March 2018, 8(1): 155-176. doi: 10.3934/mcrf.2018007

Frequency-sparse optimal quantum control

1. 

Faculty of Mathematics, Technical University Munich, Boltzmannstr. 3, D-85747 Garching, Germany

2. 

Institute of Mathematics, FU Berlin, Arnimallee 6, D-14195 Berlin, Germany

3. 

Institute of Mathematics and Scientific Computing, Karl-Franzens-Universität Graz, Heinrichstr. 36, A-8010 Graz, Austria

4. 

Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstraße 69, A-4040 Linz, Austria

* Corresponding author: Karl Kunisch

Received  April 2017 Revised  October 2017 Published  January 2018

A new class of cost functionals for optimal control of quantum systems which produces controls which are sparse in frequency and smooth in time is proposed. This is achieved by penalizing a suitable time-frequency representation of the control field, rather than the control field itself, and by employing norms which are of $L^1$ or measure form with respect to frequency but smooth with respect to time.

We prove existence of optimal controls for the resulting nonsmooth optimization problem, derive necessary optimality conditions, and rigorously establish the frequency-sparsity of the optimizers. More precisely, we show that the time-frequency representation of the control field, which a priori admits a continuum of frequencies, is supported on only finitely many frequencies. These results cover important systems of physical interest, including (infinite-dimensional) Schrödinger dynamics on multiple potential energy surfaces as arising in laser control of chemical reactions. Numerical simulations confirm that the optimal controls, unlike those obtained with the usual $L^2$ costs, concentrate on just a few frequencies, even in the infinite-dimensional case of laser-controlled chemical reactions.

Citation: Gero Friesecke, Felix Henneke, Karl Kunisch. Frequency-sparse optimal quantum control. Mathematical Control & Related Fields, 2018, 8 (1) : 155-176. doi: 10.3934/mcrf.2018007
References:
[1]

A. AugerA. BenH. YedderE. CancesC. L. BrisC. M. DionA. Keller and O. Atabek, Optimal laser control of molecular systems: Methodology and results, Math. Models Methods Appl. Sci, 12 (2012), 1281-1315.

[2] G. G. Balint-KurtiS. Zou and A. Brown, Optimal control theory for manipulating molecular processes, John Wiley & Sons, Inc., 2008.
[3]

J. M. BallJ. E. Marsden and M. Slemrod, Controllability for distributed bilinear systems, SIAM J. Control Optim., 20 (1982), 575-597.

[4]

H. BergmannH. Theuer and B. W. Shore, Coherent population transfer among quantum states of atoms and molecules, Rev. Mod. Phys., 70 (1998), 1003-1025.

[5]

E. CasasR. Herzog and G. Wachsmuth, Optimality conditions and error analysis of semilinear elliptic control problems with $L^1$ cost functional, SIAM J. Optim., 22 (2012), 795-820.

[6]

G. Ciaramella and A. Borzì, A {LONE} code for the sparse control of quantum systems, Computer Physics Communications, 200 (2016), 312-323.

[7]

G. Ciaramella and A. Borzì, Quantum optimal control problems with a sparsity cost functional, Numer. Funct. Anal. Optim., 37 (2016), 938-965.

[8]

F. H. Clarke, Optimization and Nonsmooth Analysis, vol. 5 of Classics in Applied Mathematics, 2nd edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990.

[9]

C. Clason and K. Kunisch, A duality-based approach to elliptic control problems in non-reflexive Banach spaces, ESAIM Control Optim. Calc. Var., 17 (2011), 243-266.

[10]

D. D'Alessandro, Introduction to Quantum Control and Dynamics, Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series, Chapman & Hall/CRC, Boca Raton, FL, 2008.

[11]

S. J. GlaserU. BoscainT. CalarcoC. P. KochW. KöckenbergerR. KosloffI. KuprovB. LuyS. SchirmerT. Schulte-HerbrüggenD. Sugny and F. K. Wilhelm, Training Schrödinger's cat: Quantum optimal control, The European Physical Journal D, 69 (2015), 1-24.

[12]

M. HellgrenE. Räsänen and E. K. U. Gross, Optimal control of strong-field ionization with time-dependent density-functional theory, Phys. Rev. A, 88 (2013), 013414.

