February 2019, 39(2): 841-862. doi: 10.3934/dcds.2019035

Construction of solutions for some localized nonlinear Schrödinger equations

Departamento de Matemática, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Santiago, Chile

* Corresponding author: mcourdurier@mat.uc.cl

Received  January 2018 Revised  July 2018 Published  November 2018

Fund Project: O. B. was partially supported by FONDECYT grant number 1161732. M. C. was partially supported by FONDECYT grant number 1141189. C. F. was partially supported by FONDECYT grant number 1141120

For an
$N$
-body system of linear Schrödinger equation with space dependent interaction between particles, one would expect that the corresponding one body equation, arising as a mean field approximation, would have a space dependent nonlinearity. With such motivation we consider the following model of a nonlinear reduced Schrödinger equation with space dependent nonlinearity
$\begin{align*}-\varphi''+V(x)h'(|\varphi|^2)\varphi = λ \varphi,\end{align*}$
where
$V(x) = -χ_{[-1,1]} (x)$
is minus the characteristic function of the interval
$[-1,1]$
and where
$h'$
is any continuous strictly increasing function. In this article, for any negative value of
$λ$
we present the construction and analysis of the infinitely many solutions of this equation, which are localized in space and hence correspond to bound-states of the associated time-dependent version of the equation.
Citation: Olivier Bourget, Matias Courdurier, Claudio Fernández. Construction of solutions for some localized nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 841-862. doi: 10.3934/dcds.2019035
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[1]

M. J. Ablowitz, B. Prinari and A. D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems, London Mathematical Society Lecture Note Series, 302, Cambridge University Press, 2004.

[2]

A. AmbrosettiA. Malchiodi and D. Ruiz, Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Anal. Math., 98 (2006), 317-348. doi: 10.1007/BF02790279.

[3]

G. Arioli and A. Szulkin, Semilinear Schrödinger equation in the presence of a magnetic field, Arch. Ration. Mech. Anal., 170 (2003), 277-295. doi: 10.1007/s00205-003-0274-5.

[4]

J. Bourgain, On nonlinear Schrödinger equations in Les Relations Entre Les Mathématiques et la Physique Théorique, Inst. Hautes Études Sci., Bures-sur-Yvette, (1998), 11-21.

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J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations, American Mathematical Society Colloquium Publications, 46. American Mathematical Society, 1999. doi: 10.1090/coll/046.

[6]

N. Burq and M. Zworski, Instability for the Semiclassical Non-linear Schrödinger equation, Commun. Math. Phys., 260 (2005), 45-58. doi: 10.1007/s00220-005-1402-x.

[7]

J. ByeonL. Jeanjean and K. Tanaka, Standing waves for nonlinear Schrödinger equations with a general nonlinearity: one and two dimensional cases, Comm. Partial Differential Equations, 33 (2008), 1113-1136. doi: 10.1080/03605300701518174.

[8]

C. CacciapuotiD. FincoD. Noja and A. Teta, The NLS equation in dimension one with spatially concentrated nonlinearities: the point like limit, Lett. Math. Phys., 104 (2014), 1557-1570. doi: 10.1007/s11005-014-0725-y.

[9]

R. Carretero-GonzáalezaJ. D. TalleyC. Chong and B. A. Malomed, Multistable solitons in the cubic-quintic discrete nonlinear Schrödinger equation, Physica D, 216 (2006), 77-89. doi: 10.1016/j.physd.2006.01.022.

[10]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, American Mathematical Society, 2003. doi: 10.1090/cln/010.

[11]

T. Chen and N. Pavlović, The quintic NLS as the mean field limit of a boson gas with three-body interactions, J. Funct. Anal., 260 (2011), 959-997. doi: 10.1016/j.jfa.2010.11.003.

[12]

H. ChristiansonJ. MarzuolaJ. Metcalfe and M. Taylor, Nonlinear Bound States on Weakly Homogeneous Spaces, Communications in Partial Differential Equations, 39 (2014), 34-97. doi: 10.1080/03605302.2013.845044.

[13]

S. CingolaniL. Jeanjean and K. Tanaka, Multiplicity of positive solutions of nonlinear Schrödinger equations concentrating at a potential well, Calc. Var. Partial Differential Equations, 53 (2015), 413-439. doi: 10.1007/s00526-014-0754-5.

