-
Previous Article
Free boundary problems associated with cancer treatment by combination therapy
- DCDS Home
- This Issue
-
Next Article
Minimizers of the $ p $-oscillation functional
Superfluids passing an obstacle and vortex nucleation
1. | Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY. 10012, USA |
2. | Department of Mathematics, University of British Columbia, Vancouver BC V6T 1Z2, Canada |
$ \epsilon^2 \Delta u+ u(1-|u|^2) = 0 \ \mbox{in} \ {\mathbb R}^d \backslash \Omega, \ \ \frac{\partial u}{\partial \nu} = 0 \ \mbox{on}\ \partial \Omega $ |
$ \Omega $ |
$ {\mathbb R}^d $ |
$ d\geq 2 $ |
$ \epsilon>0 $ |
$ u = \rho_\epsilon (x) e^{i \frac{\Phi_\epsilon}{\epsilon}} $ |
$ \rho_\epsilon (x) \to 1-|\nabla \Phi^\delta(x)|^2, \Phi_\epsilon (x) \to \Phi^\delta (x) $ |
$ \Phi^\delta (x) $ |
$ \nabla ( (1-|\nabla \Phi|^2)\nabla \Phi ) = 0 \ \mbox{in} \ {\mathbb R}^d \backslash \Omega, \ \frac{\partial \Phi}{\partial \nu} = 0 \ \mbox{on} \ \partial \Omega, \ \nabla \Phi (x) \to \delta \vec{e}_d \ \mbox{as} \ |x| \to +\infty $ |
$ |\delta | <\delta_{*} $ |
$ d = 2 $ |
$ |\nabla \Phi^\delta (x)|^2 $ |
References:
[1] |
A. Aftalion, Q. Du and Y. Pomeau,
Dissipative flow and vortex shedding in the Painleve boundary layer of a Bose-Einstein condensate, Phys. Rev. Lett., 91 (2003), 090407-1-4.
|
[2] |
F. Bethuel and J.-C. Saut,
Travelling waves for the Gross-Pitaevskii equation. Ⅰ, Ann. Inst. H. Poincare' Phys. The'or., 70 (1999), 147-238.
|
[3] |
F. Bethuel, G. Orlandi and D. Smets,
Vortex rings for the Gross-Pitaevskii equation, J. Eur. Math. Soc., 6 (2004), 17-94.
|
[4] |
F. Bethuel, P. Gravejat and J.-G. Saut,
Travelling waves for the Gross-Pitaevskii equation, Ⅱ, Comm. Math. Phys., 285 (2009), 567-651.
doi: 10.1007/s00220-008-0614-2. |
[5] |
F. Bethuel, H. Brezis and F. He'lein,
Asymptotics for the minimization of a Ginzburg-Landau functional, Calc. Var. and PDE., 1 (1993), 123-148.
doi: 10.1007/BF01191614. |
[6] |
F. Bethuel, H. Brezis and F. He'lein, Ginzburg-Landau Vortices, Birkha"user, Boston, 1994
doi: 10.1007/978-1-4612-0287-5. |
[7] |
F. Bethuel, P. Gravejat and J.-C. Saut,
Travelling waves for the Gross-Pitaevskii equation. Ⅱ, Comm. Math. Phys., 285 (2009), 567-651.
doi: 10.1007/s00220-008-0614-2. |
[8] |
L. Bers,
Ezistence and uniqueness of a subsonic pow past a given profile, Comm. Pure Appl. Math., 7 (1954), 441-504.
doi: 10.1002/cpa.3160070303. |
[9] |
L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics, John Wiley and Sons, New York, 1958. |
[10] |
S. Byun and L. Wang,
The conormal derivative problem for elliptic equations with BMO coefficients on Reifenberg flat domains, Proc. Lond. Math. Soc., 90 (2005), 245-272.
doi: 10.1112/S0024611504014960. |
[11] |
R. Carles, R. Danchin and J.-C. Saut,
Madelung, Gross-Pitaevskii and Korteweg, Nonlinearity, 25 (2012), 2843-2873.
doi: 10.1088/0951-7715/25/10/2843. |
[12] |
D. Chiron and M. Maris,
Rarefaction pulses for the nonlinear Schrödinger equation in the transonic limit, Comm. Math. Phys., 326 (2014), 329-392.
doi: 10.1007/s00220-013-1879-7. |
[13] |
C. Coste,
Nonlinear Schrodinger equation and superfluid hydrodynamics, Eur. Phys. J. B Condens. Matter Phys., 1 (1998), 245-253.
