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2013, 2013(special): 719-728. doi: 10.3934/proc.2013.2013.719

## Validity and dynamics in the nonlinearly excited 6th-order phase equation

 1 University of Southern Queensland, Toowoomba, Queensland 4350, Australia, Australia

Received  September 2012 Published  November 2013

A slowly varying phase of oscillators coupled by diffusion is generally described by a partial differential equation comprising infinitely many terms. We consider a particular case when the coupling is nonlocal and, as a result, the equation can be reduced to a finite form with nonlinear excitation and 6th-order dissipation. We fulfilled two tasks: (1) evaluated the range of independent parameters rendering the form valid, and (2) developed and tested the numerical code for solving the equation; some numerical solutions are presented.
Citation: Dmitry Strunin, Mayada Mohammed. Validity and dynamics in the nonlinearly excited 6th-order phase equation. Conference Publications, 2013, 2013 (special) : 719-728. doi: 10.3934/proc.2013.2013.719
##### References:
 [1] G. Sivashinsky, Nonlinear analysis of hydrodynamical instability in laminar flames, Acta Astronaut., 4 (1977), 1177-1206.  Google Scholar [2] Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Progr. Theor. Phys., 55 (1976), 356-369. Google Scholar [3] D. Tanaka and Y. Kuramoto, Complex Ginzburg-Landau equation with nonlocal coupling, Phys. Rev. E, 68 (2003), 026219. Google Scholar [4] D. Tanaka, Chemical turbulence equivalent to Nikolaevskii turbulence, Phys. Rev. E, 70 (2004), 015202(R). Google Scholar [5] D. Tanaka, Turing instability leads oscillatory systems to spatiotemporal chaos, Progr. Theor. Phys. Suppl. N, 161 (2006), 119-126. Google Scholar [6] V. Nikolaevskii, "in Recent Advances in Engineering Science," edited by S.L. Koh, C.G. Speciale, Lecture Notes in Engineering, 39 Berlin: Springer, 1989, 210. Google Scholar [7] D. Strunin, Autosoliton model of the spinning fronts of reaction, IMA J. Appl. Math., 63 (1999), 163-177.  Google Scholar [8] D. Strunin, Phase equation with nonlinear excitation for nonlocally coupled oscillators, Physica D: Nonlinear Phenomena, 238 (2009), 1909-1916.  Google Scholar [9] D. Strunin and M. Mohammed, Parametric space for nonlinearly excited phase equation, ANZIAM J. Electron. Suppl., 53 (2011), C236-C248.  Google Scholar [10] D. Strunin, Nonlinear instability in generalized nonlinear phase diffusion equation, Progr. Theor. Phys. Suppl. N, 150 (2003), 444-448.  Google Scholar [11] http:, //www.mathworks.com/matlabcentral/fileexchange/28-differential-algebraic-, equation-solvers., ().   Google Scholar

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##### References:
 [1] G. Sivashinsky, Nonlinear analysis of hydrodynamical instability in laminar flames, Acta Astronaut., 4 (1977), 1177-1206.  Google Scholar [2] Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Progr. Theor. Phys., 55 (1976), 356-369. Google Scholar [3] D. Tanaka and Y. Kuramoto, Complex Ginzburg-Landau equation with nonlocal coupling, Phys. Rev. E, 68 (2003), 026219. Google Scholar [4] D. Tanaka, Chemical turbulence equivalent to Nikolaevskii turbulence, Phys. Rev. E, 70 (2004), 015202(R). Google Scholar [5] D. Tanaka, Turing instability leads oscillatory systems to spatiotemporal chaos, Progr. Theor. Phys. Suppl. N, 161 (2006), 119-126. Google Scholar [6] V. Nikolaevskii, "in Recent Advances in Engineering Science," edited by S.L. Koh, C.G. Speciale, Lecture Notes in Engineering, 39 Berlin: Springer, 1989, 210. Google Scholar [7] D. Strunin, Autosoliton model of the spinning fronts of reaction, IMA J. Appl. Math., 63 (1999), 163-177.  Google Scholar [8] D. Strunin, Phase equation with nonlinear excitation for nonlocally coupled oscillators, Physica D: Nonlinear Phenomena, 238 (2009), 1909-1916.  Google Scholar [9] D. Strunin and M. Mohammed, Parametric space for nonlinearly excited phase equation, ANZIAM J. Electron. Suppl., 53 (2011), C236-C248.  Google Scholar [10] D. Strunin, Nonlinear instability in generalized nonlinear phase diffusion equation, Progr. Theor. Phys. Suppl. N, 150 (2003), 444-448.  Google Scholar [11] http:, //www.mathworks.com/matlabcentral/fileexchange/28-differential-algebraic-, equation-solvers., ().   Google Scholar
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