[14]

S. CingolaniL. Jeanjean and K. Tanaka, Multiple complex-valued solutions for nonlinear magnetic Schrödinger equations, J. Fixed Point Theory Appl., 19 (2017), 37-66. doi: 10.1007/s11784-016-0347-3.

[15]

D. G. Costa and H. Tehrani, On a class of asymptotically linear elliptic problems in ${\mathbb R}^N$, J. Differential Equations, 173 (2001), 470-494. doi: 10.1006/jdeq.2000.3944.

[16]

R. de Marchi and R. Ruviaro, Existence of solutions for a nonperiodic semilinear Schrödinger equation, Complex Var. Elliptic Equ., 61 (2016), 1290-1302. doi: 10.1080/17476933.2016.1167887.

[17]

F. Della Casa and A. Sacchetti, Stationary states for non linear one-dimensional Schrödinger equations with singular potential, Phys. D, 219 (2006), 60-68. doi: 10.1016/j.physd.2006.05.014.

[18]

S. DengD. Garrido and M. Musso, Multiple blow-up solutions for an exponential nonlinearity with potential in ${\mathbb R}^2$, Nonlinear Anal., 119 (2015), 419-442. doi: 10.1016/j.na.2014.10.034.

[19]

Y. Ding and Z. Wang, Bound states of nonlinear Schrödinger equations with magnetic fields, Annali di Matematica, 190 (2011), 427-451. doi: 10.1007/s10231-010-0157-y.

[20]

M. Fei and H. Yin, Bound states of asymptotically linear Schrödinger equations with compactly supported potential, Pacific J. of Math., 261 (2013), 335-367. doi: 10.2140/pjm.2013.261.335.

[21]

R. Fukuizumi and L. Jeanjean, Stability of standing waves for a nonlinear Schrödinger equation with repulsive Dirac delta potential, Disc. and Cont. Dynamical Systems, 21 (2008), 121-136. doi: 10.3934/dcds.2008.21.121.

[22]

F. GazzolaJ. Serrin and M. Tang, Existence of ground states and free boundary problems for quasilinear elliptic operators, Adv. in Diff. Eq., 5 (2000), 1-30.

[23]

J. F. Lam, B. Lippmann and F. Tappert, Self-trapped laser beams in plasma, Physics of Fluids, 20 (1977), 1176. doi: 10.1063/1.861679.

[24]

M. Lewin, Mean-field limit of Bose systems: Rigorous results, Proceedings from the International Congress of Mathematical Physics at Santiago de Chile, 2015.

[25]

M. LewinP. T. Nam and N. Rougerie, Derivation of Hartree's theory for generic mean-field Bose systems, Adv. Math., 254 (2014), 570-621. doi: 10.1016/j.aim.2013.12.010.

[26]

M. LewinP. T. Nam and N. Rougerie, The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases, Trans. Amer. Math. Soc., 368 (2016), 6131-6157. doi: 10.1090/tran/6537.

[27]

Z. LiuJ. Su and T. Weth, Compactness results for Schrödinger equations with asymptotically linear terms, J. Differential Equations, 231 (2006), 501-512. doi: 10.1016/j.jde.2006.05.007.

[28]

C. LiuZ. Wangand and H.-S. Zhou, Asymptotically linear Schrödinger equation with potential vanishing at infinity, J. Differential Equations, 245 (2008), 201-222. doi: 10.1016/j.jde.2008.01.006.

[29]

K. W. Mahmud, J. N. Kutz and W. P. Reinhardt, Bose-Einstein condensates in a one-dimensional double square well: Analytical solutions of the nonlinear Schrödinger equation, Physical Review A, 66 (2002), 063607, 11 pages.

[30]

B. Malomed and D. Pelinovsky, Persistence of the Thomas-Fermi approximation for ground states of the Gross-Pitaevskii equation supported by the nonlinear confinement, Applied Mathematics Letters, 40 (2015), 45-48. doi: 10.1016/j.aml.2014.09.004.

[31]

M. I. Molina and C. A. Bustamante, The Attractive Nonlinear Delta-function Potential, American Journal of Physics 70, 67 (2002); doi: http://dx.doi.org/10.1119/1.1417529.

