# American Institute of Mathematical Sciences

March 2019, 8(1): 73-100. doi: 10.3934/eect.2019005

## Pattern formation in flows of asymmetrically interacting particles: Peristaltic pedestrian dynamics as a case study

 1 Bogolyubov Institute for Theoretical Physics, Metrologichna str. 14B, 01413 Kiev, Ukraine 2 Continental AG, Vahrenwalder Strasse 9, D-30165 Hanover, Germany 3 Department of Applied Mathematics and Computer Science, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark 4 Department of Physics and Department of Applied Mathematics and Computer Science, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark 5 Department of Physics, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark

* Corresponding author: Mads Peter Sørensen

Received  July 2017 Revised  November 2017 Published  January 2019

Fund Project: This work is supported by Civilingeniør Frederik Leth Christiansens Almennyttige Fond, the Otto Mønsteds Fond and a special program of the National Academy of Sciences of Ukraine

The influence of asymmetry in the coupling between repulsive particles is studied. A prominent example is the social force model for pedestrian dynamics in a long corridor where the asymmetry leads to anisotropy in the repulsion such that pedestrians in front, i.e., in walking direction, have a bigger influence on the pedestrian behavior than those behind. In addition to one- and two-lane free flow situations, a new traveling regime is found that is reminiscent of peristaltic motion. We study the regimes and their respective stabilityboth analytically and numerically. First, we introduce a modified social forcemodel and compute the boundaries between different regimes analytically bya perturbation analysis of the one-lane and two-lane flow. Afterwards, theresults are verified by direct numerical simulations in the parameter plane ofpedestrian density and repulsion strength from the walls.