[13]

F. Henneke and M. Liebmann, A generalized Suzuki-Trotter type method in optimal control of coupled Schrödinger equations, Computing and Visualization in Science, 17 (2015), 277-293.

[14]

R. HerzogG. Stadler and G. Wachsmuth, Directional sparsity in optimal control of partial differential equations, SIAM J. Control Optim., 50 (2012), 943-963.

[15]

M. HintermüllerD. MarahrensP. A. Markowich and C. Sparber, Optimal bilinear control of Gross-Pitaevskii equations, SIAM J. Control Optim., 51 (2013), 2509-2543.

[16]

P. v. d. HoffS. ThallmairM. KowalewskiR. Siemering and R. D. Vivie-Riedle, Optimal control theory -closing the gap between theory and experiment, Phys. Chem. Chem. Phys., 14 (2012), 14460-14485.

[17]

K. Ito and K. Kunisch, Optimal bilinear control of an abstract Schrödinger equation, SIAM Journal on Control and Optimization, 46 (2007), 274-287.

[18]

A. F. IzmailovA. L. Pogosyan and M. V. Solodov, Semismooth SQP method for equalityconstrained optimization problems with an application to the lifted reformulation of mathematical programs with complementarity constraints, Optimization Methods and Software, 26 (2011), 847-872.

[19]

K. KormannS. Holmgren and H. Karlsson, A Fourier-coefficient based solution of an optimal control problem in quantum chemistry, Journal of Optimization Theory and Applications, 174 (2010), 491-506.

[20]

K. KunischK. Pieper and B. Vexler, Measure valued directional sparsity for parabolic optimal control problems, SIAM J. Control Optim., 52 (2014), 3078-3108.

[21]

S. Lang, Real and Functional Analysis, vol. 142 of Graduate Texts in Mathematics, 3rd edition, Springer-Verlag, New York, 1993.

[22]

M. LapertD. SugnyR. Tehini and G. Turinici, Monotonically convergent optimal control theory of quantum systems with spectral constraints on the control field, Physical Review A: Atomic, Molecular and Optical Physics, 79 (2009), 063411.

[23]

X. J. Li and J. M. Yong, Optimal Control Theory for Infinite-Dimensional Systems Systems & Control: Foundations & Applications, Birkhäuser Boston Inc., Boston, MA, 1995.

[24]

L. Meziani, On the dual space $C^*_0(S, X)$, Acta Math. Univ. Comenian. (N.S.), 78 (2009), 153-160.

[25]

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd edition, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006.

[26]

A. P. PeirceM. A. Dahleh and H. Rabitz, Optimal control of quantum-mechanical systems: Existence, numerical approximation, and applications, Phys. Rev. A (3), 37 (1988), 4950-4964.

[27] M. Reed and B. Simon, Methods of Modern Mathematical Physics. Ⅱ. Fourier analysis, Self-adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975.
[28]

Q. RenG. G. Balint-KurtiF. R. ManbyM. ArtamonovT.-S. Ho and H. Rabitz, Quantum control of molecular vibrational and rotational excitations in a homonuclear diatomic molecule: A full three-dimensional treatment with polarization forces, The Journal of Chemical Physics, 124 (2006), 014111.

[29]

S. RuetzelC. StolzenbergerF. DimlerD. J. Tannor and T. Brixner, Adaptive coherent control using the von Neumann basis, Phys. Chem. Chem. Phys., 13 (2011), 8627-8636.

[30]

J. ScheuerX. KongR. S. SaidJ. ChenA. KurzL. MarsegliaJ. DuP. R. HemmerS. MontangeroT. CalarcoB. Naydenov and F. Jelezko, Precise qubit control beyond the rotating wave approximation, New Journal of Physics, 16 (2014), 093022.

[31]

S. SharmaH. Singh and G. G. Balint-Kurti, Genetic algorithm optimization of laser pulses for molecular quantum state excitation, The Journal of Chemical Physics, 132 (2010), 064108.

[32] B. W. Shore, The Theory of Coherent Atomic Excitation, Wiley, New York, NY, 1990.
[33]

G. Stadler, Elliptic optimal control problems with $L^1$-control cost and applications for the placement of control devices, Comput. Optim. Appl., 44 (2009), 159-181.

