
-
Previous Article
Input-to-state stability of continuous-time systems via finite-time Lyapunov functions
- DCDS-B Home
- This Issue
- Next Article
Derivation of cable equation by multiscale analysis for a model of myelinated axons
1. | Faculty of Engineering and Sciences, Universidad Adolfo Ibáñez, Diagonal Las Torres 2700, Peñalolén, Santiago, Chile |
2. | Department of Electrical Engineering, Mathematics and Science, University of Gävle, SE-801 76 Gävle, Sweden |
3. | B. Verkin Institute for Low Temperature Physics and Engineering of NASU, 47 Nauky Ave., 61103 Kharkiv, Ukraine |
We derive a one-dimensional cable model for the electric potential propagation along an axon. Since the typical thickness of an axon is much smaller than its length, and the myelin sheath is distributed periodically along the neuron, we simplify the problem geometry to a thin cylinder with alternating myelinated and unmyelinated parts. Both the microstructure period and the cylinder thickness are assumed to be of order $ \varepsilon $, a small positive parameter. Assuming a nonzero conductivity of the myelin sheath, we find a critical scaling with respect to $ \varepsilon $ which leads to the appearance of an additional potential in the homogenized nonlinear cable equation. This potential contains information about the geometry of the myelin sheath in the original three-dimensional model.
References:
[1] |
G. Allaire, Commissariat 'a L'energie Atomique, Alain Damlamian, and Ulrich Hornung, Two-scale convergence on periodic surfaces and applications, Mathematical Modelling of Flow through Porous Media, 1995. Google Scholar |
[2] |
M. Amar, D. Andreucci, P. Bisegna and R. Gianni,
On a hierarchy of models for electrical conduction in biological tissues, Mathematical Methods in the Applied Sciences, 29 (2006), 767-787.
doi: 10.1002/mma.709. |
[3] |
M. Amar, D. Andreucci, P. Bisegna, R. Gianni and et al.,
A hierarchy of models for the electrical conduction in biological tissues via two-scale convergence: the nonlinear case, Differential and Integral Equations, 26 (2013), 885-912.
|
[4] |
M. Amar, D. Andreucci and R. Gianni, Asymptotic decay under nonlinear and noncoercive dissipative effects for electrical conduction in biological tissues, Nonlinear Differ. Equ. Appl., 23 (2016), Art. 48, 24 pp.
doi: 10.1007/s00030-016-0396-8. |
[5] |
H. Ammari, L. Giovangigli, H. Kwon, J. K. Seo and T. Wintz,
Spectroscopic conductivity imaging of a cell culture, Asymptotic Analysis, 100 (2016), 87-109.
doi: 10.3233/ASY-161387. |
[6] |
H. Ammari, T. Widlak and W. Zhang,
Towards monitoring critical microscopic parameters for electropermeabilization, Quart. Appl. Math., 75 (2017), 1-17.
doi: 10.1090/qam/1449. |
[7] |
T. Arbogast, J. Douglas Jr and and U. Hornung,
Derivation of the double porosity model of single phase flow via homogenization theory, SIAM Journal on Mathematical Analysis, 21 (1990), 823-836.
doi: 10.1137/0521046. |
[8] |
P. J. Basser, Cable equation for a myelinated axon derived from its microstructure, Medical & Biological Engineering & Computing, 31 (1993), S87–S92.
doi: 10.1007/BF02446655. |
[9] |
P. J. Basser and B. J. Roth,
New currents in electrical stimulation of excitable tissues, Annu. Rev. Biomed. Eng., 2 (2000), 377-397.
doi: 10.1146/annurev.bioeng.2.1.377. |
[10] |
C. Bollini and F. Cacheiro,
Peripheral nerve stimulation, Tech. Reg. Anesth. Pain Manag., 10 (2006), 79-88.
doi: 10.1053/j.trap.2006.07.007. |
[11] |
S. O. Choi, Y. C. Kim, J. W. Lee, J. H. Park, M. R. Prausnitz and M. G. Allen,
Intracellular protein delivery and gene transfection by electroporation using a microneedle electrode array, Small, 10 (2012), 1081-1091.
doi: 10.1002/smll.201101747. |
[12] |
P. Colli-Franzone, L. F. Pavarino and S. Scacchi, Mathematical and numerical methods for reaction-diffusion models in electrocardiology, In Davide Ambrosi, Alfio Quarteroni, and Gianluigi Rozza, editors, Modeling of Physiological Flows, Springer Milan, Milano, 2012,107–141. Google Scholar |
[13] |
S. Doi, J. Inoue, Z. Pan and K. Tsumoto, Computational Electrophysiology, volume 2., Tokyo, Japan: Springer Series, A First Course in On Silico Medicine, 2010.
