Semiclassical soliton ensembles for the focusing nonlinear Schrödinger equation. (English) Zbl 1057.35063

Annals of Mathematics Studies 154. Princeton, NJ: Princeton University Press (ISBN 0-691-11482-X/pbk; 0-691-11483-8/hbk). xii, 265 p. (2003).
In this research monograph the pointwise behavior of solutions to the one-dimensional focusing nonlinear Schrödinger equation \[ i\hbar \partial_t\psi+{\hbar^2\over 2}\,\partial_x^2\psi+| \psi| ^2\psi=0, \quad\psi (x, 0)=A(x), \] is considered in the limit \(\hbar\to 0\). For this purpose, first the solution \(\psi=\psi(x, t; \hbar)\) is expressed by means of a suitable \(2\times 2\)-matrix Riemann-Hilbert problem relative to a contour that surrounds the \(L^2({\mathbb R})\)-eigenvalues \(\lambda=\lambda(\hbar)\) of the associated scattering problem \[ \hbar\partial_x\left(\begin{matrix} u_1 \\ u_2\end{matrix}\right)=\left(\begin{matrix} -i\lambda & A(x) \\ -A(x) & i\lambda\end{matrix}\right)\left(\begin{matrix} u_1 \\ u_2\end{matrix}\right), \] defined by the initial data \(A=A(x)\); this function is assumed to be independent of \(\hbar\), positive, even, bell-shaped, and of rapid decay as \(| x| \to\infty\). Thereafter the matrix solution \(m=m(\lambda; \hbar)\) of the Riemann-Hilbert problem is investigated in great detail as \(\hbar\to 0\), which makes up for the largest part of the book. Since \(\psi\) can be recovered from \(m\), the desired information on \(\psi\) as \(\hbar\to 0\) can be extracted as well from these results.
The strategy as outlined above is carried through not for general initial data, but for certain potentials that are obtained as formal WKB approximations from such general initial data, see Definition 3.1.1 on p. 25 for the precise assumptions; in particular, the modified potentials are reflectionless and have a spectrum given by an explicit Bohr-Sommerfeld quantization rule.
For the special choice \(A(x)=A\,\text{sech}(x)\) it is known from work of J. Satsuma and N. Yajima [Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media, Suppl. Prog. Theor. Phys. 55, 284–306 (1974)] that along the particular sequence \(\hbar_N=A/N\) the required WKB approximations exactly agree with \(A\). Thus the rigorous semiclassical asymptotics are obtained in this case. Furthermore, a “caustic”-type curve is identified in the \((x, t)\)-plane where the microstructure of the solution changes in a phase transition from fields with smooth amplitude to oscillatory fields with intermittent concentration in amplitude.


35Q55 NLS equations (nonlinear Schrödinger equations)
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
35Q51 Soliton equations
78A60 Lasers, masers, optical bistability, nonlinear optics
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