[32]

A. R. Nahmod, The nonlinear Schrödinger equation on tori: Integrating harmonic analysis, geometry, and probability, Bull. Amer. Math. Soc. (N.S.), 53 (2016), 57-91. doi: 10.1090/bull/1516.

[33]

D. E. Pelinovsky, Localization in Periodic Potentials. From Schr ödinger operators to the Gross-Pitaevskii equation (London Mathematical Society Lecture Note Series), 1st Ed., Cambridge University Press, 2011. doi: 10.1017/CBO9780511997754.

[34]

P. Pucci and R. Servadei, Existence, non-existence and regularity of radial ground states for p-Laplacian equations with singular weights, Ann. Inst. H. Poincaré, 25 (2008), 505-537. doi: 10.1016/j.anihpc.2007.02.004.

[35]

J. J. Rasmussen and K. Rypdal, Blow-up in nonlinear Schrödinger equations-I, A general review, Physica Scripta, 33 (1986), 481-497. doi: 10.1088/0031-8949/33/6/001.

[36]

S. Sheng, F. Wang and T. An, Existence and multiplicity of positive bound states for Schrödinger equations, Boundary Value Problems, 2013 (2013), 11pp. doi: 10.1186/1687-2770-2013-271.

[37]

A. Soffer and M. I. Weinstein, Selection of the ground state for nonlinear Schrödinger equations, Reviews in Mathematical Physics, 16 (2004), 977-1071. doi: 10.1142/S0129055X04002175.

[38]

C. Sourdis, On the existence of dark solitons of the defocusing cubic nonlinear Schrödinger equation with periodic inhomogeneous nonlinearity, Applied Mathematics Letters, 46 (2015), 123-126. doi: 10.1016/j.aml.2015.02.018.

[39]

C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation. Self-focusing and Wave Collapse (Applied Mathematical Sciences), Springer, 1999.

[40]

T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis (CBMS Regional Conference Series in Mathematics), American Mathematical Society, 2006. doi: 10.1090/cbms/106.

[41]

T. Tao, A global compact attractor for high-dimensional defocusing non-linear Schrödinger equations with potential, Dynamics of PDE, 5 (2008), 101-116. doi: 10.4310/DPDE.2008.v5.n2.a1.

[42]

T. Tsai, Asymptotic dynamics of nonlinear Schrödinger equations with many bound states, J. Od Diff. Eq., 192 (2003), 225-282. doi: 10.1016/S0022-0396(03)00041-X.

[43]

C. E. Wayne and M. I. Weinstein, Dynamics of Partial Differential Equations (Frontiers in Applied Dynamical Systems: Reviews and Tutorials), Springer, 2015. doi: 10.1007/978-3-319-19935-1.

[44]

M. Willem, Minimax Theorems (Progress in Nonlinear Differential Equations and Their Applications), Birkhäuser, 1996. doi: 10.1007/978-1-4612-4146-1.

show all references

References:
[1]

M. J. Ablowitz, B. Prinari and A. D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems, London Mathematical Society Lecture Note Series, 302, Cambridge University Press, 2004.

[2]

A. AmbrosettiA. Malchiodi and D. Ruiz, Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Anal. Math., 98 (2006), 317-348. doi: 10.1007/BF02790279.

[3]

G. Arioli and A. Szulkin, Semilinear Schrödinger equation in the presence of a magnetic field, Arch. Ration. Mech. Anal., 170 (2003), 277-295. doi: 10.1007/s00205-003-0274-5.

[4]

J. Bourgain, On nonlinear Schrödinger equations in Les Relations Entre Les Mathématiques et la Physique Théorique, Inst. Hautes Études Sci., Bures-sur-Yvette, (1998), 11-21.

[5]

J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations, American Mathematical Society Colloquium Publications, 46. American Mathematical Society, 1999. doi: 10.1090/coll/046.

[6]

N. Burq and M. Zworski, Instability for the Semiclassical Non-linear Schrödinger equation, Commun. Math. Phys., 260 (2005), 45-58. doi: 10.1007/s00220-005-1402-x.

[7]

J. ByeonL. Jeanjean and K. Tanaka, Standing waves for nonlinear Schrödinger equations with a general nonlinearity: one and two dimensional cases, Comm. Partial Differential Equations, 33 (2008), 1113-1136. doi: 10.1080/03605300701518174.