doi: 10.1007/978-4-431-53862-2. |
[14] |
F. Henríquez and C. Jerez-Hanckes,
Multiple traces formulation and semi-implicit scheme for modelling biological cells under electrical stimulation, ESAIM: M2AN, 52 (2018), 659-703.
doi: 10.1051/m2an/2018019. |
[15] |
F. Henríquez, C. Jerez-Hanckes and F. Altermatt,
Boundary integral formulation and semi-implicit scheme coupling for modeling cells under electrical stimulation, Numer. Math., 136 (2016), 101-145.
doi: 10.1007/s00211-016-0835-9. |
[16] |
A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, The Journal of Physiology, 117 (1952), 500. Google Scholar |
[17] |
J. Keener and J. Sneyd, Mathematical Physiology I: Cellular Physiology, Springer-Verlag, New York, 1998. |
[18] |
E. Y. Khruslov, Homogenized models of strongly inhomogeneous media, In Proceedings of the International Congress of Mathematicians, Springer, 1995, 1270–1278. |
[19] |
A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Investigation of the equation of diffusion combined with increasing of the substance and its application to a biology problem, Bull. Moscow State Univ. Ser. A: Math. Mech, 1 (1937), 1-25. Google Scholar |
[20] |
D. C. Li and Q. Li,
Electrical stimulation of cortical neurons promotes oligodendrocyte development and remyelination in the injured spinal, Neural Regeneration Research, 12 (2017), 1613-1615.
doi: 10.4103/1673-5374.217330. |
[21] |
A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkh"auser/Springer Basel AG, Basel, 1995. |
[22] |
V. A. Marchenko and E. Y. Khruslov,
Boundary-value problems with fine-grained boundary, Matematicheskii Sbornik, 65 (1964), 458-472.
|
[23] |
H. Matano and Y. Mori,
Global existence and uniqueness of a three-dimensional model of cellular electrophysiology, Discrete Contin. Dyn. Syst, 29 (2011), 1573-1636.
doi: 10.3934/dcds.2011.29.1573. |
[24] |
V. G. Maz'ya, Sobolev Spaces, Springer-Verlag, 1985.
doi: 10.1007/978-3-662-09922-3. |
[25] |
C. C. McIntyre, A. G. Richardson and W. M. Grill,
Modeling the excitability of mammalian nerve fibers: Influence of afterpotentials on the recovery cycle, Journal of Neurophysiology, 87 (2002), 995-1006.
doi: 10.1152/jn.00353.2001. |
[26] |
H. Meffin, B. Tahayori, E. N. Sergeev, I. M. Y. Mareels, D. B. Grayden and A. N. Burkitt, Modelling extracellular electrical stimulation: Ⅲ. derivation and interpretation of neural tissue equations, Journal of Neural Engineering, 11 (2014), 065004.
doi: 10.1088/1741-2560/11/6/065004. |
[27] |
C. Meunier and B. Lamotte d'Incamps,
Extending cable theory to heterogeneous dendrites, Neural Computation, 20 (2008), 1732-1775.
doi: 10.1162/neco.2008.12-06-425. |
[28] |
L. M. Mir, M. F. Bureau, J. Gehl, R. Rangara, D. Rouy, J. M. Caillaud, P. Delaere, D. Branellec, B. Schwartz and D. Scherman,
High-efficiency gene transfer into skeletal muscle mediated by electric pulses, Proc. Nat. Acad. Sci. USA., 96 (1999), 4262-4267.
doi: 10.1073/pnas.96.8.4262. |
[29] |
J. C. Neu and W. Krassowska, Homogenization of syncytial tissues, Critical Reviews in Biomedical Engineering, 21 (1993), 137-199. Google Scholar |
[30] |
M. Neuss-Radu, Some extensions of two-scale convergence, Comptes Rendus de l'Académie des Sciences. Série 1, Mathématique, 322 (1996), 899–904. |
[31] |
P. D. O'Brien, L. M. Hinder, B. C. Callaghan and E. L. Feldman,
Neurological consequences of obesity, The Lancet Neurology, 16 (2017), 465-477.
doi: 10.1016/S1474-4422(17)30084-4. |
[32] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[33] |
M. Pennacchio, G. Savaré and P. C. Franzone,
Multiscale modeling for the bioelectric activity of the heart, SIAM Journal on Mathematical Analysis, 37 (2005), 1333-1370.
