Mathematical Physics, Analysis and Geometry最新文献

We consider a four-prototype Rossler system introduced by Otto Rössler among others as prototypes of the simplest autonomous differential equations (in the sense of minimal dimension, minimal number of parameters, minimal number of nonlinear terms) having chaotic behavior. We contribute towards the understanding of its chaotic behavior by studying its integrability from different points of view. We show that it is neither Darboux integrable, nor (C^1)-integrable.

In this paper, we consider nearest-neighbor oriented percolation with independent Bernoulli bond-occupation probability on the d-dimensional body-centered cubic (BCC) lattice ({mathbb {L}^d}) and the set of non-negative integers ({{mathbb {Z}}_+}). Thanks to the orderly structure of the BCC lattice, we prove that the infrared bound holds on ({mathbb {L}^d} times {{mathbb {Z}}_+}) in all dimensions (dge 9). As opposed to ordinary percolation, we have to deal with complex numbers due to asymmetry induced by time-orientation, which makes it hard to bound the bootstrap functions in the lace-expansion analysis. By investigating the Fourier–Laplace transform of the random-walk Green function and the two-point function, we derive the key properties to obtain the upper bounds and resolve a problematic issue in Nguyen and Yang’s bound. The issue is caused by the fact that the Fourier transform of the random-walk transition probability can take the value (-1).

We study the long time asymptotic behavior for the Cauchy problem of an integrable real nonlocal mKdV equation with nonzero initial data in the solitonic regions

where (|q_{pm }|=1) and (q_{+}=delta q_{-}), (sigma delta =-1). In our previous article, we have obtained long time asymptotics for the nonlocal mKdV equation in the solitonic region (-6<xi <6) with (xi =frac{x}{t}). In this paper, we give the asymptotic expansion of the solution q(x, t) for other solitonic regions (xi <-6) and (xi >6). Based on the Riemann–Hilbert formulation of the Cauchy problem, further using the ({bar{partial }}) steepest descent method, we derive different long time asymptotic expansions of the solution q(x, t) in above two different space-time solitonic regions. In the region (xi <-6), phase function (theta (z)) has four stationary phase points on the ({mathbb {R}}). Correspondingly, q(x, t) can be characterized with an ({mathcal {N}}(Lambda ))-soliton on discrete spectrum, the leading order term on continuous spectrum and an residual error term, which are affected by a function (textrm{Im}nu (zeta _i)). In the region (xi >6), phase function (theta (z)) has four stationary phase points on (i{mathbb {R}}), the corresponding asymptotic approximations can be characterized with an ({mathcal {N}}(Lambda ))-soliton with diverse residual error order ({mathcal {O}}(t^{-1})).

We study extensions of the classical Toda lattices at several different space–time scales. These extensions are from the classical tridiagonal phase spaces to the phase space of full Hessenberg matrices, referred to as the Full Kostant–Toda Lattice. Our formulation makes it natural to make further Lie-theoretic generalizations to dual spaces of Borel–Lie algebras. Our study brings into play factorizations of Loewner–Whitney type in terms of canonical coordinatizations due to Lusztig. Using these coordinates we formulate precise conditions for the well-posedness of the dynamics at the different space–time scales. Along the way we derive a novel, minimal box–ball system for the Full Kostant–Toda Lattice that does not involve any capacities or colorings, and which has a natural interpretation in terms of the Robinson–Schensted–Knuth algorithm. We provide as well an extension of O’Connell’s ordinary differential equations to the Full Kostant–Toda Lattice.

In some recent literature the role of non self-adjoint Hamiltonians, (Hne H^dagger ), is often considered in connection with gain-loss systems. The dynamics for these systems is, most of the times, given in terms of a Schrödinger equation. In this paper we rather focus on the Heisenberg-like picture of quantum mechanics, stressing the (few) similarities and the (many) differences with respected to the standard Heisenberg picture for systems driven by self-adjoint Hamiltonians. In particular, the role of the symmetries, *-derivations and integrals of motion is discussed.

This is the first in a series of articles about recovering the full algebraic structure of a boundary conformal field theory (CFT) from the scaling limit of the critical Ising model in slit-strip geometry. Here, we introduce spaces of holomorphic functions in continuum domains as well as corresponding spaces of discrete holomorphic functions in lattice domains. We find distinguished sets of functions characterized by their singular behavior in the three infinite directions in the slit-strip domains, and note in particular that natural subsets of these functions span analogues of Hardy spaces. We prove convergence results of the distinguished discrete holomorphic functions to the continuum ones. In the subsequent articles, the discrete holomorphic functions will be used for the calculation of the Ising model fusion coefficients (as well as for the diagonalization of the Ising transfer matrix), and the convergence of the functions is used to prove the convergence of the fusion coefficients. It will also be shown that the vertex operator algebra of the boundary conformal field theory can be recovered from the limit of the fusion coefficients via geometric transformations involving the distinguished continuum functions.

We consider the generalised Calogero–Moser–Sutherland quantum integrable system associated to the configuration of vectors (AG_2), which is a union of the root systems (A_2) and (G_2). We establish the existence of and construct a suitably defined Baker–Akhiezer function for the system, and we show that it satisfies bispectrality. We also find two corresponding dual difference operators of rational Macdonald–Ruijsenaars type in an explicit form.

We revisit the stability (instability) of the outer (inner) catenoid connecting two concentric circular rings and give an explicit new construction of the unstable mode of the inner catenoid by studying the spectrum of an exactly solvable one-dimensional Schrödinger operator with an asymmetric Darboux–Pöschl–Teller potential.

We define bi-infinite versions of four well-studied discrete integrable models, namely the ultra-discrete KdV equation, the discrete KdV equation, the ultra-discrete Toda equation, and the discrete Toda equation. For each equation, we show that there exists a unique solution to the initial value problem when the given data lies within a certain class, which includes the support of many shift ergodic measures. Our unified approach, which is also applicable to other integrable systems defined locally via lattice maps, involves the introduction of a path encoding (that is, a certain antiderivative) of the model configuration, for which we are able to describe the dynamics more generally than in previous work on finite size systems, periodic systems and semi-infinite systems. In particular, in each case we show that the behaviour of the system is characterized by a generalization of the classical ‘Pitman’s transformation’ of reflection in the past maximum, which is well-known to probabilists. The picture presented here also provides a means to identify a natural ‘carrier process’ for configurations within the given class, and is convenient for checking that the systems we discuss are all-time reversible. Finally, we investigate links between the different systems, such as showing that bi-infinite all-time solutions for the ultra-discrete KdV (resp. Toda) equation may appear as ultra-discretizations of corresponding solutions for the discrete KdV (resp. Toda) equation.

In this paper we investigate whether a quasilinear system of PDEs of first order admits Hamiltonian formulation with local and nonlocal operators. By using the theory of differential coverings, we find differential-geometric conditions necessary to write a given system with one of the three Hamiltonian operators investigated.