[34]

S. Teufel, Adiabatic Perturbation Theory in Quantum Dynamics, vol. 1821 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2003.

[35]

G. TuriniciC. Le Bris and H. Rabitz, Efficient algorithms for the laboratory discovery of optimal quantum controls, Phys. Rev. E, 70 (2004), 016704.

[36]

G. von Winckel and A. Borzì, Computational techniques for a quantum control problem with $H^1$-cost, Inverse Problems, 24 (2008), 034007, 23pp.

[37]

G. von WinckelA. Borzì and S. Volkwein, A globalized Newton method for the accurate solution of a dipole quantum control problem, SIAM J. Sci. Comput., 31 (2009/10), 4176-4203.

[38]

G. Vossen and H. Maurer, On $L^1$-minimization in optimal control and applications to robotics, Optimal Control Appl. Methods, 27 (2006), 301-321.

show all references

References:
[1]

A. AugerA. BenH. YedderE. CancesC. L. BrisC. M. DionA. Keller and O. Atabek, Optimal laser control of molecular systems: Methodology and results, Math. Models Methods Appl. Sci, 12 (2012), 1281-1315.

[2] G. G. Balint-KurtiS. Zou and A. Brown, Optimal control theory for manipulating molecular processes, John Wiley & Sons, Inc., 2008.
[3]

J. M. BallJ. E. Marsden and M. Slemrod, Controllability for distributed bilinear systems, SIAM J. Control Optim., 20 (1982), 575-597.

[4]

H. BergmannH. Theuer and B. W. Shore, Coherent population transfer among quantum states of atoms and molecules, Rev. Mod. Phys., 70 (1998), 1003-1025.

[5]

E. CasasR. Herzog and G. Wachsmuth, Optimality conditions and error analysis of semilinear elliptic control problems with $L^1$ cost functional, SIAM J. Optim., 22 (2012), 795-820.

[6]

G. Ciaramella and A. Borzì, A {LONE} code for the sparse control of quantum systems, Computer Physics Communications, 200 (2016), 312-323.

[7]

G. Ciaramella and A. Borzì, Quantum optimal control problems with a sparsity cost functional, Numer. Funct. Anal. Optim., 37 (2016), 938-965.

[8]

F. H. Clarke, Optimization and Nonsmooth Analysis, vol. 5 of Classics in Applied Mathematics, 2nd edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990.

[9]

C. Clason and K. Kunisch, A duality-based approach to elliptic control problems in non-reflexive Banach spaces, ESAIM Control Optim. Calc. Var., 17 (2011), 243-266.

[10]

D. D'Alessandro, Introduction to Quantum Control and Dynamics, Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series, Chapman & Hall/CRC, Boca Raton, FL, 2008.

[11]

S. J. GlaserU. BoscainT. CalarcoC. P. KochW. KöckenbergerR. KosloffI. KuprovB. LuyS. SchirmerT. Schulte-HerbrüggenD. Sugny and F. K. Wilhelm, Training Schrödinger's cat: Quantum optimal control, The European Physical Journal D, 69 (2015), 1-24.

[12]

M. HellgrenE. Räsänen and E. K. U. Gross, Optimal control of strong-field ionization with time-dependent density-functional theory, Phys. Rev. A, 88 (2013), 013414.

[13]

F. Henneke and M. Liebmann, A generalized Suzuki-Trotter type method in optimal control of coupled Schrödinger equations, Computing and Visualization in Science, 17 (2015), 277-293.

[14]

R. HerzogG. Stadler and G. Wachsmuth, Directional sparsity in optimal control of partial differential equations, SIAM J. Control Optim., 50 (2012), 943-963.

[15]

M. HintermüllerD. MarahrensP. A. Markowich and C. Sparber, Optimal bilinear control of Gross-Pitaevskii equations, SIAM J. Control Optim., 51 (2013), 2509-2543.

[16]

P. v. d. HoffS. ThallmairM. KowalewskiR. Siemering and R. D. Vivie-Riedle, Optimal control theory -closing the gap between theory and experiment, Phys. Chem. Chem. Phys., 14 (2012), 14460-14485.