Citation: Yuri B. Gaididei, Christian Marschler, Mads Peter Sørensen, Peter L. Christiansen, Jens Juul Rasmussen. Pattern formation in flows of asymmetrically interacting particles: Peristaltic pedestrian dynamics as a case study. Evolution Equations & Control Theory, 2019, 8 (1) : 73-100. doi: 10.3934/eect.2019005
##### References:
 [1] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover Publications, Inc., New York, 1966. [2] N. Bellomo and C. Dogbe, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Review, 53 (2011), 409-463. doi: 10.1137/090746677. [3] V. J. Blue and J. L. Adler, Cellular automata microsimulation for modeling bi-directional pedestrian walkways, Transportation Research Part B, 35 (2001), 293-312. doi: 10.1016/S0191-2615(99)00052-1. [4] C. Burstedde, K. Klauck, A. Schadschneider and J. Zittartz, Simulation of pedestrian dynamics using a two-dimensional cellular automaton, Physica A, 295 (2001), 507-525. doi: 10.1016/S0378-4371(01)00141-8. [5] O. Corradi, P. G. Hjorth and J. Starke, Equation-free detection and continuation of a Hopf bifurcation point in a particle model of pedestrian flow, SIAM Journal on Applied Dynamical Systems, 11 (2012), 1007-1032. doi: 10.1137/110854072. [6] T. Dessup, C. Coste and M. Saint Jean, Subcriticality of the zigzag transition: A nonlinear bifurcation analysis, Physical Review E, 91 (2015), 032917, 1-14. doi: 10.1103/PhysRevE.91.032917. [7] T. Dessup, T. Maimbourg, C. Coste and M. Saint Jean, Linear instability of a zigzag pattern, Physical Review E, 91 (2015), 022908, 1-12. doi: 10.1103/PhysRevE.91.022908. [8] F. Dietrich and G. Köster, Gradient navigation model for pedestrian dynamics, Physical Review E, 89 (2014), 062801, 1-8. doi: 10.1103/PhysRevE.89.062801. [9] J. E. Galván-Moya and F. M. Peeters, Ginzburg-Landau theory of the zigzag transition in quasi-one-dimensional classical Wigner crystals, Physical Review B, 84 (2011), 134106, 1-10. [10] D. Helbing, Traffic and related self-driven many-particle systems, Reviews of Modern Physics, 73 (2001), 1067-1141. doi: 10.1103/RevModPhys.73.1067. [11] D. Helbing, P. Molnar, I. J. Farkas and K. Bolay, Self-organizing pedestrian movement, Environment and Planning B-planning and Design, 28 (2001), 361-383. doi: 10.1068/b2697. [12] D. Helbing and P. Molnar, Social force model for pedestrian dynamics, Physical Review E, 51 (1995), 4282-4286. doi: 10.1103/PhysRevE.51.4282. [13] D. Helbing, I. Farkas and T. Vicsek, Simulating dynamical features of escape panic, Nature, 407 (2000), 487-490. doi: 10.1038/35035023. [14] S. P. Hoogendoorn and W. Daamen, Pedestrian behavior at bottlenecks, Transportation Science, 39 (2005), 147-288. doi: 10.1287/trsc.1040.0102. [15] A. Jelić, C. Appert-Rolland, S. Lemercier and J. Pettré, Properties of pedestrians walking in line: Fundamental diagrams, Physical Review E, 85 (2012), 036111, 1-9. [16] A. Johansson and D. Helbing, Crowd dynamics, in: Econophysics and Sociophysics. Trends and Perspectives (eds. B.K. Chakrabarti, A. Chakraborti and A. Chatterjee), Wiley-VCH, Weinheim, (2006), 449-472. doi: 10.1002/9783527610006.ch16. [17] A. Johansson, D. Helbing and P. K. Shukla, Specification of the social force pedestrian model by evolutionary adjustment to video tracking data, Advances in Complex Systems, 10 (2007), 271-288. doi: 10.1142/S0219525907001355. [18] B. S. Kerner, The physics of Traffic: Empirical Freeway Pattern Features, Engineering Applications, and Theory, 1st edition, Springer-Verlag, Berlin Heidelberg, 2004. doi: 10.1007/978-3-540-40986-1. [19] Y. G. Kevrekidis and G. Samaey, Equation-free multiscale computation: Algorithms and applications, Annual Review of Physical Chemistry, 60 (2009), 321-344. doi: 10.1146/annurev.physchem.59.032607.093610. [20] H.-K. Li, E. Urban, C. Noel, A. Chuang, Y. Xia, A. Ransford, B. Hemmerling, Y. Wang, T. Li, H. Häffner and X. Zhang, Realization of translational symmetry in trapped cold ion rings, Physical Review Letters, 118 (2017), 053001, 1-5. doi: 10.1103/PhysRevLett.118.053001. [21] C. Marschler, J. Starke, M. P. Sørensen, Yu. Gaididei and P. L. Christiansen, Pattern formation in annular systems of repulsive particles, Physics Letters A, 380 (2016), 166-170. doi: 10.1016/j.physleta.2015.10.038. [22] C. Marschler, J. Starke, P. Liu and Y. G. Kevrekidis, Coarse-grained particle model for pedestrian flow using diffusion maps, Physical Review E, 89 (2014), 013304, 1-11. doi: 10.1103/PhysRevE.89.013304. [23] C. Marschler, J. Sieber, P. G. Hjorth and J. Starke, Equation-free analysis of macroscopic behavior in traffic and pedestrian flow, in: Traffic and Granular Flow '13 (eds. M. Chraibi, M. Boltes, A. Schadschneider and A. X. Armin Seyfried) Springer-Verlag, (2015), 423-439. [24] C. Marschler, J. Sieber, R. Berkemer, A. Kawamoto and J. Starke, Implicit methods for equation-free analysis: Convergence results and analysis of emergent waves in microscopic traffic models, SIAM Journal on Applied Dynamical Systems, 13 (2014), 1202-1238, [http://arXiv.org/abs/1301.6044] doi: 10.1137/130913961. [25] J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer-Verlag, New York, 1976. [26] M. Moussaïd, E.G. Guillot, M. Moreau, J. Fehrenbach, O. Chabiron, S. Lemercier, J. Pettré, C. Appert-Rolland, P. Degond and G. Theraulaz, Traffic Instabilities in Self-Organized Pedestrian Crowds, PLoS Computational Biology, 8 (2012), e1002442, 1-10. [27] A. Schadschneider and A. Seyfried, Empirical results for pedestrian dynamics and their implications for modeling, Networks and Heterogeneous media, 6 (2011), 545-560. doi: 10.3934/nhm.2011.6.545. [28] J. P. Schiffer, Phase transitions in anisotropically confined ionic crystals, Physical Review Letters, 70 (1993), 818-821. doi: 10.1103/PhysRevLett.70.818. [29] W. Tian, W. Song, J. Ma, Z. Fang, A. Seyfried and J. Liddle, Experimental study of pedestrian behaviors in a corridor based on digital image processing, Fire Safety Journal, 47 (2012), 8-15. doi: 10.1016/j.firesaf.2011.09.005. [30] T. Vicsek and A. Zafeiris, Collective motion, Physics Reports, 517 (2012), 71-140. doi: 10.1016/j.physrep.2012.03.004. [31] D. E. Wolf, M. Schreckenberg and A. Bachem (Eds.), Traffic and Granular Flow, World Scientific, Singapore, 1996. doi: 10.1142/9789814531276. [32] Z. Xiaoping, Z. Tingkuan and L. Mengting, Modeling crowd evacuation of a building based on seven methodological approaches, Building and Environment, 44 (2009), 437-445.

