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1. Spectral stability of pattern-forming fronts in the complex Ginzburg–Landau equation with a quenching mechanism

   https://doi.org/10.1088/1361-6544/ac355b  

   Goh, Ryan ; de Rijk, Björn ( December 2021 , Nonlinearity)

   Abstract We consider pattern-forming fronts in the complex Ginzburg–Landau equation with a traveling spatial heterogeneity which destabilises, or quenches, the trivial ground state while progressing through the domain. We consider the regime where the heterogeneity propagates with speed c just below the linear invasion speed of the pattern-forming front in the associated homogeneous system. In this situation, the front locks to the interface of the heterogeneity leaving a long intermediate state lying near the unstable ground state, possibly allowing for growth of perturbations. This manifests itself in the spectrum of the linearisation about the front through the accumulation of eigenvalues onto the absolute spectrum associated with the unstable ground state. As the quench speed c increases towards the linear invasion speed, the absolute spectrum stabilises with the same rate at which eigenvalues accumulate onto it allowing us to rigorously establish spectral stability of the front in L 2 ( R ) . The presence of unstable absolute spectrum poses a technical challenge as spatial eigenvalues along the intermediate state no longer admit a hyperbolic splitting and standard tools such as exponential dichotomies are unavailable. Instead, we projectivise the linear flow, and use Riemann surface unfolding in combination with a superpositionmore »principle to study the evolution of subspaces as solutions to the associated matrix Riccati differential equation on the Grassmannian manifold. Eigenvalues can then be identified as the roots of the meromorphic Riccati–Evans function, and can be located using winding number and parity arguments.« less
2. Localized and extended patterns in the cubic-quintic Swift-Hohenberg equation on a disk

   https://doi.org/10.1103/PhysRevE.104.014208  

   Verschueren, Nicolás ; Knobloch, Edgar ; Uecker, Hannes ( July 2021 , Physical Review E)
3. Unconditionally Energy Stable DG Schemes for the Swift–Hohenberg Equation

   https://doi.org/10.1007/s10915-019-01038-6  

   Liu, Hailiang ; Yin, Peimeng ( November 2019 , Journal of Scientific Computing)
4. Localized states in the conserved Swift-Hohenberg equation with cubic nonlinearity

   https://doi.org/10.1103/PhysRevE.87.042915  

   Thiele, Uwe ; Archer, Andrew J. ; Robbins, Mark J. ; Gomez, Hector ; Knobloch, Edgar ( April 2013 , Physical Review E)
5. Complex Swift Hohenberg Equation (CSHE) Dissipative Soliton Fiber Laser

   Khanolka, A ; Zang, Y ; Chong, A ( January 2021 , Conference on Lasers and Electrooptics)

   We report dissipative solitons of CSHE in a mode-locked fiber laser by employing a unique fiber filter which produces a fourth order spectral response. The laser generates moving, asymmetric pulses with a highly asymmetric spectrum.