doi: 10.1137/040615249. |
[34] |
I. Pettersson,
Two-scale convergence in thin domains with locally periodic rapidly oscillating boundary, Differential Equations & Applications, 9 (2017), 393-412.
doi: 10.7153/dea-2017-09-28. |
[35] |
C. Pham-Dang, O. Kick, T. Collet, F. Gouin and M. Pinaud, Continuous peripheral nerve blocks with stimulating catheters, Reg. Anesth. Pain Med., 28 (2003), 83-88. Google Scholar |
[36] |
R. Plonsey and R. C. Barr, Bioelectricity: A Quantitative Approach, Springer, Boston, MA, 2007. |
[37] |
W. Rall,
Time constants and electrotonic length of membrane cylinders and neurons, Biophysical Journal, 9 (1969), 1483-1508.
doi: 10.1016/S0006-3495(69)86467-2. |
[38] |
F. Rattay, Electrical Nerve Stimulation. Theory, Experiments and Applications, Vienna, Austria: Springer–Verlag, 1990. Google Scholar |
[39] |
J. E. Riggs,
The aging population, Neurologic Clinics, 16 (1998), 555-560.
doi: 10.1016/S0733-8619(05)70079-7. |
[40] |
J. Teissié, N. Eynard, B. Gabriel and M. P. Rols, Electropermeabilization of cell membranes, Adv. Drug. Del. Rev, 35 (1999), 3-19. Google Scholar |
[41] |
V. Zhikov,
On an extension and an application of the two-scale convergence method, Sb. Math., 191 (2000), 973-1014.
doi: 10.1070/SM2000v191n07ABEH000491. |
show all references
References:
[1] |
G. Allaire, Commissariat 'a L'energie Atomique, Alain Damlamian, and Ulrich Hornung, Two-scale convergence on periodic surfaces and applications, Mathematical Modelling of Flow through Porous Media, 1995. Google Scholar |
[2] |
M. Amar, D. Andreucci, P. Bisegna and R. Gianni,
On a hierarchy of models for electrical conduction in biological tissues, Mathematical Methods in the Applied Sciences, 29 (2006), 767-787.
doi: 10.1002/mma.709. |
[3] |
M. Amar, D. Andreucci, P. Bisegna, R. Gianni and et al.,
A hierarchy of models for the electrical conduction in biological tissues via two-scale convergence: the nonlinear case, Differential and Integral Equations, 26 (2013), 885-912.
|
[4] |
M. Amar, D. Andreucci and R. Gianni, Asymptotic decay under nonlinear and noncoercive dissipative effects for electrical conduction in biological tissues, Nonlinear Differ. Equ. Appl., 23 (2016), Art. 48, 24 pp.
doi: 10.1007/s00030-016-0396-8. |
[5] |
H. Ammari, L. Giovangigli, H. Kwon, J. K. Seo and T. Wintz,
Spectroscopic conductivity imaging of a cell culture, Asymptotic Analysis, 100 (2016), 87-109.
doi: 10.3233/ASY-161387. |
[6] |
H. Ammari, T. Widlak and W. Zhang,
Towards monitoring critical microscopic parameters for electropermeabilization, Quart. Appl. Math., 75 (2017), 1-17.
doi: 10.1090/qam/1449. |
[7] |
T. Arbogast, J. Douglas Jr and and U. Hornung,
Derivation of the double porosity model of single phase flow via homogenization theory, SIAM Journal on Mathematical Analysis, 21 (1990), 823-836.
doi: 10.1137/0521046. |
[8] |
P. J. Basser, Cable equation for a myelinated axon derived from its microstructure, Medical & Biological Engineering & Computing, 31 (1993), S87–S92.
doi: 10.1007/BF02446655. |
[9] |
P. J. Basser and B. J. Roth,
New currents in electrical stimulation of excitable tissues, Annu. Rev. Biomed. Eng., 2 (2000), 377-397.
doi: 10.1146/annurev.bioeng.2.1.377. |
[10] |
C. Bollini and F. Cacheiro,
Peripheral nerve stimulation, Tech. Reg. Anesth. Pain Manag., 10 (2006), 79-88.
doi: 10.1053/j.trap.2006.07.007. |
[11] |
S. O. Choi, Y. C. Kim, J. W. Lee, J. H. Park, M. R. Prausnitz and M. G. Allen,
Intracellular protein delivery and gene transfection by electroporation using a microneedle electrode array, Small, 10 (2012), 1081-1091.
