x) U= \Gamma(x)|U|^{p-1}U}\) in \({\mathbb{R}^3}\) related to the Kerr nonlinear Maxwell equations for fully localized monochromatic fields. We search for solutions as minimizers (ground states) of the corresponding energy functional defined on subspaces (defocusing case) or natural constraints (focusing case) of \({H({\rm curl};\mathbb{R}^3)}\). Under a cylindrical symmetry assumption corresponding to a photonic fiber geometry on the functions V and \({\Gamma}\) the variational problem can be posed in a symmetric subspace of \({H({\rm curl};\mathbb{R}^3)}\). For a defocusing case \({{\rm sup} \Gamma 0}\) the concentration compactness principle produces ground states under the assumption that zero lies outside the spectrum of the linear Operator \({\Nabla \times \Nabla \times +V(x)}\). Examples of cylindrically symmetric functions V are provided for which this holds.
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Ground States of a Nonlinear Curl-Curl Problem in Cylindrically Symmetric Media
arXiv: Analysis of PDEs, 2014Co-Authors: Thomas Bartsch, Tomáš Dohnal, Michael Plum, Wolfgang ReichelAbstract:We consider the nonlinear curl-curl problem $\Nabla\times\Nabla\times U + V(x) U= \Gamma(x)|U|^{p-1}U$ in $\mathbb{R}^3$ related to the nonlinear Maxwell equations for monochromatic fields. We search for solutions as minimizers (ground states) of the corresponding energy functional defined on subspaces (defocusing case) or natural constraints (focusing case) of $H(\mathrm{curl};\mathbb{R}^3)$. Under a cylindrical symmetry assumption on the functions $V$ and $\Gamma$ the variational problem can be posed in a symmetric subspace of $H(\mathrm{curl};\mathbb{R}^3)$. For a strongly defocusing case $\mathrm{esssup}\, \Gamma 0$ the concentration compactness principle produces ground states under the assumption that zero lies outside the spectrum of the linear Operator $\Nabla \times \Nabla \times +V(x)$. Examples of cylindrically symmetric functions $V$ are provided for which this holds.
Wang Feng-yu - One of the best experts on this subject based on the ideXlab platform.
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Linearization of Nonlinear Fokker-Planck Equations and Applications
2019Co-Authors: Ren Panpan, Rockner Michael, Wang Feng-yuAbstract:Let $\mathcal P$ be the space of probability measures on $\mathbb R^d$. We associate a coupled nonlinear Fokker-Planck equation on $\mathbb R^d$, i.e. with solution paths in $\mathcal P$, to a linear Fokker-Planck equation for probability measures on the product space $\mathbb R^d\times \mathcal P$, i.e. with solution paths in $\mathcal P(\mathbb R^d\times\mathcal P)$. We explicitly determine the corresponding linear Kolmogorov Operator $\mathbf{L}_t$ using the natural tangent bundle over $\mathcal P$ with corresponding gradient Operator $\Nabla^{\mathcal P}$. Then it is proved that the diffusion process generated by $\mathbf{L}_t$ on $\mathbb R^d\times\mathcal P$ is intrinsically related to the solution of a McKean-Vlasov stochastic differential equation (SDE). We also characterize the ergodicity of the diffusion process generated by $\mathbf{L}_t$ in terms of asymptotic properties of the coupled nonlinear Fokker-Planck equation. Another main result of the paper is that the restricted well-posedness of the non-linear Fokker-Planck equation and its linearized version imply the (restricted) well-posedness of the McKean-Vlasov equation and that in this case the laws of the solutions have the Markov property. As applications, we prove the restricted weak well-posedness and the Markov property of the so-called nonlinear distorted Brownian motion, whose associated nonlinear Fokker-Planck equation is a porous media equation perturbed by a nonlinear transport term. As a further application we obtain a probabilistic representation of solutions to Schr\"odinger type PDEs on $\mathbb R^d\times\mathcal P_2$, through the Feynman-Kac formula for the corresponding diffusion processes.Comment: 36 page
Feng-yu Wang - One of the best experts on this subject based on the ideXlab platform.