doi: 10.1007/s100510050178. |
[14] |
M. del Pino, M. Kowalczyk and J. Wei,
Entire solutions of the Allen-Cahn equation and complete embedded minimal surfaces of finite total curvature, Journal of Differential Geometry, 83 (2013), 67-131.
doi: 10.4310/jdg/1357141507. |
[15] |
M. del Pino, M. Kowalczyk and M. Musso,
Variational reduction for Ginzburg-Landau vortices, J. Funct. Anal., 239 (2006), 497-541.
doi: 10.1016/j.jfa.2006.07.006. |
[16] |
G.-C. Dong and B. Ou,
Subsonic flows around a body in space, Comm. Partial Differential Equations, 18 (1993), 355-379.
doi: 10.1080/03605309308820933. |
[17] |
M. del Pino, P. Felmer and M. Kowalczyk,
Minimality and nondegeneracy of degree-one Ginzburg-Landau vortex as a Hardy's type inequality, Int. Math. Res. Not., (2004), 1511-1527.
doi: 10.1155/S1073792804133588. |
[18] |
Q. Du, J. Wei and C. Zhao,
Vortex solutions of the high-$\kappa$ high-field Ginzburg-Landau model with an applied current, SIAM J. Math. Anal., 42 (2010), 2368-2401.
doi: 10.1137/090769983. |
[19] |
R. Finn and D. Gilbarg,
Three dimensional subsonicflows and asymptotic estimates for elliptic partial differential equations, Acta Math., 98 (1957), 265-296.
doi: 10.1007/BF02404476. |
[20] |
T. Frisch, Y. Pomeau and S. Rica,
Transition to dissipation in a model of superflow, Phys. Rev. Lett., 69 (1992), 1644-1647.
doi: 10.1103/PhysRevLett.69.1644. |
[21] |
P. Gravejat,
Asymptotics for the travelling waves in the Gross-Pitaevskii equation, Asymptot. Anal., 45 (2005), 227-299.
|
[22] |
P. Gravejat,
Limit at infinity and nonexistence results for sonic travelling waves in the Gross-Pitaevskii equation, Differential Integral Equations, 17 (2004), 1213-1232.
|
[23] |
P. Gravejat,
Decay for travelling waves in the Gross-Pitaevskii equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 591-637.
doi: 10.1016/j.anihpc.2003.09.001. |
[24] |
P. Gravejat,
A non-existence result for supersonic travelling waves in the Gross-Pitaevskii equation, Comm. Math. Phys., 243 (2003), 93-103.
doi: 10.1007/s00220-003-0961-y. |
[25] |
J. Grant and P. H. Roberts,
Motions in a Bose condensate. Ⅲ. The structure and effective masses of charged and uncharged impurities, J. Phys. A: Math., Nucl. Gen., 7 (1974), 260-279.
doi: 10.1088/0305-4470/34/1/306. |
[26] |
C. Huepe and M. E. Brachet,
Scaling laws for vortical nucleation solutions in a model of superflow, Phys. D, 140 (2000), 126-140.
doi: 10.1016/S0167-2789(99)00229-8. |
[27] |
M. Abid, C. Huepe, S. Metens, C. Nore, C. T. Pham, L. S. Tuckerman and M. E. Brachet,
Gross-Pitaevskii dynamics of Bose-Einstein condensates and superfluid turbulence, Fluid Dynam. Res., 33 (2003), 509-544.
doi: 10.1016/j.fluiddyn.2003.09.001. |
[28] |
C. A. Jones, S. J. Putterman and P. H. Roberts,
Stability of wave solutions of nonlinear Schrodinger equations in two and three dimensions, J. Phys A: Math. Gen., 19 (1986), 2991-3011.
|
[29] |
C. A. Jones and P. H. Roberts,
Motion in a Bose condensate Ⅳ, Axisymmetric solitary waves, J. Phys. A, 15 (1982), 2599-2619.
doi: 10.1088/0305-4470/15/8/036. |
[30] |
C. Josserand and Y. Pomeau,
Nonlinear aspects of the theory of Bose-Einstein condensates, Nonlinearity, 14 (2001), R25-R62.
doi: 10.1088/0951-7715/14/5/201. |
[31] |
C. Josserand, Y. Pomeau and S. Rica,
Vortex shedding in a model of superflow, Phys. D, 134 (1999), 111-125.
