Swift-Hohenberg equation with broken cubic-quintic nonlinearity

Abstract

The cubic-quintic Swift-Hohenberg equation (SH35) provides a convenient order parameter description of several convective systems with reflection symmetry in the layer midplane, including binary fluid convection. We use SH35 with an additional quadratic term to determine the qualitative effects of breaking the midplane reflection symmetry on the properties of spatially localized structures in these systems. Our results describe how the snakes-and-ladders organization of localized structures in SH35 deforms with increasing symmetry breaking and show that the deformation ultimately generates the snakes-and-ladders structure familiar from the quadratic-cubic Swift-Hohenberg equation. Moreover, in nonvariational systems, such as convection, odd-parity convectons necessarily drift when the reflection symmetry is broken, permitting collisions among moving localized structures. Collisions between both identical and nonidentical traveling states are described.

Spatial localization in heterogeneous systems

“…This is in contrast to Ref. [28], where the segments r 2 < r < r 3 also correspond to stable even parity states. Figure 21 shows that for small α the analytical prediction (14) of the translation eigenvalue agrees very well with the exact eigenvalue determined numerically.…”

“…With increasing |α| this stable portion of the asymmetric states is gradually eliminated and the branch stretches into a conventional rung, but now with half as many rungs as when α = 0, in a process that once again follows that identified in Ref. [28]. In view of this similarity we expect to find S-shaped branches of rung states as well.…”

“…The asymmetry between the α > 0 and α < 0 branches develops in the same way as in Ref. [28]. In both cases one ends up with four-limit snaking.…”

“…These solutions are all unstable; as α increases and the saddle nodes r 2 and r 3 annihilate the S-shaped rung branch shrinks to zero, again as in Ref. [28]. Figure 12 shows the corresponding rung states connecting odd parity states.…”

“…7(b)]. The whole process is highly reminiscent of a similar sequence of transformations that takes place when the symmetry u → −u of SH35 is progressively broken [28,29]. The reason for this is the following.…”