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Quintic function
About: Quintic function is a research topic. Over the lifetime, 1677 publications have been published within this topic receiving 26780 citations.
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TL;DR: In this paper, the authors improved the Strichartz estimates obtained in A. de Bouard, A. Debussche, and A. De Bouard for the Schrodinger equation with white noise dispersion in one dimension, and proved global well posedness when a quintic critical nonlinearity is added to the equation.
54 citations
TL;DR: In this article, the growth of higher Sobolev norms of solutions of the onedimensional periodic nonlinear Schrodinger equation (NLS) was studied, and by a combination of the normal form reduction and the upside-down I-method, they established α = 1 for a general power nonlinearity.
Abstract: We study growth of higher Sobolev norms of solutions of the onedimensional periodic nonlinear Schrodinger equation (NLS). By a combination of the normal form reduction and the upside-down I-method, we establish
$${\left\| {u(t)} \right\|_{{H^s}}} \le {(1 + \left| t \right|)^{a(s - 1) + }}$$
with α = 1 for a general power nonlinearity. In the quintic case, we obtain the above estimate with α = 1/2 via the space-time estimate due to Bourgain [4, 5]. In the cubic case, we compute concretely the terms arising in the first few steps of the normal form reduction and prove the above estimate with α = 4/9. These results improve the previously known results (except for the quintic case). In the Appendix, we also show how Bourgain’s idea in [4] on the normal form reduction for the quintic nonlinearity can be applied to other powers.
54 citations
TL;DR: In this paper, the quintic nonlinear Schrodinger equation (NLS) on the circle was considered and it was shown that there exist solutions corresponding to an initial datum built on four Fourier modes which form a resonant set, which have a non trivial dynamic that involves periodic energy exchanges between the modes initially excited.
Abstract: We consider the quintic nonlinear Schr\"odinger equation (NLS) on the circle. We prove that there exist solutions corresponding to an initial datum built on four Fourier modes which form a resonant set, which have a non trivial dynamic that involves periodic energy exchanges between the modes initially excited. It is noticeable that this nonlinear phenomena does not depend on the choice of the resonant set. The dynamical result is obtained by calculating a resonant normal form up to order 10 of the Hamiltonian of the quintic NLS and then by isolating an effective term of order 6. Notice that this phenomena can not occur in the cubic NLS case for which the amplitudes of the Fourier modes are almost actions, i.e. they are almost constant.
54 citations
TL;DR: The traveling wave solutions involving parameters of nonlinear evolution equations, via, the perturbed nonlinear Schrodinger equation and the nonlinear cubic–quintic Ginzburg Landau equation are constructed using the modified ( G ′ / G ) -expansion method, where G satisfies a second order linear ODE.
53 citations
TL;DR: The problem is analyzed in terms of cubic splines first and then extended to the use of quintic and septic splines to numerically solve two-point boundary-value problems.
Abstract: : The report is concerned with the use of collocation by splines to numerically solve two-point boundary-value problems. The problem is analyzed in terms of cubic splines first and then extended to the use of quintic and septic splines. Consideration is given both to convergences as the mesh is refined and to the bandwidth of the matrices involved. Comparisons are made to a similar approach using the Galerkin method rather than collocation. (Author)
53 citations