[8]

C. CacciapuotiD. FincoD. Noja and A. Teta, The NLS equation in dimension one with spatially concentrated nonlinearities: the point like limit, Lett. Math. Phys., 104 (2014), 1557-1570. doi: 10.1007/s11005-014-0725-y.

[9]

R. Carretero-GonzáalezaJ. D. TalleyC. Chong and B. A. Malomed, Multistable solitons in the cubic-quintic discrete nonlinear Schrödinger equation, Physica D, 216 (2006), 77-89. doi: 10.1016/j.physd.2006.01.022.

[10]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, American Mathematical Society, 2003. doi: 10.1090/cln/010.

[11]

T. Chen and N. Pavlović, The quintic NLS as the mean field limit of a boson gas with three-body interactions, J. Funct. Anal., 260 (2011), 959-997. doi: 10.1016/j.jfa.2010.11.003.

[12]

H. ChristiansonJ. MarzuolaJ. Metcalfe and M. Taylor, Nonlinear Bound States on Weakly Homogeneous Spaces, Communications in Partial Differential Equations, 39 (2014), 34-97. doi: 10.1080/03605302.2013.845044.

[13]

S. CingolaniL. Jeanjean and K. Tanaka, Multiplicity of positive solutions of nonlinear Schrödinger equations concentrating at a potential well, Calc. Var. Partial Differential Equations, 53 (2015), 413-439. doi: 10.1007/s00526-014-0754-5.

[14]

S. CingolaniL. Jeanjean and K. Tanaka, Multiple complex-valued solutions for nonlinear magnetic Schrödinger equations, J. Fixed Point Theory Appl., 19 (2017), 37-66. doi: 10.1007/s11784-016-0347-3.

[15]

D. G. Costa and H. Tehrani, On a class of asymptotically linear elliptic problems in ${\mathbb R}^N$, J. Differential Equations, 173 (2001), 470-494. doi: 10.1006/jdeq.2000.3944.

[16]

R. de Marchi and R. Ruviaro, Existence of solutions for a nonperiodic semilinear Schrödinger equation, Complex Var. Elliptic Equ., 61 (2016), 1290-1302. doi: 10.1080/17476933.2016.1167887.

[17]

F. Della Casa and A. Sacchetti, Stationary states for non linear one-dimensional Schrödinger equations with singular potential, Phys. D, 219 (2006), 60-68. doi: 10.1016/j.physd.2006.05.014.

[18]

S. DengD. Garrido and M. Musso, Multiple blow-up solutions for an exponential nonlinearity with potential in ${\mathbb R}^2$, Nonlinear Anal., 119 (2015), 419-442. doi: 10.1016/j.na.2014.10.034.

[19]

Y. Ding and Z. Wang, Bound states of nonlinear Schrödinger equations with magnetic fields, Annali di Matematica, 190 (2011), 427-451. doi: 10.1007/s10231-010-0157-y.

[20]

M. Fei and H. Yin, Bound states of asymptotically linear Schrödinger equations with compactly supported potential, Pacific J. of Math., 261 (2013), 335-367. doi: 10.2140/pjm.2013.261.335.

[21]

R. Fukuizumi and L. Jeanjean, Stability of standing waves for a nonlinear Schrödinger equation with repulsive Dirac delta potential, Disc. and Cont. Dynamical Systems, 21 (2008), 121-136. doi: 10.3934/dcds.2008.21.121.

[22]

F. GazzolaJ. Serrin and M. Tang, Existence of ground states and free boundary problems for quasilinear elliptic operators, Adv. in Diff. Eq., 5 (2000), 1-30.

[23]

J. F. Lam, B. Lippmann and F. Tappert, Self-trapped laser beams in plasma, Physics of Fluids, 20 (1977), 1176. doi: 10.1063/1.861679.

[24]

M. Lewin, Mean-field limit of Bose systems: Rigorous results, Proceedings from the International Congress of Mathematical Physics at Santiago de Chile, 2015.

[25]

M. LewinP. T. Nam and N. Rougerie, Derivation of Hartree's theory for generic mean-field Bose systems, Adv. Math., 254 (2014), 570-621. doi: 10.1016/j.aim.2013.12.010.

