| Softcover ISBN: | 978-2-85629-854-1 |
| Product Code: | AST/389 |
| List Price: | $52.00 |
| AMS Member Price: | $41.60 |
| Softcover ISBN: | 978-2-85629-854-1 |
| Product Code: | AST/389 |
| List Price: | $52.00 |
| AMS Member Price: | $41.60 |
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Book DetailsAstérisqueVolume: 389; 2017; 114 ppMSC: Primary 35; 47; 37
This monograph is devoted to the dynamics on Sobolev spaces of the cubic Szegő equation on the circle \(\mathbb{S}^1\), \[i\partial_t u=\Pi(\vert u\vert^{2} u).\] Here \(\Pi\) denotes the orthogonal projector from \(L^2(\mathbb{S}^1)\) onto the subspace \(L^{2}_+(\mathbb{S}^1)\) of functions with nonnegative Fourier modes. The authors construct a nonlinear Fourier transformation on \(H^{1/2}(\mathbb{S}^1)\cap L^2_+(\mathbb{S}^1)\), allowing them to describe explicitly the solutions of this equation with data in \(H^{1/2}(\mathbb{S}^1)\cap L^2_+(\mathbb{S}^1)\).
This explicit description implies almost-periodicity of every solution in this space. Furthermore, it allows the authors to display the following turbulence phenomenon. For a dense \(G_\delta\) subset of initial data in \(C^\infty(\mathbb{S}^1)\cap L^2_+(\mathbb{S}^1)\), the solutions tend to infinity in \(H^s\) for every \(s>\frac 12\) with super-polynomial growth on some sequence of times, while they go back to their initial data on another sequence of times tending to infinity.
This transformation is defined by solving a general inverse spectral problem involving singular values of a Hilbert–Schmidt Hankel operator and of its shifted Hankel operator.
A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.
ReadershipGraduate students and research mathematicians.
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This monograph is devoted to the dynamics on Sobolev spaces of the cubic Szegő equation on the circle \(\mathbb{S}^1\), \[i\partial_t u=\Pi(\vert u\vert^{2} u).\] Here \(\Pi\) denotes the orthogonal projector from \(L^2(\mathbb{S}^1)\) onto the subspace \(L^{2}_+(\mathbb{S}^1)\) of functions with nonnegative Fourier modes. The authors construct a nonlinear Fourier transformation on \(H^{1/2}(\mathbb{S}^1)\cap L^2_+(\mathbb{S}^1)\), allowing them to describe explicitly the solutions of this equation with data in \(H^{1/2}(\mathbb{S}^1)\cap L^2_+(\mathbb{S}^1)\).
This explicit description implies almost-periodicity of every solution in this space. Furthermore, it allows the authors to display the following turbulence phenomenon. For a dense \(G_\delta\) subset of initial data in \(C^\infty(\mathbb{S}^1)\cap L^2_+(\mathbb{S}^1)\), the solutions tend to infinity in \(H^s\) for every \(s>\frac 12\) with super-polynomial growth on some sequence of times, while they go back to their initial data on another sequence of times tending to infinity.
This transformation is defined by solving a general inverse spectral problem involving singular values of a Hilbert–Schmidt Hankel operator and of its shifted Hankel operator.
A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.
Graduate students and research mathematicians.
