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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

Dmitry Strunin 1, and  Mayada Mohammed 1,

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.
Keywords: dissipation., nonlinear excitation, Partial differential equation.
Mathematics Subject Classification: Primary: 35K55; Secondary: 65M0.
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.   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.   Google Scholar

[3]

D. Tanaka and Y. Kuramoto, Complex Ginzburg-Landau equation with nonlocal coupling,, Phys. Rev. E, 68 (2003).   Google Scholar

[4]

D. Tanaka, Chemical turbulence equivalent to Nikolaevskii turbulence,, Phys. Rev. E, 70 (2004).   Google Scholar

[5]

D. Tanaka, Turing instability leads oscillatory systems to spatiotemporal chaos,, Progr. Theor. Phys. Suppl. N, 161 (2006), 119.   Google Scholar

[6]

V. Nikolaevskii, "in Recent Advances in Engineering Science,", edited by S.L. Koh, (1989).   Google Scholar

[7]

D. Strunin, Autosoliton model of the spinning fronts of reaction,, IMA J. Appl. Math., 63 (1999), 163.   Google Scholar

[8]

D. Strunin, Phase equation with nonlinear excitation for nonlocally coupled oscillators,, Physica D: Nonlinear Phenomena, 238 (2009), 1909.   Google Scholar

[9]

D. Strunin and M. Mohammed, Parametric space for nonlinearly excited phase equation,, ANZIAM J. Electron. Suppl., 53 (2011).   Google Scholar

[10]

D. Strunin, Nonlinear instability in generalized nonlinear phase diffusion equation,, Progr. Theor. Phys. Suppl. N, 150 (2003), 444.   Google Scholar

[11]

http:, //www.mathworks.com/matlabcentral/fileexchange/28-differential-algebraic-, equation-solvers., ().   Google Scholar

show all references

References:
[1]

G. Sivashinsky, Nonlinear analysis of hydrodynamical instability in laminar flames,, Acta Astronaut., 4 (1977), 1177.   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.   Google Scholar

[3]

D. Tanaka and Y. Kuramoto, Complex Ginzburg-Landau equation with nonlocal coupling,, Phys. Rev. E, 68 (2003).   Google Scholar

[4]

D. Tanaka, Chemical turbulence equivalent to Nikolaevskii turbulence,, Phys. Rev. E, 70 (2004).   Google Scholar

[5]

D. Tanaka, Turing instability leads oscillatory systems to spatiotemporal chaos,, Progr. Theor. Phys. Suppl. N, 161 (2006), 119.   Google Scholar

[6]

V. Nikolaevskii, "in Recent Advances in Engineering Science,", edited by S.L. Koh, (1989).   Google Scholar

[7]

D. Strunin, Autosoliton model of the spinning fronts of reaction,, IMA J. Appl. Math., 63 (1999), 163.   Google Scholar

[8]

D. Strunin, Phase equation with nonlinear excitation for nonlocally coupled oscillators,, Physica D: Nonlinear Phenomena, 238 (2009), 1909.   Google Scholar

[9]

D. Strunin and M. Mohammed, Parametric space for nonlinearly excited phase equation,, ANZIAM J. Electron. Suppl., 53 (2011).   Google Scholar

[10]

D. Strunin, Nonlinear instability in generalized nonlinear phase diffusion equation,, Progr. Theor. Phys. Suppl. N, 150 (2003), 444.   Google Scholar

[11]

http:, //www.mathworks.com/matlabcentral/fileexchange/28-differential-algebraic-, equation-solvers., ().   Google Scholar

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