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Short Time Behavior of Solutions to Linear and Nonlinear Schrödinger Equations

Published online by Cambridge University Press:  20 November 2018

Michael Taylor*
Affiliation:
Mathematics Deptartment, University of North Carolina at Chapel Hill, Chapel Hill, NC, 27599, U.S.A. e-mail:met@math.unc.edu
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Abstract

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We examine the fine structure of the short time behavior of solutions to various linear and nonlinear Schrödinger equations ${{u}_{t}}=i\Delta u+q(u)$ on $I\times {{\mathbb{R}}^{n}}$, with initial data $u(0,x)=f(x)$. Particular attention is paid to cases where $f$ is piecewise smooth, with jump across an $(n-1)$-dimensional surface. We give detailed analyses of Gibbs-like phenomena and also focusing effects, including analogues of the Pinsky phenomenon. We give results for general $n$ in the linear case. We also have detailed analyses for a broad class of nonlinear equations when $n=1$ and 2, with emphasis on the analysis of the first order correction to the solution of the corresponding linear equation. This work complements estimates on the error in this approximation.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2008

References

[1] Arnold, V., Gusein-Zade, S., and Varchenko, A., Singularities of DifferentiableMaps. I. Classification of Critical Points, Caustics, and Wave Fronts. Birkhäuser Boston, Boston, MA, 1985.Google Scholar
[2] Arnold, V., Gusein-Zade, S., and Varchenko, A., Singularities of DifferentiableMappings. II. Monodromy and Asymptotics of Integrals. Birkhäuser Boston, Boston,MA, 1988.Google Scholar
[3] Beale, T., Kato, T., and Majda, A., Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94(1984), 6166.Google Scholar
[4] Brézis, H. and Gallouet, T., Nonlinear Schrödinger evolution equations Nonlin. Anal. 4(1980), no. 4, 677681.Google Scholar
[5] Brézis, H. and Wainger, S., A note on limiting cases of Sobolev embeddings and convolution inequalities. Comm. Partial Differential Equations 5(1980), no. 7, 773789.Google Scholar
[6] Bump, D., Diaconis, P., and Keller, J., Unitary correlations and the Fejér kernel. Math. Phys. Anal. Geom. 5(2002), no. 2, 101123.Google Scholar
[7] Christ, F. M. and Weinstein, M., Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equations. J. Funct. Anal. 100(1991), no. 1, 87109.Google Scholar
[8] Colzani, L. and Vignati, M., Gibbs phenomenon for multiple Fourier integrals. J. Approx. Theory 80(1995), no. 1, 119131.Google Scholar
[9] DiFranco, J., Gibbs Phenomenon for the Defocusing Nonlinear Schrödinger Equation. Ph.D. Thesis, Univ. of North Carolina, 2004.Google Scholar
[10] Duistermaat, J., Oscillatory integrals, Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math. 27(1974), 207281.Google Scholar
[11] Duistermaat, J., Fourier Integral Operators. Second edition. Progress in Mathematics 130, Birkhäuser Boston, Boston,MA, 1996.Google Scholar
[12] Gao, J., Short time behavior of solutions to nonlinear Schrödinger equations in n dimensions. PhD. Thesis, Univ. of North Carolina, 2007.Google Scholar
[13] Guillemin, V. and Sternberg, S., Geometric Asymptotics, Mathematical Surveys 14, American Mathematical Society, Providence, RI, 1977.Google Scholar
[14] H, L.örmander, The Weyl calculus of pseudodifferential operators. Comm. Pure Appl. Math. 32(1979), no. 3, 360444.Google Scholar
[15] Lebedev, N., Special Functions and Their Applications. Dover, New York, 1972.Google Scholar
[16] Pinsky, M., Pointwise Fourier inversion and related eigenfunction expansions. Comm. Pure Appl. Math. 47(1994), no. 5, 653681.Google Scholar
[17] Pinsky, M. and Taylor, M., Pointwise Fourier inversion: a wave equation approach. J. Fourier Anal. Appl. 3(1997), no. 6, 647703.Google Scholar
[18] Runst, T.,Mapping properties of non-linear operators in spaces of Triebel-Lizorkin and Besov type. Anal. Math. 12(1986), no. 4, 313346.Google Scholar
[19] Taylor, M., Pseudodifferential Operators and Nonlinear PDE. Progress in Mathematics 100, Birkhäuser Boston, boston, MA, 1991.Google Scholar
[20] Taylor, M., Partial Differential Equations. Texts in Applied Mathematics 23, Springer-Verlag, New York, 1996.Google Scholar
[21] Taylor, M., Tools for PDE. Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials. Mathematical Surveys and Monographs 81, American Mathematical Society, Providence, RI, 2000.Google Scholar
[22] Taylor, M., Multi-dimensional Fejér kernel asymptotics. In: Harmonic Analysis at Mount Holyoke. ContemporaryMath. 320, American Mathematical Society, Providence, RI, 2003, pp. 411–434.Google Scholar
[23] Taylor, M., Fourier series and lattice point problems. Houston J. Math. 30(2004), no. 1, 117135.Google Scholar
[24] Taylor, M., Short time behavior of solutions to nonlinear Schrödinger equations in one and two space dimensions. Comm. Partial Differential Equations, 31(2006), no. 4-6, 945957.Google Scholar
[25] Triebel, H., Theory of Function Spaces. Monographs in Mathematics 78, Birkhäuser Verlag, Basel, 1983.Google Scholar
[26] Vega, L., Schrödinger equations: pointwise convergence to the initial data. Proc. Amer. Math. Soc. 102(1988), no. 4, 874878.Google Scholar