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Register a new userLemniscates do not survive Laplacian growth
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The large class of moving boundary processes in the plane modeled by the so-called Laplacian growth, which describes, e.g., solidification, electrodeposition, viscous fingering, bacterial growth, etc., is known to be integrable and to exhibit a large number of exact solutions. In this work, the boundaries are assumed to be in the class of lemniscates with all zeros inside the bounded component of the complex plane. We prove that for any initial boundary taken from this class, the evolving boundary instantly stops being in the class, or else Laplacian growth destroys lemniscates instantly.
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We review here particular aspects of the connection between Laplacian growth problems and classical integrable systems. In addition, we put forth a possible relation between quantum integrable systems and Laplacian growth problems. Such a connection, if confirmed, has the potential to allow for a theoretical prediction of the fractal properties of Laplacian growth clusters, through the representation theory of conformal field theory.
We develop a physical model for how galactic disks survive and/or are destroyed in interactions. Based on dynamical arguments, we show gas primarily loses angular momentum to internal torques in a merger. Gas within some characteristic radius (a function of the orbital parameters, mass ratio, and gas fraction of the merging galaxies), will quickly lose angular momentum to the stars sharing the perturbed disk, fall to the center and be consumed in a starburst. A similar analysis predicts where violent relaxation of the stellar disks is efficient. Our model allows us to predict the stellar and gas content that will survive to re-form a disk in the remnant, versus being violently relaxed or contributing to a starburst. We test this in hydrodynamic simulations and find good agreement as a function of mass ratio, orbital parameters, and gas fraction, in simulations spanning a wide range in these properties and others, including different prescriptions for gas physics and feedback. In an immediate sense, the amount of disk that re-forms can be understood in terms of well-understood gravitational physics, independent of details of ISM gas physics or feedback. This allows us to explicitly quantify the requirements for such feedback to (indirectly) enable disk survival, by changing the pre-merger gas content and distribution. The efficiency of disk destruction is a strong function of gas content: we show how and why sufficiently gas-rich major mergers can, under general conditions, yield systems with small bulges (B/T<0.2). We provide prescriptions for inclusion of our results in semi-analytic models.
The Laplacian growth (the Hele-Shaw problem) of multi-connected domains in the case of zero surface tension is proven to be equivalent to an integrable systems of Whitham equations known in soliton theory. The Whitham equations describe slowly modulated periodic solutions of integrable hierarchies of nonlinear differential equations. Through this connection the Laplacian growth is understood as a flow in the moduli space of Riemann surfaces.
We theoretically study the topological robustness of the surface physics induced by Weyl Fermi-arc surface states in the presence of short-ranged quenched disorder and surface-bulk hybridization. This is investigated with numerically exact calculations on a lattice model exhibiting Weyl Fermi-arcs. We find that the Fermi-arc surface states, in addition to having a finite lifetime from disorder broadening, hybridize with nonperturbative bulk rare states making them no longer bound to the surface (i.e. they lose their purely surface spectral character). Thus, we provide strong numerical evidence that the Weyl Fermi-arcs are not topologically protected from disorder. Nonetheless, the surface chiral velocity is robust and survives in the presence of strong disorder, persisting all the way to the Anderson-localized phase by forming localized current loops that live within the localization length of the surface. Thus, the Weyl semimetal is not topologically robust to the presence of disorder, but the surface chiral velocity is.
We study the effect of adding to a directed chain of interconnected systems a directed feedback from the last element in the chain to the first. The problem is closely related to the fundamental question of how a change in network topology may influence the behavior of coupled systems. We begin the analysis by investigating a simple linear system. The matrix that specifies the system dynamics is the transpose of the network Laplacian matrix, which codes the connectivity of the network. Our analysis shows that for any nonzero complex eigenvalue $lambda$ of this matrix, the following inequality holds: $frac{|Im lambda |}{|Re lambda |} leq cotfrac{pi}{n}$. This bound is sharp, as it becomes an equality for an eigenvalue of a simple directed cycle with uniform interaction weights. The latter has the slowest decay of oscillations among all other network configurations with the same number of states. The result is generalized to directed rings and chains of identical nonlinear oscillators. For directed rings, a lower bound $sigma_c$ for the connection strengths that guarantees asymptotic synchronization is found to follow a similar pattern: $sigma_c=frac{1}{1-cosleft( 2pi /nright)} $. Numerical analysis revealed that, depending on the network size $n$, multiple dynamic regimes co-exist in the state space of the system. In addition to the fully synchronous state a rotating wave solution occurs. The effect is observed in networks exceeding a certain critical size. The emergence of a rotating wave highlights the importance of long chains and loops in networks of oscillators: the larger the size of chains and loops, the more sensitive the network dynamics becomes to removal or addition of a single connection.
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