[17]

K. Ito and K. Kunisch, Optimal bilinear control of an abstract Schrödinger equation, SIAM Journal on Control and Optimization, 46 (2007), 274-287.

[18]

A. F. IzmailovA. L. Pogosyan and M. V. Solodov, Semismooth SQP method for equalityconstrained optimization problems with an application to the lifted reformulation of mathematical programs with complementarity constraints, Optimization Methods and Software, 26 (2011), 847-872.

[19]

K. KormannS. Holmgren and H. Karlsson, A Fourier-coefficient based solution of an optimal control problem in quantum chemistry, Journal of Optimization Theory and Applications, 174 (2010), 491-506.

[20]

K. KunischK. Pieper and B. Vexler, Measure valued directional sparsity for parabolic optimal control problems, SIAM J. Control Optim., 52 (2014), 3078-3108.

[21]

S. Lang, Real and Functional Analysis, vol. 142 of Graduate Texts in Mathematics, 3rd edition, Springer-Verlag, New York, 1993.

[22]

M. LapertD. SugnyR. Tehini and G. Turinici, Monotonically convergent optimal control theory of quantum systems with spectral constraints on the control field, Physical Review A: Atomic, Molecular and Optical Physics, 79 (2009), 063411.

[23]

X. J. Li and J. M. Yong, Optimal Control Theory for Infinite-Dimensional Systems Systems & Control: Foundations & Applications, Birkhäuser Boston Inc., Boston, MA, 1995.

[24]

L. Meziani, On the dual space $C^*_0(S, X)$, Acta Math. Univ. Comenian. (N.S.), 78 (2009), 153-160.

[25]

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd edition, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006.

[26]

A. P. PeirceM. A. Dahleh and H. Rabitz, Optimal control of quantum-mechanical systems: Existence, numerical approximation, and applications, Phys. Rev. A (3), 37 (1988), 4950-4964.

[27] M. Reed and B. Simon, Methods of Modern Mathematical Physics. Ⅱ. Fourier analysis, Self-adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975.
[28]

Q. RenG. G. Balint-KurtiF. R. ManbyM. ArtamonovT.-S. Ho and H. Rabitz, Quantum control of molecular vibrational and rotational excitations in a homonuclear diatomic molecule: A full three-dimensional treatment with polarization forces, The Journal of Chemical Physics, 124 (2006), 014111.

[29]

S. RuetzelC. StolzenbergerF. DimlerD. J. Tannor and T. Brixner, Adaptive coherent control using the von Neumann basis, Phys. Chem. Chem. Phys., 13 (2011), 8627-8636.

[30]

J. ScheuerX. KongR. S. SaidJ. ChenA. KurzL. MarsegliaJ. DuP. R. HemmerS. MontangeroT. CalarcoB. Naydenov and F. Jelezko, Precise qubit control beyond the rotating wave approximation, New Journal of Physics, 16 (2014), 093022.

[31]

S. SharmaH. Singh and G. G. Balint-Kurti, Genetic algorithm optimization of laser pulses for molecular quantum state excitation, The Journal of Chemical Physics, 132 (2010), 064108.

[32] B. W. Shore, The Theory of Coherent Atomic Excitation, Wiley, New York, NY, 1990.
[33]

G. Stadler, Elliptic optimal control problems with $L^1$-control cost and applications for the placement of control devices, Comput. Optim. Appl., 44 (2009), 159-181.

[34]

S. Teufel, Adiabatic Perturbation Theory in Quantum Dynamics, vol. 1821 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2003.

[35]

G. TuriniciC. Le Bris and H. Rabitz, Efficient algorithms for the laboratory discovery of optimal quantum controls, Phys. Rev. E, 70 (2004), 016704.

[36]

G. von Winckel and A. Borzì, Computational techniques for a quantum control problem with $H^1$-cost, Inverse Problems, 24 (2008), 034007, 23pp.

[37]

G. von WinckelA. Borzì and S. Volkwein, A globalized Newton method for the accurate solution of a dipole quantum control problem, SIAM J. Sci. Comput., 31 (2009/10), 4176-4203.