doi: 10.1002/smll.201101747. |
[12] |
P. Colli-Franzone, L. F. Pavarino and S. Scacchi, Mathematical and numerical methods for reaction-diffusion models in electrocardiology, In Davide Ambrosi, Alfio Quarteroni, and Gianluigi Rozza, editors, Modeling of Physiological Flows, Springer Milan, Milano, 2012,107–141. Google Scholar |
[13] |
S. Doi, J. Inoue, Z. Pan and K. Tsumoto, Computational Electrophysiology, volume 2., Tokyo, Japan: Springer Series, A First Course in On Silico Medicine, 2010.
doi: 10.1007/978-4-431-53862-2. |
[14] |
F. Henríquez and C. Jerez-Hanckes,
Multiple traces formulation and semi-implicit scheme for modelling biological cells under electrical stimulation, ESAIM: M2AN, 52 (2018), 659-703.
doi: 10.1051/m2an/2018019. |
[15] |
F. Henríquez, C. Jerez-Hanckes and F. Altermatt,
Boundary integral formulation and semi-implicit scheme coupling for modeling cells under electrical stimulation, Numer. Math., 136 (2016), 101-145.
doi: 10.1007/s00211-016-0835-9. |
[16] |
A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, The Journal of Physiology, 117 (1952), 500. Google Scholar |
[17] |
J. Keener and J. Sneyd, Mathematical Physiology I: Cellular Physiology, Springer-Verlag, New York, 1998. |
[18] |
E. Y. Khruslov, Homogenized models of strongly inhomogeneous media, In Proceedings of the International Congress of Mathematicians, Springer, 1995, 1270–1278. |
[19] |
A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Investigation of the equation of diffusion combined with increasing of the substance and its application to a biology problem, Bull. Moscow State Univ. Ser. A: Math. Mech, 1 (1937), 1-25. Google Scholar |
[20] |
D. C. Li and Q. Li,
Electrical stimulation of cortical neurons promotes oligodendrocyte development and remyelination in the injured spinal, Neural Regeneration Research, 12 (2017), 1613-1615.
doi: 10.4103/1673-5374.217330. |
[21] |
A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkh"auser/Springer Basel AG, Basel, 1995. |
[22] |
V. A. Marchenko and E. Y. Khruslov,
Boundary-value problems with fine-grained boundary, Matematicheskii Sbornik, 65 (1964), 458-472.
|
[23] |
H. Matano and Y. Mori,
Global existence and uniqueness of a three-dimensional model of cellular electrophysiology, Discrete Contin. Dyn. Syst, 29 (2011), 1573-1636.
doi: 10.3934/dcds.2011.29.1573. |
[24] |
V. G. Maz'ya, Sobolev Spaces, Springer-Verlag, 1985.
doi: 10.1007/978-3-662-09922-3. |
[25] |
C. C. McIntyre, A. G. Richardson and W. M. Grill,
Modeling the excitability of mammalian nerve fibers: Influence of afterpotentials on the recovery cycle, Journal of Neurophysiology, 87 (2002), 995-1006.
doi: 10.1152/jn.00353.2001. |
[26] |
H. Meffin, B. Tahayori, E. N. Sergeev, I. M. Y. Mareels, D. B. Grayden and A. N. Burkitt, Modelling extracellular electrical stimulation: Ⅲ. derivation and interpretation of neural tissue equations, Journal of Neural Engineering, 11 (2014), 065004.
doi: 10.1088/1741-2560/11/6/065004. |
[27] |
C. Meunier and B. Lamotte d'Incamps,
Extending cable theory to heterogeneous dendrites, Neural Computation, 20 (2008), 1732-1775.
doi: 10.1162/neco.2008.12-06-425. |
[28] |
L. M. Mir, M. F. Bureau, J. Gehl, R. Rangara, D. Rouy, J. M. Caillaud, P. Delaere, D. Branellec, B. Schwartz and D. Scherman,
High-efficiency gene transfer into skeletal muscle mediated by electric pulses, Proc. Nat. Acad. Sci. USA., 96 (1999), 4262-4267.
doi: 10.1073/pnas.96.8.4262. |
[29] |
J. C. Neu and W. Krassowska, Homogenization of syncytial tissues, Critical Reviews in Biomedical Engineering, 21 (1993), 137-199. Google Scholar |
[30] |
M. Neuss-Radu, Some extensions of two-scale convergence, Comptes Rendus de l'Académie des Sciences. Série 1, Mathématique, 322 (1996), 899–904. |
[31] |
P. D. O'Brien, L. M. Hinder, B. C. Callaghan and E. L. Feldman,
Neurological consequences of obesity, The Lancet Neurology, 16 (2017), 465-477.