doi: 10.1016/S0167-2789(99)00066-4. |
[32] |
L. Landau and E. Lifshitz,
On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sowj, 8 (1935), 153.
|
[33] |
F.-H. Lin and J. Wei,
Traveling wave solutions of Schrödinger map equation, Comm. Pure Appl. Math., 63 (2010), 1585-1621.
doi: 10.1002/cpa.20338. |
[34] |
F.-H. Lin and P. Zhang,
Semiclassical limit of the Gross-Pitaevskii equation in an exterior domain, Arch. Ration. Mech. Anal., 179 (2006), 79-107.
doi: 10.1007/s00205-005-0383-4. |
[35] |
Y. Liu and J. Wei, Adler-Moser polynomials and traveling waves solutions of Gross-Pitaevskii, preprint. |
[36] |
P. I. Lizorkin,
Multipliers of Fourier integrals, Proc. Steklov Inst. Math., 89 (1967), 269-290.
|
[37] |
M. Maris,
Existence of nonstationary bubbles in higher dimensions, J. Math. Pures Appl., 81 (2002), 1207-1239.
doi: 10.1016/S0021-7824(02)01274-6. |
[38] |
M. Maris,
Nonexistence of supersonic traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity, SIAM J. Math. Anal., 40 (2008), 1076-1103.
doi: 10.1137/070711189. |
[39] |
M. Maris,
Traveling waves for nonlinear Schrodinger equations with nonzero conditions at infinity, Ann. Math., 178 (2013), 107-182.
doi: 10.4007/annals.2013.178.1.2. |
[40] |
C.-T. Pham, C. Nore and M. E. Brachet,
Boundary layers and emitted excitations in nonlinear Schröinger superflow past a disk, Phys. D, 210 (2005), 203-226.
doi: 10.1016/j.physd.2005.07.013. |
[41] |
S. Rica,
A remark on the critical speed of vortex nucleation in the nonlinear Schrodinger equation, Phys. D, 148 (2001), 221-226.
doi: 10.1016/S0167-2789(00)00168-8. |
[42] |
O. Rey and J. Wei,
Blowing up solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. Part Ⅱ: $N \geq 4$, Ann. Non linearie, Annoles de l'Institut H. Poincaré, 22 (2005), 459-484.
doi: 10.1016/j.anihpc.2004.07.004. |
[43] |
M. Shiffman,
On the ezistence of subsonic flows of a compressible fluid, Arch. Rational Mech. Anal., 2 (1952), 605-652.
doi: 10.1512/iumj.1952.1.51020. |
[44] |
E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, vol. 32, Princeton University Press, Princeton, NJ, 1971. |
[45] |
J. Wei,
Uniqueness and critical spectrum of boundary spike solutions, Proc. Royal Soc. Edin. A, 131 (2001), 1457-1480.
doi: 10.1017/S0308210500001487. |
show all references
References:
[1] |
A. Aftalion, Q. Du and Y. Pomeau,
Dissipative flow and vortex shedding in the Painleve boundary layer of a Bose-Einstein condensate, Phys. Rev. Lett., 91 (2003), 090407-1-4.
|
[2] |
F. Bethuel and J.-C. Saut,
Travelling waves for the Gross-Pitaevskii equation. Ⅰ, Ann. Inst. H. Poincare' Phys. The'or., 70 (1999), 147-238.
|
[3] |
F. Bethuel, G. Orlandi and D. Smets,
Vortex rings for the Gross-Pitaevskii equation, J. Eur. Math. Soc., 6 (2004), 17-94.
|
[4] |
F. Bethuel, P. Gravejat and J.-G. Saut,
Travelling waves for the Gross-Pitaevskii equation, Ⅱ, Comm. Math. Phys., 285 (2009), 567-651.
doi: 10.1007/s00220-008-0614-2. |
[5] |
F. Bethuel, H. Brezis and F. He'lein,
Asymptotics for the minimization of a Ginzburg-Landau functional, Calc. Var. and PDE., 1 (1993), 123-148.
doi: 10.1007/BF01191614. |
[6] |
F. Bethuel, H. Brezis and F. He'lein, Ginzburg-Landau Vortices, Birkha"user, Boston, 1994
doi: 10.1007/978-1-4612-0287-5. |
[7] |
F. Bethuel, P. Gravejat and J.-C. Saut,
Travelling waves for the Gross-Pitaevskii equation. Ⅱ, Comm. Math. Phys., 285 (2009), 567-651.