[38]

G. Vossen and H. Maurer, On $L^1$-minimization in optimal control and applications to robotics, Optimal Control Appl. Methods, 27 (2006), 301-321.

Figure 1.  Schematic representation of laser-controlled chemical reaction dynamics. The nuclei of a molecule move on different potential energy surfaces depending on the electronic state, and the laser induces transitions between these states. Blue: Potential energy surfaces. Magenta: Initial wave function of the nuclei. Cyan: Target region
Figure 2.  Optimal control when the cost is chosen as a measure norm with respect to frequency and the $H^1_0$ norm with respect to time (system: Example 2.1, cost: Example 3.1). (A) Control field $Bu(t)$ as a function of time. (B) The contributions due to the two active frequencies of the optimal field. (C) Time-frequency representation $u(\omega, t)$ (color indicates absolute value).
Figure 3.  Detailed numerical illustration of frequency sparsity of the optimal control from Figure 2. In Figure 3(A) it is depicted that the numerical optimal control (dots in (A)), drops by three orders of magnitude below the threshold given by the numerical regularization parameter $\theta$ (dashed line in (A)), away from the coincidence set $\lVert (B^* {\mathop{\rm Re}\nolimits}\langle \varphi, \tilde H \psi \rangle)(\omega) \rVert_{\mathcal U} = \alpha$, precisely as theoretically predicted by equation (34). In Figure 3(B) we illustrate the quantifiers of Theorem 5.1, which asserts that the optimal control should vanish off the frequencies where the norm $\lVert (B^* {\mathop{\rm Re}\nolimits}\langle \varphi, \tilde H \psi \rangle)(\omega) \rVert_{\mathcal U}$ (dots in (B)) reaches the cost parameter $\alpha$ (solid line in (B))
Figure 4.  Optimal controls for Schrödinger dynamics on two potential energy surfaces (Example 2.2). Rows: Different choices of cost functionals and control operators. Columns: Time, frequency, and time-frequency representation of the optimal controls (i.e. $(Bu)(t)$, $\lvert u \rvert(\omega)$ and $u(\omega, t)$). The dashed red line in the middle column, rows 1 to 3, indicates the Huber threshold, and the nonzero contributions below it are an artifact of the Huber regularization (see Theorem 5.1). In the rightmost column, the absolute values of the optimal measures are plotted in the time-frequency plane. Note that in column 1, 2, and 4 far fewer frequencies are active compared to the standard $L^2$ control (column 5)
Table 1.  Cost parameters $\alpha$ and expectation values $f(\bar u) = \frac{1}{2} \langle \bar \psi, \mathcal O \bar \psi \rangle$ for different control spaces. A value of $f(\bar u) = 2.5 \cdot 10^{-2}$ corresponds to a $95\%$ achievement of the control goal
control space $\alpha$ $f(\bar u)$
$\mathcal M(\Omega; H^1_0)$ $0.03$ $4.49 \cdot 10^{-3}$
$\mathcal M(\Omega; L^2)$ $0.03$ $2.69 \cdot 10^{-2}$
$\mathcal M(\Omega; \mathbb {C})$ $0.06$ $2.24 \cdot 10^{-2}$
$\mathcal M(\Omega\times[0, T]; \mathbb {C})$ $0.02$ $1.89 \cdot 10^{-2}$
$L^2(0, T)$ $0.0001$ $5.39 \cdot 10^{-6}$
control space $\alpha$ $f(\bar u)$
$\mathcal M(\Omega; H^1_0)$ $0.03$ $4.49 \cdot 10^{-3}$
$\mathcal M(\Omega; L^2)$ $0.03$ $2.69 \cdot 10^{-2}$
$\mathcal M(\Omega; \mathbb {C})$ $0.06$ $2.24 \cdot 10^{-2}$
$\mathcal M(\Omega\times[0, T]; \mathbb {C})$ $0.02$ $1.89 \cdot 10^{-2}$
$L^2(0, T)$ $0.0001$ $5.39 \cdot 10^{-6}$
[1]

Roberta Ghezzi, Benedetto Piccoli. Optimal control of a multi-level dynamic model for biofuel production. Mathematical Control & Related Fields, 2017, 7 (2) : 235-257. doi: 10.3934/mcrf.2017008

[2]