doi: 10.1016/S1474-4422(17)30084-4. |
[32] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[33] |
M. Pennacchio, G. Savaré and P. C. Franzone,
Multiscale modeling for the bioelectric activity of the heart, SIAM Journal on Mathematical Analysis, 37 (2005), 1333-1370.
doi: 10.1137/040615249. |
[34] |
I. Pettersson,
Two-scale convergence in thin domains with locally periodic rapidly oscillating boundary, Differential Equations & Applications, 9 (2017), 393-412.
doi: 10.7153/dea-2017-09-28. |
[35] |
C. Pham-Dang, O. Kick, T. Collet, F. Gouin and M. Pinaud, Continuous peripheral nerve blocks with stimulating catheters, Reg. Anesth. Pain Med., 28 (2003), 83-88. Google Scholar |
[36] |
R. Plonsey and R. C. Barr, Bioelectricity: A Quantitative Approach, Springer, Boston, MA, 2007. |
[37] |
W. Rall,
Time constants and electrotonic length of membrane cylinders and neurons, Biophysical Journal, 9 (1969), 1483-1508.
doi: 10.1016/S0006-3495(69)86467-2. |
[38] |
F. Rattay, Electrical Nerve Stimulation. Theory, Experiments and Applications, Vienna, Austria: Springer–Verlag, 1990. Google Scholar |
[39] |
J. E. Riggs,
The aging population, Neurologic Clinics, 16 (1998), 555-560.
doi: 10.1016/S0733-8619(05)70079-7. |
[40] |
J. Teissié, N. Eynard, B. Gabriel and M. P. Rols, Electropermeabilization of cell membranes, Adv. Drug. Del. Rev, 35 (1999), 3-19. Google Scholar |
[41] |
V. Zhikov,
On an extension and an application of the two-scale convergence method, Sb. Math., 191 (2000), 973-1014.
doi: 10.1070/SM2000v191n07ABEH000491. |


[1] |
Yongjie Wang, Nan Gao. Some properties for almost cellular algebras. Electronic Research Archive, 2021, 29 (1) : 1681-1689. doi: 10.3934/era.2020086 |
[2] |
Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366 |
[3] |
Alain Damlamian, Klas Pettersson. Homogenization of oscillating boundaries. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 197-219. doi: 10.3934/dcds.2009.23.197 |
[4] |
Thomas Y. Hou, Dong Liang. Multiscale analysis for convection dominated transport equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 281-298. doi: 10.3934/dcds.2009.23.281 |
[5] |
Leslaw Skrzypek, Yuncheng You. Feedback synchronization of FHN cellular neural networks. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021001 |
[6] |
Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079 |
[7] |
Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020448 |
[8] |
Vo Van Au, Hossein Jafari, Zakia Hammouch, Nguyen Huy Tuan. On a final value problem for a nonlinear fractional pseudo-parabolic equation. Electronic Research Archive, 2021, 29 (1) : 1709-1734. doi: 10.3934/era.2020088 |
[9] |
Patrick Martinez, Judith Vancostenoble. Lipschitz stability for the growth rate coefficients in a nonlinear Fisher-KPP equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 695-721. doi: 10.3934/dcdss.2020362 |
[10] |
Mohammad Ghani, Jingyu Li, Kaijun Zhang. Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021017 |
[11] |
Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076 |
[12] |
Monia Capanna, Jean C. Nakasato, Marcone C. Pereira, Julio D. Rossi. Homogenization for nonlocal problems with smooth kernels. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020385 |
[13] |
Eduard Marušić-Paloka, Igor Pažanin. Homogenization and singular perturbation in porous media. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020279 |
[14] |
Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020317 |
[15] |
Gi-Chan Bae, Christian Klingenberg, Marlies Pirner, Seok-Bae Yun. BGK model of the multi-species Uehling-Uhlenbeck equation. Kinetic & Related Models, 2021, 14 (1) : 25-44. doi: 10.3934/krm.2020047 |
[16] |
Olivier Pironneau, Alexei Lozinski, Alain Perronnet, Frédéric Hecht. Numerical zoom for multiscale problems with an application to flows through porous media. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 265-280. doi: 10.3934/dcds.2009.23.265 |
[17] |
Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247 |
[18] |
José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020376 |
[19] |
Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276 |
[20] |
Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259 |
2019 Impact Factor: 1.27
Tools
Metrics
Other articles
by authors
[Back to Top]