Related papers: Growth processes related to the dispersionless Lax…
We study one-parameter expanding evolution families of simply connected domains in the complex plane described by infinite systems of evolution parameters. These evolution parameters in some cases admit Hamiltonian formulation and lead to…
Generalizing the graded commutator in superalgebras, we propose a new bracket operation on the space of graded operators with an involution. We study properties of this operation and show that the Lax representation of the two-dimensional…
We construct a two-dimensional family of integrable coupled Dirac--scalar field theories in $1+1$ dimensions, parameterized by $(\thz,\alpha)\in[0,\pi/2]^2$, whose Lax connection takes values in $\slC$ throughout. The family arises as the…
The theory of random matrices with eigenvalues distributed in the complex plane and more general "beta-ensembles" (logarithmic gases in 2D) is reviewed. The distribution and correlations of the eigenvalues are investigated in the large N…
The reaction-diffusion processes in a growing domain involves a dilution term that modifies the properties of the homogeneous state that, in contrast to a fixed domain, depends on time. We study how the dilution term changes the steady…
Equations of the Loewner class subject to non-constant boundary conditions along the real axis, are formulated and solved giving the geodesic paths of slits growing in the upper half complex plane. The problem is motivated by Laplacian…
Stochastic Loewner evolutions (SLE) are random growth processes of sets, called hulls, embedded in the two dimensional upper half plane. We elaborate and develop a relation between SLE evolutions and conformal field theories (CFT) which is…
We study the link between the degree growth of integrable birational mappings of order higher than two and their singularity structures. The higher order mappings we use in this study are all obtained by coupling mappings that are…
SLE_k stochastic processes describe growth of random curves which, in some cases, may be identified with boundaries of two dimensional critical percolating clusters. By generalizing SLE_k growths to formal Markov processes on the central…
An iterative algorithm for determining a class of solutions of the dispersionful 2-Toda hierarchy characterized by string equations is developed. This class includes the solution which underlies the large N-limit of the Hermitian matrix…
Laplacian growth is the study of interfaces that move in proportion to harmonic measure. Physically, it arises in fluid flow and electrical problems involving a moving boundary. We survey progress over the last decade on discrete models of…
We review the recent approach to the construction of (3+1)-dimensional integrable dispersionless partial differential systems based on their contact Lax pairs and the related $R$-matrix theory for the Lie algebra of functions with respect…
Surface growth is a crucial component of many natural and artificial processes from cell proliferation to additive manufacturing. In elastic systems surface growth is usually accompanied by the development of geometrical incompatibility…
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…
Given a dynamical system with $m$ independent conserved quantities, we construct a multi-parameter family of new systems in which these quantities evolve monotonically and proportionally, and are replaced by $m-1$ conserved linear…
We report a new algorithm to generate Laplacian Growth Patterns using iterated conformal maps. The difficulty of growing a complete layer with local width proportional to the gradient of the Laplacian field is overcome. The resulting growth…
Analogues of the KP and the Toda lattice hierarchy called dispersionless KP and Toda hierarchy are studied. Dressing operations in the dispersionless hierarchies are introduced as a canonical transformation, quantization of which is…
We generalize the Toda lattice hierarchy by considering N+M dependent variables. We construct roots and logarithms of the Lax operator which are uniquely defined operators with coefficients that are $\epsilon$-series of differential…
We present a simple, unified approach to determining the growth law for the characteristic length scale, $L(t)$, in the phase ordering kinetics of a system quenched from a disordered phase to within an ordered phase. This approach, based on…
This is a simplification of our prior work on the existence theory for the Rosseland-type equations. Inspired by the Rosseland equation in the conduction-radiation coupled heat transfer, we use the locally arbitrary growth conditions…