show all references

##### References:
 [1] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover Publications, Inc., New York, 1966. [2] N. Bellomo and C. Dogbe, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Review, 53 (2011), 409-463. doi: 10.1137/090746677. [3] V. J. Blue and J. L. Adler, Cellular automata microsimulation for modeling bi-directional pedestrian walkways, Transportation Research Part B, 35 (2001), 293-312. doi: 10.1016/S0191-2615(99)00052-1. [4] C. Burstedde, K. Klauck, A. Schadschneider and J. Zittartz, Simulation of pedestrian dynamics using a two-dimensional cellular automaton, Physica A, 295 (2001), 507-525. doi: 10.1016/S0378-4371(01)00141-8. [5] O. Corradi, P. G. Hjorth and J. Starke, Equation-free detection and continuation of a Hopf bifurcation point in a particle model of pedestrian flow, SIAM Journal on Applied Dynamical Systems, 11 (2012), 1007-1032. doi: 10.1137/110854072. [6] T. Dessup, C. Coste and M. Saint Jean, Subcriticality of the zigzag transition: A nonlinear bifurcation analysis, Physical Review E, 91 (2015), 032917, 1-14. doi: 10.1103/PhysRevE.91.032917. [7] T. Dessup, T. Maimbourg, C. Coste and M. Saint Jean, Linear instability of a zigzag pattern, Physical Review E, 91 (2015), 022908, 1-12. doi: 10.1103/PhysRevE.91.022908. [8] F. Dietrich and G. Köster, Gradient navigation model for pedestrian dynamics, Physical Review E, 89 (2014), 062801, 1-8. doi: 10.1103/PhysRevE.89.062801. [9] J. E. Galván-Moya and F. M. Peeters, Ginzburg-Landau theory of the zigzag transition in quasi-one-dimensional classical Wigner crystals, Physical Review B, 84 (2011), 134106, 1-10. [10] D. Helbing, Traffic and related self-driven many-particle systems, Reviews of Modern Physics, 73 (2001), 1067-1141. doi: 10.1103/RevModPhys.73.1067. [11] D. Helbing, P. Molnar, I. J. Farkas and K. Bolay, Self-organizing pedestrian movement, Environment and Planning B-planning and Design, 28 (2001), 361-383. doi: 10.1068/b2697. [12] D. Helbing and P. Molnar, Social force model for pedestrian dynamics, Physical Review E, 51 (1995), 4282-4286. doi: 10.1103/PhysRevE.51.4282. [13] D. Helbing, I. Farkas and T. Vicsek, Simulating dynamical features of escape panic, Nature, 407 (2000), 487-490. doi: 10.1038/35035023. [14] S. P. Hoogendoorn and W. Daamen, Pedestrian behavior at bottlenecks, Transportation Science, 39 (2005), 147-288. doi: 10.1287/trsc.1040.0102. [15] A. Jelić, C. Appert-Rolland, S. Lemercier and J. Pettré, Properties of pedestrians walking in line: Fundamental diagrams, Physical Review E, 85 (2012), 036111, 1-9. [16] A. Johansson and D. Helbing, Crowd dynamics, in: Econophysics and Sociophysics. Trends and Perspectives (eds. B.K. Chakrabarti, A. Chakraborti and A. Chatterjee), Wiley-VCH, Weinheim, (2006), 449-472. doi: 10.1002/9783527610006.ch16. [17] A. Johansson, D. Helbing and P. K. Shukla, Specification of the social force pedestrian model by evolutionary adjustment to video tracking data, Advances in Complex Systems, 10 (2007), 271-288. doi: 10.1142/S0219525907001355. [18] B. S. Kerner, The physics of Traffic: Empirical Freeway Pattern Features, Engineering Applications, and Theory, 1st edition, Springer-Verlag, Berlin Heidelberg, 2004. doi: 10.1007/978-3-540-40986-1. [19] Y. G. Kevrekidis and G. Samaey, Equation-free multiscale computation: Algorithms and applications, Annual Review of Physical Chemistry, 60 (2009), 321-344. doi: 10.1146/annurev.physchem.59.032607.093610. [20] H.