Jean-Philippe Cointet, David Chavalarias. Multi-level science mapping with asymmetrical paradigmatic proximity. Networks & Heterogeneous Media, 2008, 3 (2) : 267-276. doi: 10.3934/nhm.2008.3.267

[3]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[4]

Kai Wang, Dun Zhao, Binhua Feng. Optimal nonlinearity control of Schrödinger equation. Evolution Equations & Control Theory, 2018, 7 (2) : 317-334. doi: 10.3934/eect.2018016

[5]

Yohei Sato. Sign-changing multi-peak solutions for nonlinear Schrödinger equations with critical frequency. Communications on Pure & Applied Analysis, 2008, 7 (4) : 883-903. doi: 10.3934/cpaa.2008.7.883

[6]

Kai-Ling Mak, Pai Peng, Ka-Fai Cedric Yiu. Fabric defect detection using multi-level tuned-matched Gabor filters. Journal of Industrial & Management Optimization, 2012, 8 (2) : 325-341. doi: 10.3934/jimo.2012.8.325

[7]

Andrew J. Majda, Yuan Yuan. Fundamental limitations of Ad hoc linear and quadratic multi-level regression models for physical systems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1333-1363. doi: 10.3934/dcdsb.2012.17.1333

[8]

Gökçe Dİlek Küçük, Gabil Yagub, Ercan Çelİk. On the existence and uniqueness of the solution of an optimal control problem for Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 503-512. doi: 10.3934/dcdss.2019033

[9]

François Genoud. Existence and stability of high frequency standing waves for a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1229-1247. doi: 10.3934/dcds.2009.25.1229

[10]

Claudianor O. Alves, Minbo Yang. Existence of positive multi-bump solutions for a Schrödinger-Poisson system in $\mathbb{R}^{3}$. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5881-5910. doi: 10.3934/dcds.2016058

[11]

Harald Friedrich. Semiclassical and large quantum number limits of the Schrödinger equation. Conference Publications, 2003, 2003 (Special) : 288-294. doi: 10.3934/proc.2003.2003.288

[12]

Shi Jin, Peng Qi. A hybrid Schrödinger/Gaussian beam solver for quantum barriers and surface hopping. Kinetic & Related Models, 2011, 4 (4) : 1097-1120. doi: 10.3934/krm.2011.4.1097

[13]

Camille Laurent. Internal control of the Schrödinger equation. Mathematical Control & Related Fields, 2014, 4 (2) : 161-186. doi: 10.3934/mcrf.2014.4.161

[14]

Bernard Bonnard, Jean-Baptiste Caillau, Olivier Cots. Energy minimization in two-level dissipative quantum control: Th e integrable case. Conference Publications, 2011, 2011 (Special) : 198-208. doi: 10.3934/proc.2011.2011.198

[15]

Fengping Yao. Optimal regularity for parabolic Schrödinger operators. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1407-1414. doi: 10.3934/cpaa.2013.12.1407

[16]

Giulia Cavagnari, Antonio Marigonda, Benedetto Piccoli. Optimal synchronization problem for a multi-agent system. Networks & Heterogeneous Media, 2017, 12 (2) : 277-295. doi: 10.3934/nhm.2017012

[17]

Yohei Yamazaki. Transverse instability for a system of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 565-588. doi: 10.3934/dcdsb.2014.19.565

[18]

José Luis López, Jesús Montejo-Gámez. On viscous quantum hydrodynamics associated with nonlinear Schrödinger-Doebner-Goldin models. Kinetic & Related Models, 2012, 5 (3) : 517-536. doi: 10.3934/krm.2012.5.517

[19]

David Damanik, Serguei Tcheremchantsev. A general description of quantum dynamical spreading over an orthonormal basis and applications to Schrödinger operators. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1381-1412. doi: 10.3934/dcds.2010.28.1381

[20]

Vianney Combet. Multi-existence of multi-solitons for the supercritical nonlinear Schrödinger equation in one dimension. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1961-1993. doi: 10.3934/dcds.2014.34.1961

2017 Impact Factor: 0.631

Metrics

  • PDF downloads (47)
  • HTML views (220)
  • Cited by (0)

Other articles
by authors

[Back to Top]