A review of dispersive limits of (non)linear schrödinger-type equations

Taiwanese Journal of Mathematics

Volumn 4, Issue 4, 2000, Pages 501-529

  Gasser, Ingenuin ^a   Lin, Chi Kun ^b   Markowich, Peter A ^c  

^a UNIVERSITY OF HAMBURG   (Germany)

^b NATIONAL CHENG KUNG UNIVERSITY   (Taiwan)

^c UNIVERSITY OF VIENNA   (Austria)

Author keywords

Compressible Euler equation; Dispersive limit; KdV equation; Quantum hydrodynamics; Wigner transform

Indexed keywords

EID: 0000538578     PISSN: 10275487     EISSN: None     Source Type: Journal    
DOI: 10.11650/twjm/1500407291     Document Type: Review

References (37)

1. M. G. Ancona and G. J. Iafrate, Quantum correction to the equation of state of an electron gas in a semiconductor, Phys. Rev. B 39 (1989), 9536-9540.
2. U. Ascher, J. Christiansen and R. D. Russel, Collocation software for boundary value ODEs, Trans. Math. Soft. 7 (1981), 209-222.
3. U. Ascher, P. A. Markowich, P. Pietra and C. Schmeiser, A phase plane analysis of transonic solutions for the hydrodynamic semiconductor model, Math. Models Methods Appl. Sci. 1 (1991), 347-376.
4. N. Ben Abdallah and A. Unterreiter, On a nonlinear Schrödinger Poisson system, in preparation.
5. C. Bardos, J.-M. Ghidaglia and S. Kamvissis, Weak convergence and deterministic approach to turbulent diffusion, math.AP/9904119.
6. B. Desjardins, C.-K. Lin and T.-C. Tso, Semiclassical limit of the derivative nonlinear Schrödinger equation, Math. Models Methods Appl. Sci. 10 (2000), 261-285.
7. B. Desjardins and C.-K. Lin, On the semiclassical limit of the general modified NLS equation, submitted, 1999.
8. P. Deift and D. T-R. McLaughlin, A continuum limit of the Toda lattice, Mem. Amer. Math. Soc. 624 (1998).
9. N. Ercolani, S. Jin, C. D. Levermore and W. MacEvoy, The zero dispersionlimit of the NLS/mKdV hierarchy for the nonselfadjoint ZS operator, preprint, 1997.
10. R. P. Feynman, Statistical Mechanics, A set of lectures, Frontiers in Physics, W. A. Benjamin, Inc., 1972.
11. C. L. Gardner, The quantum hydrodynamic model for semiconductor devices, SIAM J. Appl. Math. 54 (1994), 409-427.
12. C. L. Gardner, Resonant tunneling in the quantum hydrodynamic model, VLSI Design 3 (1995), 201-210.
13. I. Gasser and P. A. Markowich, Quantum hydrodynamics, Wigner transforms and the classical limit, Asymptotic Anal. 14 (1997), 97-116.
14. J. Ginibre and G. Velo, On the global Cauchy problem for some nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 309-323.
15. E. Grenier, Semiclassical limit of the nonlinear Schrödinger equation in small time, Proc. Amer. Math. Soc. 126 (1998), 523-530.
16. S. Jin, C. D. Levermore and D. W. McLaughlin, The behavior of solutions of the NLS equation in the semiclassical limit, in: Singular Limits of Dispersive Waves, N. Ercolani, I. Gabitov, C. D. Levermore and D. Serre, eds., NATO ASI Series, Series B, Physics 320, Plenum, New York, 1994, pp. 235-256.
17. S. Jin, C. D. Levermore and D. W. McLaughlin, The semiclassical limit of the defocusing NLS hierarchy, Comm. Pure Appl. Math. 52 (1999), 613-654.
18. L. D. Landau and E. M. Lifschitz, Lehrbuch der Theoretischen Physik III - Quantenmechanik, Akademie-Verlag, 1985.
19. P. D. Lax, On dispersive difference schemes, Phys. D 18 (1986), 250-254.
20. P. D. Lax, The zero dispersion limit, a deterministic analogue of turbulences, Comm. Pure Appl. Math. 54 (1991), 1047-1056.
21. P. D. Lax and C. D. Levermore, The small dispersion limit of the Korteweg-deVries equation I, II, III, Comm. Pure Appl. Math. 36 (1983), 253-290, 571-593, 809-830.
22. P.-L. Lions and T. Paul, Sur les mesures de Wigner, Rev. Mat. Iberoamericana 9 (1993), 553-618.
23. M. I. Loffredo and L. M. Morato, Self-consistent hydrodynamical model for Hell near absolute zero in the frame work of stochastic mechanics, Phys. Rev. B 35 (1987), 1742-1747.
24. M. I. Loffredo and L. M. Morato, On the creation of quantized vortex lines in rotating Hell, Nuovo Cimento B(II) 108 (1993), 205-215.
25. P. A. Markowich, The Stationary Semiconductor Device Equations, Springer-Verlag, Vienna, 1986.
26. P. A. Markowich and N. J. Mauser, The classical limit of a self-consistent quantum-Vlasov equation in 3d, Math. Models Methods Appl. Sci. 3 (1993), 109-124.
27. P. A. Markowich and C. Ringhofer, Quantum hydrodynamics for semiconductors in the high-field case, Appl. Math. Lett. 7 (1994), 37-41.
28. P. A. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990.
29. P. A. Markowich and A. Unterreiter, Vacuum solution of a stationary drift-diffusion model, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 20 (1993), 371-386.
30. D. W. McLaughlin and J. Strain, Calculating the weak limit of KdV, Comm. Pure Appl. Math. 47 (1994), 1-46.
31. P. D. Miller and S. Kamvissis, On the semiclassical limit of the focusing nonlinear Schrödinger equation, Phys. Lett. A 247 (1998), 75-86.
32. F. Pacard and A. Unterreiter, A variational analysis of the thermal equilibrium state of charged quantum fluids, Comm. Partial Differential Equations 20 (1995), 885-900.
33. W. V. van Roosbroeck, Theory of flow of electrons and holes in Germanium and other semiconductors, Bell System Tech. J. 29 (1950), 560-604.
34. S. Shipman, A Continuum Limit of a Finite Discrete Nonlinear Schrödinger System, Ph.D. Dissertation, University of Arizona, 1997.
35. W. A. Strauss, The Nonlinear Wave Equation, Regional Conference Series in Mathematics, 73, Amer. Math. Soc., 1989.
36. A. Unterreiter, The thermal equilibrium states of semiconductor devices, Appl. Math. Lett. 7 (1994), 39-43.
37. A. Unterreiter, The thermal equilibrium solution of a generic bipolar quantum hydrodynamic model, Comm. Math. Phys. 188 (1997), 69-88.