-K. Li, E. Urban, C. Noel, A. Chuang, Y. Xia, A. Ransford, B. Hemmerling, Y. Wang, T. Li, H. Häffner and X. Zhang, Realization of translational symmetry in trapped cold ion rings, Physical Review Letters, 118 (2017), 053001, 1-5. doi: 10.1103/PhysRevLett.118.053001. [21] C. Marschler, J. Starke, M. P. Sørensen, Yu. Gaididei and P. L. Christiansen, Pattern formation in annular systems of repulsive particles, Physics Letters A, 380 (2016), 166-170. doi: 10.1016/j.physleta.2015.10.038. [22] C. Marschler, J. Starke, P. Liu and Y. G. Kevrekidis, Coarse-grained particle model for pedestrian flow using diffusion maps, Physical Review E, 89 (2014), 013304, 1-11. doi: 10.1103/PhysRevE.89.013304. [23] C. Marschler, J. Sieber, P. G. Hjorth and J. Starke, Equation-free analysis of macroscopic behavior in traffic and pedestrian flow, in: Traffic and Granular Flow '13 (eds. M. Chraibi, M. Boltes, A. Schadschneider and A. X. Armin Seyfried) Springer-Verlag, (2015), 423-439. [24] C. Marschler, J. Sieber, R. Berkemer, A. Kawamoto and J. Starke, Implicit methods for equation-free analysis: Convergence results and analysis of emergent waves in microscopic traffic models, SIAM Journal on Applied Dynamical Systems, 13 (2014), 1202-1238, [http://arXiv.org/abs/1301.6044] doi: 10.1137/130913961. [25] J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer-Verlag, New York, 1976. [26] M. Moussaïd, E.G. Guillot, M. Moreau, J. Fehrenbach, O. Chabiron, S. Lemercier, J. Pettré, C. Appert-Rolland, P. Degond and G. Theraulaz, Traffic Instabilities in Self-Organized Pedestrian Crowds, PLoS Computational Biology, 8 (2012), e1002442, 1-10. [27] A. Schadschneider and A. Seyfried, Empirical results for pedestrian dynamics and their implications for modeling, Networks and Heterogeneous media, 6 (2011), 545-560. doi: 10.3934/nhm.2011.6.545. [28] J. P. Schiffer, Phase transitions in anisotropically confined ionic crystals, Physical Review Letters, 70 (1993), 818-821. doi: 10.1103/PhysRevLett.70.818. [29] W. Tian, W. Song, J. Ma, Z. Fang, A. Seyfried and J. Liddle, Experimental study of pedestrian behaviors in a corridor based on digital image processing, Fire Safety Journal, 47 (2012), 8-15. doi: 10.1016/j.firesaf.2011.09.005. [30] T. Vicsek and A. Zafeiris, Collective motion, Physics Reports, 517 (2012), 71-140. doi: 10.1016/j.physrep.2012.03.004. [31] D. E. Wolf, M. Schreckenberg and A. Bachem (Eds.), Traffic and Granular Flow, World Scientific, Singapore, 1996. doi: 10.1142/9789814531276. [32] Z. Xiaoping, Z. Tingkuan and L. Mengting, Modeling crowd evacuation of a building based on seven methodological approaches, Building and Environment, 44 (2009), 437-445.
Patterns emerging in the pedestrian model. Color indicates pedestrian index. Numerical solution of Eqs. (9) for parameters specified in the text.
Transverse stationary distance $b$ between pedestrians in the two-lane zig-zag flow shown in Fig. 1(b). Panel (a): $b$ vs. density $\rho$ for fixed $\nu = 1$, panel (b): $b$ vs. interaction strength $\nu$ for fixed $\rho = 1$, $\rho$ being pedestrian density, $\nu$ being strength of pedestrian wall interaction. Panel (a): in the region to the left (right) of the curve the flow is single (two-) lane. Panel (b): in the region to the left (right) of the curve the flow is two- (single) lane. Direct numerical simulations (circles) and analytical predictions (curves) are in agreement.
Panel (a): the growth rate $\Re(z_2)$ of the linear mode $\mu = 2$ vs the wave number $k$. Panel (b): the growth rate of the first three harmonics of the linear mode $\mu = 2$ vs the mean interparticle distance $a$. In both figures $\epsilon = 0.5, \nu = 0.05, N = 32$
Bifurcation diagrams obtained from the linear stability analysis for $N = 32$ and the asymmetry parameter $\epsilon = 0$ (panel (a)) and $\epsilon = 0.5$ (panel (b)). Insets show details of the diagram in the vicinity of the two-lane regime instability. In the white (orange) area the one- (two-) lane flow is stable. In the blue area we observe the peristaltic regime and the distance between lanes is spatially and time modulated, in the green area the two-lane flow is linearly unstable, and the instability leads to unsorted motion as shown in Fig. 1(d)
The same as in Fig. 4 for $N = 128$
The order parameter $R$ of the peristaltic phase vs. mean headway, $a$, for fixed $\nu = 0.05$. Panel (a): $R$ vs. $a$, in the case of totally symmetric social interaction $\epsilon = 0$: panel (b) $R$ vs. $a$, in the case of partially asymmetric social interaction: $\epsilon = 0.5$. Panel (a): in the region between arrows a hysteretic behavior takes place: red-dot-curve presents downsweep stable branch, black-dot-curve presents upsweep stable branch. Panel (b): as in panel (a); the inset shows the hysteretic behavior near the right boundary of the peristaltic phase $a = 3.198$.
