The Experts below are selected from a list of 140046 Experts worldwide ranked by ideXlab platform
Monica Visan - One of the best experts on this subject based on the ideXlab platform.
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the defocusing energy supercritical Nonlinear Wave equation in three space dimensions
Transactions of the American Mathematical Society, 2011Co-Authors: Rowan Killip, Monica VisanAbstract:We consider the defocusing Nonlinear Wave equation u tt ― Δu + |u| p u = 0 in the energy-supercritical regime p > 4. For even values of the power p, we show that blowup (or failure to scatter) must be accompanied by blowup of the critical Sobolev norm. An equivalent formulation is that solutions with bounded critical Sobolev norm are global and scatter. The impetus to consider this problem comes from recent work of Kenig and Merle who treated the case of spherically-symmetric solutions.
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the radial defocusing energy supercritical Nonlinear Wave equation in all space dimensions
Proceedings of the American Mathematical Society, 2011Co-Authors: Rowan Killip, Monica VisanAbstract:We consider the defocusing Nonlinear Wave equation u tt ― Δu + |u| p u = 0 with spherically-symmetric initial data in the regime 4 d―2 7, but for a smaller range of p > 4 d―2. The principal result is that blowup (or failure to scatter) must be accompanied by blowup of the critical Sobolev norm. An equivalent formulation is that maximal-lifespan solutions with bounded critical Sobolev norm are global and scatter.
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the radial defocusing energy supercritical Nonlinear Wave equation in all space dimensions
arXiv: Analysis of PDEs, 2010Co-Authors: Rowan Killip, Monica VisanAbstract:We consider the defocusing Nonlinear Wave equation $u_{tt}-\Delta u + |u|^p u=0$ with spherically-symmetric initial data in the regime $\frac4{d-2} \frac4{d-2}$. The principal result is that blowup (or failure to scatter) must be accompanied by blowup of the critical Sobolev norm. An equivalent formulation is that maximal-lifespan solutions with bounded critical Sobolev norm are global and scatter.
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the defocusing energy supercritical Nonlinear Wave equation in three space dimensions
arXiv: Analysis of PDEs, 2010Co-Authors: Rowan Killip, Monica VisanAbstract:We consider the defocusing Nonlinear Wave equation $u_{tt}-\Delta u + |u|^p u=0$ in the energy-supercritical regime p>4. For even values of the power p, we show that blowup (or failure to scatter) must be accompanied by blowup of the critical Sobolev norm. An equivalent formulation is that solutions with bounded critical Sobolev norm are global and scatter. The impetus to consider this problem comes from recent work of Kenig and Merle who treated the case of spherically-symmetric solutions.
E A Kuznetsov - One of the best experts on this subject based on the ideXlab platform.
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solitons and collapses two evolution scenarios of Nonlinear Wave systems
Physics-Uspekhi, 2012Co-Authors: V E Zakharov, E A KuznetsovAbstract:Two alternative scenarios pertaining to the evolution of Nonlinear Wave systems are considered: solitons and Wave collapses. For the former, it suffices that the Hamiltonian be boundedfrombelow(orabove),andthenthesolitonrealizingits minimum (or maximum) is Lyapunov stable. The extremum is approached via the radiation of small-amplitude Waves, a pro- cess absent in systems with finitely many degrees of freedom. The framework of the Nonlinear Schrequation and the three-Wave system is used to show how the boundedness of the Hamiltonian—andhencethestabilityofthesolitonminimizing it—can be proved rigorously using the integral estimate meth- od based on the Sobolev embedding theorems. Wave systems with the Hamiltonians unbounded from below must evolve to a collapse, which can be considered as the fall of a particle in an unbounded potential. The radiation of small-amplitude Waves promotes collapse in this case.
Yoshiharu Omura - One of the best experts on this subject based on the ideXlab platform.
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Nonlinear Wave growth theory of coherent hiss emissions in the plasmasphere
Journal of Geophysical Research, 2015Co-Authors: Yoshiharu Omura, C A Kletzing, Satoko Nakamura, Danny Summers, M HikishimaAbstract:Recent observations of plasmaspheric hiss emissions by the Van Allen Probes show that broadband hiss emissions in the plasmasphere comprise short-time coherent elements with rising and falling tone frequencies. Based on Nonlinear Wave growth theory of whistler mode chorus emissions, we have examined the applicability of the Nonlinear theory to the coherent hiss emissions. We have generalized the derivation of the optimum Wave amplitude for triggering rising tone chorus emissions to the cases of both rising and falling tone hiss elements. The amplitude profiles of the hiss emissions are well approximated by the optimum Wave amplitudes for triggering rising or falling tones. Through the formation of electron holes for rising tones and electron hills for falling tones, the coherent Waves evolve to attain the optimum amplitudes. An electromagnetic particle simulation confirms the Nonlinear Wave growth mechanism as the initial phase of the hiss generation process. We find very good agreement between the theoretical optimum amplitudes and the observed amplitudes as a function of instantaneous frequency. We calculate Nonlinear growth rates at the equator and find that Nonlinear growth rates for rising tone emissions are much larger than the linear growth rates. The time scales of observed hiss emissions also agree with those predicted by the Nonlinear theory. Based on the theory, we can infer properties of energetic electrons generating hiss emissions in the equatorial region of the plasmasphere.
