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Hexagonal spike clusters for some PDE's in 2D
1. | Dalhousie University, Halifax, Canada |
2. | University of British Columbia, Vancouver, Canada |
We study hexagonal spike cluster patterns for Gierer-Meinhardt reaction-diffusion system with a precursor on all of $ \mathbb R^2 $. These clusters consist of $ N $ spikes which form a nearly hexagonal lattice of a finite size. The lattice density is locally nearly constant, but globally non-uniform. We also characterize a similar hexagonal spike cluster steady state for a simple elliptic PDE $ 0 = \Delta u - u +u^2 + \varepsilon |x|^2 $ with a small "confinement well" $ \varepsilon |x|^2 $. The key idea is to explicitly exploit the local hexagonality structure to asymptotically approximate the solution using certain lattice sums. In the limit of many spikes, we derive the effective spike density as well as the cluster radius. This effective density is a solution to a certain separable first-order ODE coupled to an integral boundary condition.
References:
[1] |
A. J. Bernoff and C. M. Topaz,
Nonlocal aggregation models: A primer of swarm equilibria, SIAM Rev., 55 (2013), 709-747.
doi: 10.1137/130925669. |
[2] |
Y. Chen and T. Kolokolnikov, A minimal model of predator-swarm interactions, J. Royal Soc. Interface, 11 (2014).
doi: 10.1098/rsif.2013.1208. |
[3] |
M. R. D'Orsogna, Y.-L. Chuang, A. L. Bertozzi and L. S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Phys. Rev. Lett., 96 (2006).
doi: 10.1103/PhysRevLett.96.104302. |
[4] |
R. C. Fetecau, Y. Huang and T. Kolokolnikov,
Swarm dynamics and equilibria for a nonlocal aggregation model, Nonlinearity, 24 (2011), 2681-2716.
doi: 10.1088/0951-7715/24/10/002. |
[5] |
G. Flierl, D. Grünbaum, S. Levins and D. Olson,
From individuals to aggregations: The interplay between behavior and physics, J. Theoret. Biol., 196 (1999), 397-454.
doi: 10.1006/jtbi.1998.0842. |
[6] |
B. Gidas, Symmetry of positive solutions of nonlinear elliptic equations in Rn, in Mathematical Analysis and Applications, Part A, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981,369-402. |
[7] |
A. Gierer and H. Meinhardt,
A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.
doi: 10.1007/BF00289234. |
[8] |
C. Gui and J. Wei,
Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differential Equations, 158 (1999), 1-27.
doi: 10.1016/S0022-0396(99)80016-3. |
[9] |
D. Iron, M. J. Ward and J. Wei,
The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, Phys. D, 150 (2001), 25-62.
doi: 10.1016/S0167-2789(00)00206-2. |
[10] |
T. Kolokolnikov, P. Kevrekidis and R. Carretero-González, A tale of two distributions: From few to many vortices in quasi-two-dimensional Bose-Einstein condensates, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), 18pp.
doi: 10.1098/rspa.2014.0048. |
[11] |
T. Kolokolnikov, H. Sun, D. Uminsky and A. L. Bertozzi, Stability of ring patterns arising from two-dimensional particle interactions, Phys. Rev. E, 84 (2011).
doi: 10.1103/PhysRevE.84.015203. |
[12] |
T. Kolokolnikov and M. J. Ward,
Reduced wave Green's functions and their effect on the dynamics of a spike for the Gierer-Meinhardt model, European J. Appl. Math., 14 (2003), 513-545.
doi: 10.1017/S0956792503005254. |
[13] |
T. Kolokolnikov, M. J. Ward and J. Wei,
Spot self-replication and dynamics for the Schnakenburg model in a two-dimensional domain, J. Nonlinear Sci., 19 (2009), 1-56.
doi: 10.1007/s00332-008-9024-z. |
[14] |
T. Kolokolnikov and S. Xie, Spike density distribution for the Gierer-Meinhardt model with precursor, Phys. D, in press.
doi: 10.1016/j.physd.2019.132247. |
[15] |
M. K. Kwong and L. Zhang,
Uniqueness of the positive solution of $\delta u+f(u) = 0$ in an annulus, Differential Integral Equations, 4 (1991), 583-599.
|
[16] |
W.-M. Ni and J. Wei,
On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math., 48 (1995), 731-768.
doi: 10.1002/cpa.3160480704. |
[17] |
M. J. Ward, D. McInerney, P. Houston, D. Gavaghan and P. Maini,
The dynamics and pinning of a spike for a reaction-diffusion system, SIAM J. Appl. Math., 62 (2002), 1297-1328.
doi: 10.1137/S0036139900375112. |
[18] |
J. Wei and M. Winter,
Stable spike clusters for the one-dimensional Gierer-Meinhardt system, European J. Appl. Math., 28 (2017), 576-635.
doi: 10.1017/S0956792516000450. |
[19] |
J. Wei and M. Winter,
Stationary multiple spots for reaction-diffusion systems, J. Math. Biol., 57 (2008), 53-89.
doi: 10.1007/s00285-007-0146-y. |
[20] |
J. Wei and M. Winter,
Spikes for the Gierer-Meinhardt system in two dimensions: The strong coupling case, J. Differential Equations, 178 (2002), 478-518.
doi: 10.1006/jdeq.2001.4019. |
[21] |
J. Wei and M. Winter,
Existence and stability analysis of asymmetric patterns for the Gierer-Meinhardt system, J. Math. Pures Appl., 83 (2004), 433-476.
doi: 10.1016/j.matpur.2003.09.006. |
[22] |
J. Wei, M. Winter and W. Yang, Stable spike clusters for the precursor Gierer-Meinhardt system in $R^{2}$, Calc. Var. Partial Differential Equations, 56 (2017), 40pp.