Staggered transversal coordinates $(-1)^n y_n$ in the mixed phase state for two different values of the pedestrian headway: $a = 2.93$ (panel (a)) and $a = 3.13$ (panel (b)). The social interaction is symmetric: $\epsilon = 0$. Other parameters are chosen inside the domain of mixed phases state: $~\nu = 0.05, ~N = 128$. The solid lines represent the results obtained in the frame of the analytical approach, the dots represent the results of numerical solutions of Eqs. (9)
Longitudinal distances between nearest neighbors $x_{n+1}-x_n$ in the mixed phase state. All parameters are the same as in Fig. 7.
Energy difference between the mixed state and the spatially homogeneous two-lane state as a function of the mean headway $a$ in the interval $a\in (a_2,a_3)$. The social interaction is symmetric $\epsilon=0$, the pedestrian-wall interaction is fixed: $\nu =0.05$
The stationary value of the inverse width $\kappa$ vs the mean distance $a$ obtained from Eq. (81). The solid (dashed) curve presents a stable (unstable) solution. The curves are plotted in the mean distance interval $a\in(a_2,a_l)$, where the mixed phase is unstable in the linear anaylsis aproach and it is stable in the frame of the variational approach. The solid line gives the contour, where $\partial^2_\kappa {\mathcal E}_b=0$. The social interaction is symmetric $\epsilon=0$, the pedestrian-wall interaction is fixed: $\nu =0.05$
Spatio-temporal evolution of the local distance between lanes $\Delta y_n = |y_{n+1}-y_n|$ (panel (a)) and the excess density $\Delta \rho_n = \frac{1}{x_{n+1}-x_n}-\frac{1}{a}$ (panel (b)) for totally asymmetric social interaction ($\epsilon = 1$). Other parameters are chosen inside the domain of peristaltic motion: $~ a = 1.4, ~\nu = 0.65$. The two profiles are separated by the time difference $\Delta t = 250$
Velocity of the peristaltic pulse as a function of the inverse density. Comparison of the analytical results obtained from Eq. (90) (solid curve) and full scale numerical results (dots). The social interaction is weakly asymmetric $\epsilon = 0.01$, the pedestrian-wall interaction is fixed: $\nu = 0.05$, the number of pedestrian $N = 128$
Panel (a): The modulus of elliptic function $m$ vs. mean headway $a$, dashed line presents an energetically unstable branch. Panel (b): The dimensionless energy difference between the spatially homogeneous two-lane state and the peristaltic state $\delta E = (E_{per}-E_{two-lane})/|E_{two-lane}|$ vs. mean headway $a$. The critical headway $a_r$ gives the right boundary of the peristaltic state stability interval. The two-lane state looses its stability and the peristaltic state is established for $a<a_r$. The solid and dashed lines correspond to two branches presented in panel (a).
Two stationary localized solutions $Y(n)$ of Eq. (93) in the case of symmetric interparticle interaction for the mean headway $a = a_r-0.0015$, the pedestrian-wall interaction $\nu = 0.05$. The number of pedestrian is $N = 128$. The solid line corresponds to the energetically more favorable state.
Staggered transversal coordinate $(-1)^n y_n$ profile obtained by numerical simulations (dots) and analytically from Eq. (103). The social interaction is symmetric $\epsilon = 0$, the number of particles is $N = 128$, the pedestrian-wall interaction is fixed: $\nu = 0.05$, the mean headway $a = a_r-0.001$ (panel(a)), and $a = a_r-0.0003$ (panel(b))
Pulse velocity in the vicinity of the bifurcation point $a_r$ obtained from numerical solutions of Eq. (9) (dots) and from analysis (see Eq. (111)) in the case of weakly asymmetric interparticle interaction $\epsilon = 0.01$ for the mean headway $0<a_r-a\ll 1$. The pedestrian-wall interaction is $\nu = 0.05$. The number of pedestrian is $N = 128$. See also Fig. 12
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