doi: 10.1007/s00220-008-0614-2. |
[8] |
L. Bers,
Ezistence and uniqueness of a subsonic pow past a given profile, Comm. Pure Appl. Math., 7 (1954), 441-504.
doi: 10.1002/cpa.3160070303. |
[9] |
L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics, John Wiley and Sons, New York, 1958. |
[10] |
S. Byun and L. Wang,
The conormal derivative problem for elliptic equations with BMO coefficients on Reifenberg flat domains, Proc. Lond. Math. Soc., 90 (2005), 245-272.
doi: 10.1112/S0024611504014960. |
[11] |
R. Carles, R. Danchin and J.-C. Saut,
Madelung, Gross-Pitaevskii and Korteweg, Nonlinearity, 25 (2012), 2843-2873.
doi: 10.1088/0951-7715/25/10/2843. |
[12] |
D. Chiron and M. Maris,
Rarefaction pulses for the nonlinear Schrödinger equation in the transonic limit, Comm. Math. Phys., 326 (2014), 329-392.
doi: 10.1007/s00220-013-1879-7. |
[13] |
C. Coste,
Nonlinear Schrodinger equation and superfluid hydrodynamics, Eur. Phys. J. B Condens. Matter Phys., 1 (1998), 245-253.
doi: 10.1007/s100510050178. |
[14] |
M. del Pino, M. Kowalczyk and J. Wei,
Entire solutions of the Allen-Cahn equation and complete embedded minimal surfaces of finite total curvature, Journal of Differential Geometry, 83 (2013), 67-131.
doi: 10.4310/jdg/1357141507. |
[15] |
M. del Pino, M. Kowalczyk and M. Musso,
Variational reduction for Ginzburg-Landau vortices, J. Funct. Anal., 239 (2006), 497-541.
doi: 10.1016/j.jfa.2006.07.006. |
[16] |
G.-C. Dong and B. Ou,
Subsonic flows around a body in space, Comm. Partial Differential Equations, 18 (1993), 355-379.
doi: 10.1080/03605309308820933. |
[17] |
M. del Pino, P. Felmer and M. Kowalczyk,
Minimality and nondegeneracy of degree-one Ginzburg-Landau vortex as a Hardy's type inequality, Int. Math. Res. Not., (2004), 1511-1527.
doi: 10.1155/S1073792804133588. |
[18] |
Q. Du, J. Wei and C. Zhao,
Vortex solutions of the high-$\kappa$ high-field Ginzburg-Landau model with an applied current, SIAM J. Math. Anal., 42 (2010), 2368-2401.
doi: 10.1137/090769983. |
[19] |
R. Finn and D. Gilbarg,
Three dimensional subsonicflows and asymptotic estimates for elliptic partial differential equations, Acta Math., 98 (1957), 265-296.
doi: 10.1007/BF02404476. |
[20] |
T. Frisch, Y. Pomeau and S. Rica,
Transition to dissipation in a model of superflow, Phys. Rev. Lett., 69 (1992), 1644-1647.
doi: 10.1103/PhysRevLett.69.1644. |
[21] |
P. Gravejat,
Asymptotics for the travelling waves in the Gross-Pitaevskii equation, Asymptot. Anal., 45 (2005), 227-299.
|
[22] |
P. Gravejat,
Limit at infinity and nonexistence results for sonic travelling waves in the Gross-Pitaevskii equation, Differential Integral Equations, 17 (2004), 1213-1232.
|
[23] |
P. Gravejat,
Decay for travelling waves in the Gross-Pitaevskii equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 591-637.
doi: 10.1016/j.anihpc.2003.09.001. |
[24] |
P. Gravejat,
A non-existence result for supersonic travelling waves in the Gross-Pitaevskii equation, Comm. Math. Phys., 243 (2003), 93-103.
doi: 10.1007/s00220-003-0961-y. |
[25] |
J. Grant and P. H. Roberts,
Motions in a Bose condensate. Ⅲ. The structure and effective masses of charged and uncharged impurities, J. Phys. A: Math., Nucl. Gen., 7 (1974), 260-279.
doi: 10.1088/0305-4470/34/1/306. |
[26] |
C. Huepe and M. E. Brachet,
Scaling laws for vortical nucleation solutions in a model of superflow, Phys. D, 140 (2000), 126-140.
