Freezing Relative Equilibria in Equivariant First-Order Evolution Equations (with Comsol Multiphysics 5.2)
1.1 Traveling waves in reaction diffusion systems
1.2 Freezing method for traveling waves
1.3 Numerical approximation of traveling waves via freezing method
1.4 Spectra and eigenfunctions of traveling waves
1.4.1 Point spectrum of traveling waves on the imaginary axis
1.4.2 Essential spectrum of traveling waves
2.1 Oscillating waves in reaction diffusion systems
2.2 Freezing method for oscillating waves
2.3 Numerical approximation of oscillating waves via freezing method
2.4 Spectra and eigenfunctions of oscillating waves
2.4.1 Point spectrum of oscillating waves on the imaginary axis
2.4.2 Essential spectrum of localized oscillating waves
3.1 Rotating waves in reaction diffusion systems
3.2 Freezing method for rotating waves
3.3 Numerical approximation of rotating waves via freezing method
4. Freezing Waves with Several Symmetries in Reaction Diffusion Systems
5. Freezing Multistructures (Multifronts, Multipulses and Multisolitons) and Wave Interactions in Reaction Diffusion Systems
6. Further coherent structures in Reaction Diffusion Systems
Freezing Relative Equilibria in Equivariant Second-Order Evolution Equations (with Comsol Multiphysics 5.2)
7.1 Traveling waves in systems of damped wave equations
7.2 Freezing method for traveling waves
7.3 Numerical approximation of traveling waves via freezing method
7.4 Spectra and eigenfunctions of traveling waves
7.4.1 Point spectrum of traveling waves on the imaginary axis
7.4.2 Essential spectrum of traveling waves
8. Freezing Rotating Waves in Damped Wave Equations
8.1 Rotating waves in systems of damped wave equations
8.2 Freezing method for rotating waves
8.3 Numerical approximation of rotating waves via freezing method
The model library of the examples below is summerized in the following file
For the implementation of the results below we suggest the following Comsol tutorial
Some further tutorials created by the older version Comsol 3.5a can be found here.
Freezing Relative Equilibria in Equivariant First-Order Evolution Equations (with Comsol Multiphysics 5.1 and 5.2)
1. Freezing Traveling Waves in Reaction Diffusion Equations
2. Freezing Oscillating Waves in Reaction Diffusion Equations
Equation
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Space
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Wave Phenomena
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Oscillating wave
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Space-Time
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Profile
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Space-Time
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Velocities
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Spectrum
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Eigenfunction
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Files
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Schrödinger equation
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1D
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oscillating pulse
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mph
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Gross-Pitaevskii equation
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1D
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oscillating pulse
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mph
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cubic-quintic complex Ginzburg-Landau equation
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1D
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oscillating pulse
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mph
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3. Freezing Rotating Waves in Reaction Diffusion Systems
Equation
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Space
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Wave Phenomena
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3D-View
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Space-Time
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Profile
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Space-Time
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Velocities
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Spectrum
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Eigenfunction
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Files
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cubic-quintic complex Ginzburg-Landau equation
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2D
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spinning soliton, dissipative soliton, rotating soliton
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mph
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cubic-quintic complex Ginzburg-Landau equation
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2D
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rotating spiral wave
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mph
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Lambda-omega system
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2D
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rotating spiral wave
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mph
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Barkley model
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2D
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rotating spiral wave
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mph
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cubic-quintic complex Ginzburg-Landau equation
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3D
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spinning soliton, dissipative soliton, rotating soliton
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mph
mph
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Lambda-omega system
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3D
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scroll wave, scroll ring
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4. Freezing Waves with Several Symmetries in Reaction Diffusion Systems
Equation
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Space
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Wave Phenomena
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Wave Phenomena
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Space-Time
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Profile
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Space-Time
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Velocities
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Spectrum
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Eigenfunction
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Files
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Schrödinger equation
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1D
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traveling oscillating pulse
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mph
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cubic-quintic complex Ginzburg-Landau equation
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1D
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traveling oscillating front
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mph
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5. Freezing Multistructures (Multifronts, Multipulses and Multisolitons) and Wave Interactions in Reaction Diffusion Systems
Equation
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Space
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Wave Phenomena
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Wave Phenomena
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Space-Time
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Profile
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Space-Time
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Velocities, Positions
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Spectrum
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Eigenfunction
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Files
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Nagumo equation
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1D
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Traveling 2-front (traveling multifront, repelling fronts)
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mph
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Nagumo equation
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1D
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Traveling 2-front (collision, colliding fronts)
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mph
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quintic Nagumo equation
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1D
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Traveling 2-front (traveling multifront)
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mph
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quintic Nagumo equation
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1D
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Traveling 2-front (collision)
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mph
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quintic Nagumo equation
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1D
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Traveling 3-front (traveling multifront)
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quintic Nagumo equation
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1D
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Traveling 4-front (traveling multifront)
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quintic Nagumo equation
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1D
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Traveling 4-front (collision)
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FitzHugh-Nagumo system
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1D
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Traveling 2-pulse (traveling multipulse)
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cubic-quintic complex Ginzburg-Landau equation
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1D
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Oscillating 2-pulse (oscillating multipulse)
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cubic-quintic complex Ginzburg-Landau equation
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1D
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Oscillating pulse and traveling oscillating front
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cubic-quintic complex Ginzburg-Landau equation
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1D
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Traveling oscillating 2-front (traveling oscillating multifront)
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cubic-quintic complex Ginzburg-Landau equation
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2D
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Spinning 2-soliton (rotating multisoliton)
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cubic-quintic complex Ginzburg-Landau equation
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2D
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Spinning 3-soliton (rotating multisoliton)
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6. Further coherent structures in Reaction Diffusion Systems
Equation
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Space
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Wave Phenomena
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Wave Phenomena
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Space-Time
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Profile
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Space-Time
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Velocities
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Spectrum
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Eigenfunction
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Files
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cubic-quintic complex Ginzburg-Landau equation
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1D
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pulsating soliton
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mph
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cubic-quintic complex Ginzburg-Landau equation
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1D
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creeping soliton
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mph
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Freezing Relative Equilibria in Equivariant Second-Order Evolution Equations (with Comsol Multiphysics 5.1 and 5.2)
7. Freezing Traveling Waves in Damped Wave Equations
Equation
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Space
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Wave Phenomena
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Traveling wave
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Space-Time
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Profile
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Space-Time
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Velocity, Acceleration
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Spectrum
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Eigenfunction
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Files
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Damped Fisher's equation
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1D
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traveling front
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Damped Nagumo equation
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1D
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traveling front
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mph
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Damped quintic Nagumo equation
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1D
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traveling front
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mph
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Damped FitzHugh-Nagumo system
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1D
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traveling front
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mph
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Damped FitzHugh-Nagumo system
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1D
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traveling pulse
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mph
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Damped Barkley model
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1D
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traveling pulse
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8. Freezing Rotating Waves in Damped Wave Equations
Equation
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Space
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Wave Phenomena
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3D-View
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Space-Time
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Profile
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Space-Time
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Velocities, Accelerations
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Spectrum
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Eigenfunction
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Files
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damped cubic-quintic complex Ginzburg-Landau equation
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2D
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spinning soliton, dissipative soliton, rotating soliton
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Literature
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