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The non-linear Schrödinger equation with non-periodic potential: infinite-bump solutions and non-degeneracy
1. | University Science Institute, University of Iceland, Dunhaga 3, 107--Reykjavik, Iceland |
2. | Reykjavik Junior College, Lækjargata 7, 101--Reykjavik, Iceland |
References:
[1] |
N. Ackermann, A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations, J. Funct. Analysis, 234 (2006), 277-320.
doi: 10.1016/j.jfa.2005.11.010. |
[2] |
S. Angenent, The shadowing lemma for elliptic PDE, in "Dynamics of Infinite-dimensional Systems," (eds. S. N. Chow and J. K. Hale), Springer-Verlag, New York, (1987), 7-22. |
[3] |
V. Coti Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $\R^n$, Comm. Pure. Appl. Math., 45 (1992), 1217-1270.
doi: 10.1002/cpa.3160451002. |
[4] |
A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Analysis, 69 (1986), 397-408.
doi: 10.1016/0022-1236(86)90096-0. |
[5] |
C. Gui, Existence of multi-bump solutions for non-linear Schrödinger equations via variational method, Comm. in PDE, 21 (1996), 787-820.
doi: 10.1080/03605309608821208. |
[6] |
M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\R^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[7] |
R. Magnus, The implicit function theorem and multibump solutions of periodic partial differential equations, Proceedings of the Royal Society of Edinburgh, 136A (2006), 559-583.
doi: 10.1017/S0308210500005060. |
[8] |
R. Magnus, A scaling approach to bumps and multi-bumps for nonlinear partial differential equations, Proceedings of the Royal Society of Edinburgh, 136A (2006), 585-614.
doi: 10.1017/S0308210500005072. |
[9] |
R. Magnus and O. Moschetta, Non-degeneracy of perturbed solutions of semilinear partial differential equations, Ann. Aca. Scient. Fennicae, 35 (2010), 75-86.
doi: 10.5186/aasfm.2010.3505. |
[10] |
O. Moschetta, "The Non-linear Schrödinger Equation: Non-degeneracy and Infinite-bump Solutions," Ph.D thesis, University of Iceland, 2010. |
[11] |
Y. G. Oh, Existence of semi-classical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$, Commun. Partial Diff. Eq., 13 (1988), 1499-1519.
doi: 10.1080/03605308808820585. |
[12] |
Y. G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys., 131 (1990), 223-255.
doi: 10.1007/BF02161413. |
[13] |
J. M. Ortega, The Newton-Kantorovich theorem, Amer. Math. Monthly, 75 (1968), 658-660.
doi: 10.2307/2313800. |
[14] |
M. del Pino and P. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 15 (1998), 127-149. |
[15] |
M. del Pino and P. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method, Math. Ann., 324 (2002), 1-32.
doi: 10.1007/s002080200327. |
[16] |
M. del Pino, P. Felmer and K. Tanaka, An elementary construction of complex patterns in nonlinear Schrödinger equations, Nonlinearity, 15 (2002), 1653-1671. |
[17] |
P. Rabier and C. Stuart, Exponential decay of the solutions of quasilinear second-order equations and Pohozaev identities, Journal of Differential Equations, 165 (2000), 199-234.
doi: 10.1006/jdeq.1999.3749. |
[18] |
L. Schwartz, "Théorie des distributions," Hermann, Paris, 1966. |
[19] |
C. Stuart, An introduction to elliptic equations in $\R^n$, in "Nonlinear Functional Analysis and Applications to Differential Equations," World Sci. Publ. River Edge, NJ, (1998), 237-285. |
[20] |
N. Thandi, "On the Existence of Infinite Bump Solutions of Nonlinear Schrödinger Equations with Periodic Potentials," Ph.D thesis, University of Wisconsin-Madison, 1995. |
[21] |
J-L. Verger-Gaugry, Covering a ball with smaller equal balls in $\mathbbR^n$, Discrete and Computational Geometry, 33 (2005), 143-155.
doi: 10.1007/s00454-004-2916-2. |
[22] |
X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.