doi: 10.1016/S0167-2789(99)00229-8. |
[27] |
M. Abid, C. Huepe, S. Metens, C. Nore, C. T. Pham, L. S. Tuckerman and M. E. Brachet,
Gross-Pitaevskii dynamics of Bose-Einstein condensates and superfluid turbulence, Fluid Dynam. Res., 33 (2003), 509-544.
doi: 10.1016/j.fluiddyn.2003.09.001. |
[28] |
C. A. Jones, S. J. Putterman and P. H. Roberts,
Stability of wave solutions of nonlinear Schrodinger equations in two and three dimensions, J. Phys A: Math. Gen., 19 (1986), 2991-3011.
|
[29] |
C. A. Jones and P. H. Roberts,
Motion in a Bose condensate Ⅳ, Axisymmetric solitary waves, J. Phys. A, 15 (1982), 2599-2619.
doi: 10.1088/0305-4470/15/8/036. |
[30] |
C. Josserand and Y. Pomeau,
Nonlinear aspects of the theory of Bose-Einstein condensates, Nonlinearity, 14 (2001), R25-R62.
doi: 10.1088/0951-7715/14/5/201. |
[31] |
C. Josserand, Y. Pomeau and S. Rica,
Vortex shedding in a model of superflow, Phys. D, 134 (1999), 111-125.
doi: 10.1016/S0167-2789(99)00066-4. |
[32] |
L. Landau and E. Lifshitz,
On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sowj, 8 (1935), 153.
|
[33] |
F.-H. Lin and J. Wei,
Traveling wave solutions of Schrödinger map equation, Comm. Pure Appl. Math., 63 (2010), 1585-1621.
doi: 10.1002/cpa.20338. |
[34] |
F.-H. Lin and P. Zhang,
Semiclassical limit of the Gross-Pitaevskii equation in an exterior domain, Arch. Ration. Mech. Anal., 179 (2006), 79-107.
doi: 10.1007/s00205-005-0383-4. |
[35] |
Y. Liu and J. Wei, Adler-Moser polynomials and traveling waves solutions of Gross-Pitaevskii, preprint. |
[36] |
P. I. Lizorkin,
Multipliers of Fourier integrals, Proc. Steklov Inst. Math., 89 (1967), 269-290.
|
[37] |
M. Maris,
Existence of nonstationary bubbles in higher dimensions, J. Math. Pures Appl., 81 (2002), 1207-1239.
doi: 10.1016/S0021-7824(02)01274-6. |
[38] |
M. Maris,
Nonexistence of supersonic traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity, SIAM J. Math. Anal., 40 (2008), 1076-1103.
doi: 10.1137/070711189. |
[39] |
M. Maris,
Traveling waves for nonlinear Schrodinger equations with nonzero conditions at infinity, Ann. Math., 178 (2013), 107-182.
doi: 10.4007/annals.2013.178.1.2. |
[40] |
C.-T. Pham, C. Nore and M. E. Brachet,
Boundary layers and emitted excitations in nonlinear Schröinger superflow past a disk, Phys. D, 210 (2005), 203-226.
doi: 10.1016/j.physd.2005.07.013. |
[41] |
S. Rica,
A remark on the critical speed of vortex nucleation in the nonlinear Schrodinger equation, Phys. D, 148 (2001), 221-226.
doi: 10.1016/S0167-2789(00)00168-8. |
[42] |
O. Rey and J. Wei,
Blowing up solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. Part Ⅱ: $N \geq 4$, Ann. Non linearie, Annoles de l'Institut H. Poincaré, 22 (2005), 459-484.
doi: 10.1016/j.anihpc.2004.07.004. |
[43] |
M. Shiffman,
On the ezistence of subsonic flows of a compressible fluid, Arch. Rational Mech. Anal., 2 (1952), 605-652.
doi: 10.1512/iumj.1952.1.51020. |
[44] |
E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, vol. 32, Princeton University Press, Princeton, NJ, 1971. |
[45] |
J. Wei,
Uniqueness and critical spectrum of boundary spike solutions, Proc. Royal Soc. Edin. A, 131 (2001), 1457-1480.