[26]

M. LewinP. T. Nam and N. Rougerie, The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases, Trans. Amer. Math. Soc., 368 (2016), 6131-6157. doi: 10.1090/tran/6537.

[27]

Z. LiuJ. Su and T. Weth, Compactness results for Schrödinger equations with asymptotically linear terms, J. Differential Equations, 231 (2006), 501-512. doi: 10.1016/j.jde.2006.05.007.

[28]

C. LiuZ. Wangand and H.-S. Zhou, Asymptotically linear Schrödinger equation with potential vanishing at infinity, J. Differential Equations, 245 (2008), 201-222. doi: 10.1016/j.jde.2008.01.006.

[29]

K. W. Mahmud, J. N. Kutz and W. P. Reinhardt, Bose-Einstein condensates in a one-dimensional double square well: Analytical solutions of the nonlinear Schrödinger equation, Physical Review A, 66 (2002), 063607, 11 pages.

[30]

B. Malomed and D. Pelinovsky, Persistence of the Thomas-Fermi approximation for ground states of the Gross-Pitaevskii equation supported by the nonlinear confinement, Applied Mathematics Letters, 40 (2015), 45-48. doi: 10.1016/j.aml.2014.09.004.

[31]

M. I. Molina and C. A. Bustamante, The Attractive Nonlinear Delta-function Potential, American Journal of Physics 70, 67 (2002); doi: http://dx.doi.org/10.1119/1.1417529.

[32]

A. R. Nahmod, The nonlinear Schrödinger equation on tori: Integrating harmonic analysis, geometry, and probability, Bull. Amer. Math. Soc. (N.S.), 53 (2016), 57-91. doi: 10.1090/bull/1516.

[33]

D. E. Pelinovsky, Localization in Periodic Potentials. From Schr ödinger operators to the Gross-Pitaevskii equation (London Mathematical Society Lecture Note Series), 1st Ed., Cambridge University Press, 2011. doi: 10.1017/CBO9780511997754.

[34]

P. Pucci and R. Servadei, Existence, non-existence and regularity of radial ground states for p-Laplacian equations with singular weights, Ann. Inst. H. Poincaré, 25 (2008), 505-537. doi: 10.1016/j.anihpc.2007.02.004.

[35]

J. J. Rasmussen and K. Rypdal, Blow-up in nonlinear Schrödinger equations-I, A general review, Physica Scripta, 33 (1986), 481-497. doi: 10.1088/0031-8949/33/6/001.

[36]

S. Sheng, F. Wang and T. An, Existence and multiplicity of positive bound states for Schrödinger equations, Boundary Value Problems, 2013 (2013), 11pp. doi: 10.1186/1687-2770-2013-271.

[37]

A. Soffer and M. I. Weinstein, Selection of the ground state for nonlinear Schrödinger equations, Reviews in Mathematical Physics, 16 (2004), 977-1071. doi: 10.1142/S0129055X04002175.

[38]

C. Sourdis, On the existence of dark solitons of the defocusing cubic nonlinear Schrödinger equation with periodic inhomogeneous nonlinearity, Applied Mathematics Letters, 46 (2015), 123-126. doi: 10.1016/j.aml.2015.02.018.

[39]

C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation. Self-focusing and Wave Collapse (Applied Mathematical Sciences), Springer, 1999.

[40]

T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis (CBMS Regional Conference Series in Mathematics), American Mathematical Society, 2006. doi: 10.1090/cbms/106.

[41]

T. Tao, A global compact attractor for high-dimensional defocusing non-linear Schrödinger equations with potential, Dynamics of PDE, 5 (2008), 101-116. doi: 10.4310/DPDE.2008.v5.n2.a1.

[42]

T. Tsai, Asymptotic dynamics of nonlinear Schrödinger equations with many bound states, J. Od Diff. Eq., 192 (2003), 225-282. doi: 10.1016/S0022-0396(03)00041-X.

[43]

C. E. Wayne and M. I. Weinstein, Dynamics of Partial Differential Equations (Frontiers in Applied Dynamical Systems: Reviews and Tutorials), Springer, 2015. doi: 10.1007/978-3-319-19935-1.

[44]

M. Willem, Minimax Theorems (Progress in Nonlinear Differential Equations and Their Applications), Birkhäuser, 1996. doi: 10.1007/978-1-4612-4146-1.

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