doi: 10.1007/BF02096642. |
[23] |
M. Weinstein, Modulational stability of the ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 567-576. |
show all references
References:
[1] |
N. Ackermann, A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations, J. Funct. Analysis, 234 (2006), 277-320.
doi: 10.1016/j.jfa.2005.11.010. |
[2] |
S. Angenent, The shadowing lemma for elliptic PDE, in "Dynamics of Infinite-dimensional Systems," (eds. S. N. Chow and J. K. Hale), Springer-Verlag, New York, (1987), 7-22. |
[3] |
V. Coti Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $\R^n$, Comm. Pure. Appl. Math., 45 (1992), 1217-1270.
doi: 10.1002/cpa.3160451002. |
[4] |
A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Analysis, 69 (1986), 397-408.
doi: 10.1016/0022-1236(86)90096-0. |
[5] |
C. Gui, Existence of multi-bump solutions for non-linear Schrödinger equations via variational method, Comm. in PDE, 21 (1996), 787-820.
doi: 10.1080/03605309608821208. |
[6] |
M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\R^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[7] |
R. Magnus, The implicit function theorem and multibump solutions of periodic partial differential equations, Proceedings of the Royal Society of Edinburgh, 136A (2006), 559-583.
doi: 10.1017/S0308210500005060. |
[8] |
R. Magnus, A scaling approach to bumps and multi-bumps for nonlinear partial differential equations, Proceedings of the Royal Society of Edinburgh, 136A (2006), 585-614.
doi: 10.1017/S0308210500005072. |
[9] |
R. Magnus and O. Moschetta, Non-degeneracy of perturbed solutions of semilinear partial differential equations, Ann. Aca. Scient. Fennicae, 35 (2010), 75-86.
doi: 10.5186/aasfm.2010.3505. |
[10] |
O. Moschetta, "The Non-linear Schrödinger Equation: Non-degeneracy and Infinite-bump Solutions," Ph.D thesis, University of Iceland, 2010. |
[11] |
Y. G. Oh, Existence of semi-classical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$, Commun. Partial Diff. Eq., 13 (1988), 1499-1519.
doi: 10.1080/03605308808820585. |
[12] |
Y. G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys., 131 (1990), 223-255.
doi: 10.1007/BF02161413. |
[13] |
J. M. Ortega, The Newton-Kantorovich theorem, Amer. Math. Monthly, 75 (1968), 658-660.
doi: 10.2307/2313800. |
[14] |
M. del Pino and P. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 15 (1998), 127-149. |
[15] |
M. del Pino and P. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method, Math. Ann., 324 (2002), 1-32.
doi: 10.1007/s002080200327. |
[16] |
M. del Pino, P. Felmer and K. Tanaka, An elementary construction of complex patterns in nonlinear Schrödinger equations, Nonlinearity, 15 (2002), 1653-1671. |
[17] |
P. Rabier and C. Stuart, Exponential decay of the solutions of quasilinear second-order equations and Pohozaev identities, Journal of Differential Equations, 165 (2000), 199-234.
doi: 10.1006/jdeq.1999.3749. |
[18] |
L. Schwartz, "Théorie des distributions," Hermann, Paris, 1966. |
[19] |
C. Stuart, An introduction to elliptic equations in $\R^n$, in "Nonlinear Functional Analysis and Applications to Differential Equations," World Sci. Publ. River Edge, NJ, (1998), 237-285. |
[20] |
N. Thandi, "On the Existence of Infinite Bump Solutions of Nonlinear Schrödinger Equations with Periodic Potentials," Ph.D thesis, University of Wisconsin-Madison, 1995. |
[21] |
J-L. Verger-Gaugry, Covering a ball with smaller equal balls in $\mathbbR^n$, Discrete and Computational Geometry, 33 (2005), 143-155.
doi: 10.1007/s00454-004-2916-2. |
[22] |
X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.
doi: 10.1007/BF02096642. |
[23] |
M. Weinstein, Modulational stability of the ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 567-576. |
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