doi: 10.1017/S0308210500001487. |
[1] |
Ko-Shin Chen, Peter Sternberg. Dynamics of Ginzburg-Landau and Gross-Pitaevskii vortices on manifolds. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1905-1931. doi: 10.3934/dcds.2014.34.1905 |
[2] |
Yue Zhang, Jian Zhang. Stability and instability of standing waves for Gross-Pitaevskii equations with double power nonlinearities. Mathematical Control and Related Fields, 2022 doi: 10.3934/mcrf.2022007 |
[3] |
Norman E. Dancer. On the converse problem for the Gross-Pitaevskii equations with a large parameter. Discrete and Continuous Dynamical Systems, 2014, 34 (6) : 2481-2493. doi: 10.3934/dcds.2014.34.2481 |
[4] |
Philipp Bader, Sergio Blanes, Fernando Casas, Mechthild Thalhammer. Efficient time integration methods for Gross-Pitaevskii equations with rotation term. Journal of Computational Dynamics, 2019, 6 (2) : 147-169. doi: 10.3934/jcd.2019008 |
[5] |
André de Laire, Pierre Mennuni. Traveling waves for some nonlocal 1D Gross–Pitaevskii equations with nonzero conditions at infinity. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 635-682. doi: 10.3934/dcds.2020026 |
[6] |
Dong Deng, Ruikuan Liu. Bifurcation solutions of Gross-Pitaevskii equations for spin-1 Bose-Einstein condensates. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3175-3193. doi: 10.3934/dcdsb.2018306 |
[7] |
Yujin Guo, Xiaoyu Zeng, Huan-Song Zhou. Blow-up solutions for two coupled Gross-Pitaevskii equations with attractive interactions. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3749-3786. doi: 10.3934/dcds.2017159 |
[8] |
Jeremy L. Marzuola, Michael I. Weinstein. Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii equations. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1505-1554. doi: 10.3934/dcds.2010.28.1505 |
[9] |
Xiaoyu Zeng, Yimin Zhang. Asymptotic behaviors of ground states for a modified Gross-Pitaevskii equation. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5263-5273. doi: 10.3934/dcds.2019214 |
[10] |
E. Norman Dancer. On a degree associated with the Gross-Pitaevskii system with a large parameter. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 1835-1839. doi: 10.3934/dcdss.2019120 |
[11] |
Patrick Henning, Johan Wärnegård. Numerical comparison of mass-conservative schemes for the Gross-Pitaevskii equation. Kinetic and Related Models, 2019, 12 (6) : 1247-1271. doi: 10.3934/krm.2019048 |
[12] |
Thomas Chen, Nataša Pavlović. On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies. Discrete and Continuous Dynamical Systems, 2010, 27 (2) : 715-739. doi: 10.3934/dcds.2010.27.715 |
[13] |
Georgy L. Alfimov, Pavel P. Kizin, Dmitry A. Zezyulin. Gap solitons for the repulsive Gross-Pitaevskii equation with periodic potential: Coding and method for computation. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1207-1229. doi: 10.3934/dcdsb.2017059 |
[14] |
Roy H. Goodman, Jeremy L. Marzuola, Michael I. Weinstein. Self-trapping and Josephson tunneling solutions to the nonlinear Schrödinger / Gross-Pitaevskii equation. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 225-246. doi: 10.3934/dcds.2015.35.225 |
[15] |
Shuai Li, Jingjing Yan, Xincai Zhu. Constraint minimizers of perturbed gross-pitaevskii energy functionals in $\mathbb{R}^N$. Communications on Pure and Applied Analysis, 2019, 18 (1) : 65-81. doi: 10.3934/cpaa.2019005 |
[16] |
Weiran Sun, Min Tang. A relaxation method for one dimensional traveling waves of singular and nonlocal equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1459-1491. doi: 10.3934/dcdsb.2013.18.1459 |
[17] |
Adèle Bourgeois, Victor LeBlanc, Frithjof Lutscher. Dynamical stabilization and traveling waves in integrodifference equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3029-3045. doi: 10.3934/dcdss.2020117 |
[18] |
Andrea Corli, Lorenzo di Ruvo, Luisa Malaguti, Massimiliano D. Rosini. Traveling waves for degenerate diffusive equations on networks. Networks and Heterogeneous Media, 2017, 12 (3) : 339-370. doi: 10.3934/nhm.2017015 |
[19] |
Hua Chen, Ling-Jun Wang. A perturbation approach for the transverse spectral stability of small periodic traveling waves of the ZK equation. Kinetic and Related Models, 2012, 5 (2) : 261-281. doi: 10.3934/krm.2012.5.261 |
[20] |
Wei Wang, Yan Lv. Limit behavior of nonlinear stochastic wave equations with singular perturbation. Discrete and Continuous Dynamical Systems - B, 2010, 13 (1) : 175-193. doi: 10.3934/dcdsb.2010.13.175 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]