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Spatially localized structures are key components of turbulence and otherspatio-temporally chaotic systems. From a dynamical systems viewpoint, it isdesirable to obtain corresponding exact solutions, though their existence isnot guaranteed. A damping filter method is introduced to obtain variouslylocalized solutions, and adopted into two typical cases. This method introducesa spatially selective damping effect to make a good guess at the exactsolution, and we can obtain an exact solution through a continuation with thedamping amplitude. First target is a steady solution to Swift-Hohenbergequation, which is a representative of bi-stable systems in which localizedsolutions coexist, and a model for span-wisely localized cases. Not onlysolutions belonging to the well-known snaking branches but also those belongingto an isolated branch known as "isolas" are found with a continuation pathsbetween them in phase space extended with the damping amplitude. This indicatesthat this spatially selective excitation mechanism has an advantage insearching spatially localized solutions. Second target is a spatially localizedtraveling-wave solution to Kuramoto-Sivashinsky equation, which is a model forstream-wisely localized cases. Since the spatially selective damping effectbreaks Galilean and translational invariances, the propagation velocity cannotbe determined uniquely while the damping is active, and a singularity ariseswhen these invariances are recovered. We demonstrate that this singularity canbe avoided by imposing a simple condition, and a localized traveling-wavesolution is obtained with a specific propagation speed.

Damping filter method for obtaining spatially localized solutions