doi: 10.1007/s00526-017-1233-6. |
[23] |
S. Xie, P. G. Kevrekidis and T. Kolokolnikov, Multi-vortex crystal lattices in Bose-Einstein condensates with a rotating trap, Proc. A, 474 (2018), 21pp.
doi: 10.1098/rspa.2017.0553. |
show all references
References:
[1] |
A. J. Bernoff and C. M. Topaz,
Nonlocal aggregation models: A primer of swarm equilibria, SIAM Rev., 55 (2013), 709-747.
doi: 10.1137/130925669. |
[2] |
Y. Chen and T. Kolokolnikov, A minimal model of predator-swarm interactions, J. Royal Soc. Interface, 11 (2014).
doi: 10.1098/rsif.2013.1208. |
[3] |
M. R. D'Orsogna, Y.-L. Chuang, A. L. Bertozzi and L. S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Phys. Rev. Lett., 96 (2006).
doi: 10.1103/PhysRevLett.96.104302. |
[4] |
R. C. Fetecau, Y. Huang and T. Kolokolnikov,
Swarm dynamics and equilibria for a nonlocal aggregation model, Nonlinearity, 24 (2011), 2681-2716.
doi: 10.1088/0951-7715/24/10/002. |
[5] |
G. Flierl, D. Grünbaum, S. Levins and D. Olson,
From individuals to aggregations: The interplay between behavior and physics, J. Theoret. Biol., 196 (1999), 397-454.
doi: 10.1006/jtbi.1998.0842. |
[6] |
B. Gidas, Symmetry of positive solutions of nonlinear elliptic equations in Rn, in Mathematical Analysis and Applications, Part A, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981,369-402. |
[7] |
A. Gierer and H. Meinhardt,
A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.
doi: 10.1007/BF00289234. |
[8] |
C. Gui and J. Wei,
Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differential Equations, 158 (1999), 1-27.
doi: 10.1016/S0022-0396(99)80016-3. |
[9] |
D. Iron, M. J. Ward and J. Wei,
The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, Phys. D, 150 (2001), 25-62.
doi: 10.1016/S0167-2789(00)00206-2. |
[10] |
T. Kolokolnikov, P. Kevrekidis and R. Carretero-González, A tale of two distributions: From few to many vortices in quasi-two-dimensional Bose-Einstein condensates, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), 18pp.
doi: 10.1098/rspa.2014.0048. |
[11] |
T. Kolokolnikov, H. Sun, D. Uminsky and A. L. Bertozzi, Stability of ring patterns arising from two-dimensional particle interactions, Phys. Rev. E, 84 (2011).
doi: 10.1103/PhysRevE.84.015203. |
[12] |
T. Kolokolnikov and M. J. Ward,
Reduced wave Green's functions and their effect on the dynamics of a spike for the Gierer-Meinhardt model, European J. Appl. Math., 14 (2003), 513-545.
doi: 10.1017/S0956792503005254. |
[13] |
T. Kolokolnikov, M. J. Ward and J. Wei,
Spot self-replication and dynamics for the Schnakenburg model in a two-dimensional domain, J. Nonlinear Sci., 19 (2009), 1-56.
doi: 10.1007/s00332-008-9024-z. |
[14] |
T. Kolokolnikov and S. Xie, Spike density distribution for the Gierer-Meinhardt model with precursor, Phys. D, in press.
doi: 10.1016/j.physd.2019.132247. |
[15] |
M. K. Kwong and L. Zhang,
Uniqueness of the positive solution of $\delta u+f(u) = 0$ in an annulus, Differential Integral Equations, 4 (1991), 583-599.
|
[16] |
W.-M. Ni and J. Wei,
On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math., 48 (1995), 731-768.
doi: 10.1002/cpa.3160480704. |
[17] |
M. J. Ward, D. McInerney, P. Houston, D. Gavaghan and P. Maini,
The dynamics and pinning of a spike for a reaction-diffusion system, SIAM J. Appl. Math., 62 (2002), 1297-1328.
doi: 10.1137/S0036139900375112. |
[18] |
J. Wei and M. Winter,
Stable spike clusters for the one-dimensional Gierer-Meinhardt system, European J. Appl. Math., 28 (2017), 576-635.
doi: 10.1017/S0956792516000450. |
[19] |
J. Wei and M. Winter,
Stationary multiple spots for reaction-diffusion systems, J. Math. Biol., 57 (2008), 53-89.
doi: 10.1007/s00285-007-0146-y. |
[20] |
J. Wei and M. Winter,
Spikes for the Gierer-Meinhardt system in two dimensions: The strong coupling case, J. Differential Equations, 178 (2002), 478-518.
doi: 10.1006/jdeq.2001.4019. |
[21] |
J. Wei and M. Winter,
Existence and stability analysis of asymmetric patterns for the Gierer-Meinhardt system, J. Math. Pures Appl., 83 (2004), 433-476.
doi: 10.1016/j.matpur.2003.09.006. |
[22] |
J. Wei, M. Winter and W. Yang, Stable spike clusters for the precursor Gierer-Meinhardt system in $R^{2}$, Calc. Var. Partial Differential Equations, 56 (2017), 40pp.
doi: 10.1007/s00526-017-1233-6. |
[23] |
S. Xie, P. G. Kevrekidis and T. Kolokolnikov, Multi-vortex crystal lattices in Bose-Einstein condensates with a rotating trap, Proc. A, 474 (2018), 21pp.
doi: 10.1098/rspa.2017.0553. |





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