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a computational and theoretical investigation of Nonlinear Wave particle interactions in oblique whistlers
Journal of Geophysical Research, 2015Co-Authors: D Nunn, Yoshiharu OmuraAbstract:Most previous work on Nonlinear Wave-particle interactions between energetic electrons and VLF Waves in the Earth's magnetosphere has assumed parallel propagation, the underlying mechanism being Nonlinear trapping of cyclotron resonant electrons in a parabolic magnetic field inhomogeneity. Here Nonlinear Wave-particle interaction in oblique whistlers in the Earth's magnetosphere is investigated. The study is nonself-consistent and assumes an arbitrarily chosen Wave field. We employ a “continuous Wave” Wave field with constant frequency and amplitude, and a model for an individual VLF chorus element. We derive the equations of motion and trapping conditions in oblique whistlers. The resonant particle distribution function, resonant current, and Nonlinear growth rate are computed as functions of position and time. For all resonances of order n, resonant electrons obey the trapping equation, and provided the Wave amplitude is big enough for the prevailing obliquity, Nonlinearity manifests itself by a “hole” or “hill” in distribution function, depending on the zero-order distribution function and on position. A key finding is that the n = 1 resonance is relatively unaffected by moderate obliquity up to 25°, but growth rates roll off rapidly at high obliquity. The n = 1 resonance saturates due to the adiabatic effect and here reaches a maximum growth at ~20 pT, 2000 km from the equator. Damping due to the n = 0 resonance is not subject to adiabatic effects and maximizes at some 8000 km from the equator at an obliquity ~55°.
D Nunn - One of the best experts on this subject based on the ideXlab platform.
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a computational and theoretical investigation of Nonlinear Wave particle interactions in oblique whistlers
Journal of Geophysical Research, 2015Co-Authors: D Nunn, Yoshiharu OmuraAbstract:Most previous work on Nonlinear Wave-particle interactions between energetic electrons and VLF Waves in the Earth's magnetosphere has assumed parallel propagation, the underlying mechanism being Nonlinear trapping of cyclotron resonant electrons in a parabolic magnetic field inhomogeneity. Here Nonlinear Wave-particle interaction in oblique whistlers in the Earth's magnetosphere is investigated. The study is nonself-consistent and assumes an arbitrarily chosen Wave field. We employ a “continuous Wave” Wave field with constant frequency and amplitude, and a model for an individual VLF chorus element. We derive the equations of motion and trapping conditions in oblique whistlers. The resonant particle distribution function, resonant current, and Nonlinear growth rate are computed as functions of position and time. For all resonances of order n, resonant electrons obey the trapping equation, and provided the Wave amplitude is big enough for the prevailing obliquity, Nonlinearity manifests itself by a “hole” or “hill” in distribution function, depending on the zero-order distribution function and on position. A key finding is that the n = 1 resonance is relatively unaffected by moderate obliquity up to 25°, but growth rates roll off rapidly at high obliquity. The n = 1 resonance saturates due to the adiabatic effect and here reaches a maximum growth at ~20 pT, 2000 km from the equator. Damping due to the n = 0 resonance is not subject to adiabatic effects and maximizes at some 8000 km from the equator at an obliquity ~55°.
Rowan Killip - One of the best experts on this subject based on the ideXlab platform.
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the defocusing energy supercritical Nonlinear Wave equation in three space dimensions
Transactions of the American Mathematical Society, 2011Co-Authors: Rowan Killip, Monica VisanAbstract:We consider the defocusing Nonlinear Wave equation u tt ― Δu + |u| p u = 0 in the energy-supercritical regime p > 4. For even values of the power p, we show that blowup (or failure to scatter) must be accompanied by blowup of the critical Sobolev norm. An equivalent formulation is that solutions with bounded critical Sobolev norm are global and scatter. The impetus to consider this problem comes from recent work of Kenig and Merle who treated the case of spherically-symmetric solutions.
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the radial defocusing energy supercritical Nonlinear Wave equation in all space dimensions
Proceedings of the American Mathematical Society, 2011Co-Authors: Rowan Killip, Monica VisanAbstract:We consider the defocusing Nonlinear Wave equation u tt ― Δu + |u| p u = 0 with spherically-symmetric initial data in the regime 4 d―2 7, but for a smaller range of p > 4 d―2. The principal result is that blowup (or failure to scatter) must be accompanied by blowup of the critical Sobolev norm. An equivalent formulation is that maximal-lifespan solutions with bounded critical Sobolev norm are global and scatter.
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the radial defocusing energy supercritical Nonlinear Wave equation in all space dimensions
arXiv: Analysis of PDEs, 2010Co-Authors: Rowan Killip, Monica VisanAbstract:We consider the defocusing Nonlinear Wave equation $u_{tt}-\Delta u + |u|^p u=0$ with spherically-symmetric initial data in the regime $\frac4{d-2} \frac4{d-2}$. The principal result is that blowup (or failure to scatter) must be accompanied by blowup of the critical Sobolev norm. An equivalent formulation is that maximal-lifespan solutions with bounded critical Sobolev norm are global and scatter.
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the defocusing energy supercritical Nonlinear Wave equation in three space dimensions
arXiv: Analysis of PDEs, 2010Co-Authors: Rowan Killip, Monica VisanAbstract:We consider the defocusing Nonlinear Wave equation $u_{tt}-\Delta u + |u|^p u=0$ in the energy-supercritical regime p>4. For even values of the power p, we show that blowup (or failure to scatter) must be accompanied by blowup of the critical Sobolev norm. An equivalent formulation is that solutions with bounded critical Sobolev norm are global and scatter. The impetus to consider this problem comes from recent work of Kenig and Merle who treated the case of spherically-symmetric solutions.