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Linearization of Nonlinear Fokker-Planck Equations and Applications
arXiv: Probability, 2019Co-Authors: Panpan Ren, Michael Röckner, Feng-yu WangAbstract:Let $\mathcal P$ be the space of probability measures on $\mathbb R^d$. We associate a coupled nonlinear Fokker-Planck equation on $\mathbb R^d$, i.e. with solution paths in $\mathcal P$, to a linear Fokker-Planck equation for probability measures on the product space $\mathbb R^d\times \mathcal P$, i.e. with solution paths in $\mathcal P(\mathbb R^d\times\mathcal P)$. We explicitly determine the corresponding linear Kolmogorov Operator $\mathbf{L}_t$ using the natural tangent bundle over $\mathcal P$ with corresponding gradient Operator $\Nabla^{\mathcal P}$. Then it is proved that the diffusion process generated by $\mathbf{L}_t$ on $\mathbb R^d\times\mathcal P$ is intrinsically related to the solution of a McKean-Vlasov stochastic differential equation (SDE). We also characterize the ergodicity of the diffusion process generated by $\mathbf{L}_t$ in terms of asymptotic properties of the coupled nonlinear Fokker-Planck equation. Another main result of the paper is that the restricted well-posedness of the non-linear Fokker-Planck equation and its linearized version imply the (restricted) well-posedness of the McKean-Vlasov equation and that in this case the laws of the solutions have the Markov property. As applications, we prove the restricted weak well-posedness and the Markov property of the so-called nonlinear distorted Brownian motion, whose associated nonlinear Fokker-Planck equation is a porous media equation perturbed by a nonlinear transport term. As a further application we obtain a probabilistic representation of solutions to Schrodinger type PDEs on $\mathbb R^d\times\mathcal P_2$, through the Feynman-Kac formula for the corresponding diffusion processes.
Thomas Bartsch - One of the best experts on this subject based on the ideXlab platform.
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Ground states of a nonlinear curl-curl problem in cylindrically symmetric media
Nonlinear Differential Equations and Applications NoDEA, 2016Co-Authors: Thomas Bartsch, Tomáš Dohnal, Michael Plum, Wolfgang ReichelAbstract:We consider the nonlinear curl-curl problem \({\Nabla\times\Nabla\times U + V(x) U= \Gamma(x)|U|^{p-1}U}\) in \({\mathbb{R}^3}\) related to the Kerr nonlinear Maxwell equations for fully localized monochromatic fields. We search for solutions as minimizers (ground states) of the corresponding energy functional defined on subspaces (defocusing case) or natural constraints (focusing case) of \({H({\rm curl};\mathbb{R}^3)}\). Under a cylindrical symmetry assumption corresponding to a photonic fiber geometry on the functions V and \({\Gamma}\) the variational problem can be posed in a symmetric subspace of \({H({\rm curl};\mathbb{R}^3)}\). For a defocusing case \({{\rm sup} \Gamma 0}\) the concentration compactness principle produces ground states under the assumption that zero lies outside the spectrum of the linear Operator \({\Nabla \times \Nabla \times +V(x)}\). Examples of cylindrically symmetric functions V are provided for which this holds.
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Ground States of a Nonlinear Curl-Curl Problem in Cylindrically Symmetric Media
arXiv: Analysis of PDEs, 2014Co-Authors: Thomas Bartsch, Tomáš Dohnal, Michael Plum, Wolfgang ReichelAbstract:We consider the nonlinear curl-curl problem $\Nabla\times\Nabla\times U + V(x) U= \Gamma(x)|U|^{p-1}U$ in $\mathbb{R}^3$ related to the nonlinear Maxwell equations for monochromatic fields. We search for solutions as minimizers (ground states) of the corresponding energy functional defined on subspaces (defocusing case) or natural constraints (focusing case) of $H(\mathrm{curl};\mathbb{R}^3)$. Under a cylindrical symmetry assumption on the functions $V$ and $\Gamma$ the variational problem can be posed in a symmetric subspace of $H(\mathrm{curl};\mathbb{R}^3)$. For a strongly defocusing case $\mathrm{esssup}\, \Gamma 0$ the concentration compactness principle produces ground states under the assumption that zero lies outside the spectrum of the linear Operator $\Nabla \times \Nabla \times +V(x)$. Examples of cylindrically symmetric functions $V$ are provided for which this holds.
G. V. S. R. Deekshitulu - One of the best experts on this subject based on the ideXlab platform.
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Solutions of Nabla Fractional Difference Equations Using N-Transforms
Communications in Mathematics and Statistics, 2014Co-Authors: J. Jagan Mohan, G. V. S. R. DeekshituluAbstract:In the present paper, we present some important properties of N-transform, which is the Laplace transform for the Nabla derivative on the time scale of integers (Bohner and Peterson in Dynamic equations on time scales, Birkhauser, Boston, 2001 ; Advances in dynamic equations on time scales, Birkhauser, Boston, 2002 ). We obtain the N-transform of Nabla fractional sums and differences and then apply this transform to solve some Nabla fractional difference equations with initial value problems. Finally, using N-transforms, we prove that discrete Mittag-Leffler function is the eigen function of Caputo type Nabla fractional difference Operator $$\Nabla ^{\alpha }$$ ∇ α .