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Register a new userModulating traveling fronts for the Swift-Hohenberg equation in the case of an additional conservation law
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Bastian Hilder
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We consider the one-dimensional Swift-Hohenberg equation coupled to a conservation law. As a parameter increases the system undergoes a Turing bifurcation. We study the dynamics near this bifurcation. First, we show that stationary, periodic solutions bifurcate from a homogeneous ground state. Second, we construct modulating traveling fronts which model an invasion of the unstable ground state by the periodic solutions. This provides a mechanism of pattern formation for the studied system. The existence proof uses center manifold theory for a reduction to a finite-dimensional problem. This is possible despite the presence of infinitely many imaginary eigenvalues for vanishing bifurcation parameter since the eigenvalues leave the imaginary axis with different velocities if the parameter increases. Furthermore, compared to non-conservative systems, we address new difficulties arising from an additional neutral mode at Fourier wave number $k=0$ by exploiting that the amplitude of the conserved variable is small compared to the other variables.
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A theoretical model for studying pattern formation in electroconvection is proposed in the form of a modified Swift-Hohenberg equation. A localized state is found in two dimension, in agreement with the experimentally observed ``worm state. The corresponding one dimensional model is also studied, and a novel stationary localized state due to nonadiabatic effect is found. The existence of the 1D localized state is shown to be responsible for the formation of the two dimensional ``worm state in our model.
We consider the thin-film equation $partial_t h + partial_y left(m(h) partial_y^3 hright) = 0$ in ${h > 0}$ with partial-wetting boundary conditions and inhomogeneous mobility of the form $m(h) = h^3+lambda^{3-n}h^n$, where $h ge 0$ is the film height, $lambda > 0$ is the slip length, $y > 0$ denotes the lateral variable, and $n in (0,3)$ is the mobility exponent parameterizing the nonlinear slip condition. The partial-wetting regime implies the boundary condition $partial_y h = mathrm{const.} > 0$ at the triple junction $partial{h > 0}$ (nonzero microscopic contact angle). Existence and uniqueness of traveling-wave solutions to this problem under the constraint $partial_y^2 h to 0$ as $h to infty$ have been proved in previous work by Chiricotto and Giacomelli in [Commun. Appl. Ind. Math., 2(2):e-388, 16, 2011]. We are interested in the asymptotics as $h downarrow 0$ and $h to infty$. By reformulating the problem as $h downarrow 0$ as a dynamical system for the error between the solution and the microscopic contact angle, values for $n$ are found for which linear as well as nonlinear resonances occur. These resonances lead to a different asymptotic behavior of the solution as $hdownarrow0$ depending on $n$. Together with the asymptotics as $htoinfty$ characterizing Tanners law for the velocity-dependent macroscopic contact angle as found by Giacomelli, the first author of this work, and Otto in [Nonlinearity, 29(9):2497-2536, 2016], the rigorous asymptotics of the traveling-wave solution to the thin-film equation in partial wetting can be characterized. Furthermore, our approach enables us to analyze the relation between the microscopic and macroscopic contact angle. It is found that Tanners law for the macroscopic contact angle depends continuously differentiably on the microscopic contact angle.
We show that all meromorphic solutions of the stationary reduction of the real cubic Swift-Hohenberg equation are elliptic or degenerate elliptic. We then obtain them all explicitly by the subequation method, and one of them appears to be a new elliptic solution.
We study traveling wave solutions of the nonlinear variational wave equation. In particular, we show how to obtain global, bounded, weak traveling wave solutions from local, classical ones. The resulting waves consist of monotone and constant segments, glued together at points where at least one one-sided derivative is unbounded. Applying the method of proof to the Camassa--Holm equation, we recover some well-known results on its traveling wave solutions.
Generalizing the L-Kuramoto-Sivashinsky (L-KS) kernel from our earlier work, we give a novel explicit-kernel formulation useful for a large class of fourth order deterministic, stochastic, linear, and nonlinear PDEs in multispatial dimensions. These include pattern formation equations like the Swift-Hohenberg and many other prominent and new PDEs. We first establish existence, uniqueness, and sharp dimension-dependent spatio-temporal Holder regularity for the canonical (zero drift) L-KS SPDE, driven by white noise on ${RptimesRd}_{d=1}^{3}$. The spatio-temporal Holder exponents are exactly the same as the striking ones we proved for our recently introduced Brownian-time Brownian motion (BTBM) stochastic integral equation, associated with time-fractional PDEs. The challenge here is that, unlike the positive BTBM density, the L-KS kernel is the Gaussian average of a modified, highly oscillatory, and complex Schrodinger propagator. We use a combination of harmonic and delicate analysis to get the necessary estimates. Second, attaching order parameters $vepo$ to the L-KS spatial operator and $vept$ to the noise term, we show that the dimension-dependent critical ratio $vept/vepo^{d/8}$ controls the limiting behavior of the L-KS SPDE, as $vepo,veptsearrow0$; and we compare this behavior to that of the less regular second order heat SPDEs. Finally, we give a change-of-measure equivalence between the canonical L-KS SPDE and nonlinear L-KS SPDEs. In particular, we prove uniqueness in law for the Swift-Hohenberg and the law equivalence---and hence the same Holder regularity---of the Swift-Hohenberg SPDE and the canonical L-KS SPDE on compacts in one-to-three dimensions.
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