Abstract:
This paper concerns the topology of the isospectral {\it real} manifold of the ${\mathfrak sl}(N)$ periodic Toda lattice consisting of $2^{N-1}$ different systems. The solutions of those systems contain blow-ups, and the set of those singular points defines a devisor of the manifold. Then adding the divisor, the manifold is compactified as the real part of the $(N-1)$-dimensional Jacobi variety associated with a hyperelliptic Riemann surface of genus $g=N-1$. We also study the real structure of the divisor, and then provide conjectures on the topology of the affine part of the real Jacobian and on the gluing rule over the divisor to compactify the manifold based upon the sign-representation of the Weyl group of ${\mathfrak sl}(N)$.

Abstract:
Winter CO2 flux is an important element to assess when estimating the annual carbon budget on regional and global scales. However, winter observation frequency is limited due to the extreme cold weather in sub-Arctic and Arctic ecosystems. In this study, the continuous monitoring of winter CO2 flux in black spruce forest soil of interior Alaska was performed using NDIR CO2 sensors at 10, 20, and 30 cm above the soil surface during the snow-covered period (DOY 357 to 466) of 2006/2007. The atmospheric pressure was divided into four phases: >1000 hPa (HP: high pressure); 985

Abstract:
We present a review of the normal form theory for weakly dispersive nonlinear wave equations where the leading order phenomena can be described by the KdV equation. This is an infinite dimensional extension of the well-known Poincar\'e-Dulac normal form theory for ordinary differential equations. We also provide a detailed analysis of the interaction problem of solitary wavesas an important application of the normal form theory. Several explicit examples are discussed based on the normal form theory, and the results are compared with their numerical simulations. Those examples include the ion acoustic wave equation, the Boussinesq equation as a model of the shallow water waves, the regularized long wave equation and the Hirota bilinear equation having a 7th order linear dispersion.

Abstract:
We present a new class of topological conformal field theories (TCFT) characterized by a rational $W$ potential, which includes the minimal models of A and D types as its subclasses. An explicit form of the $W$ potential is found by solving the underlying dispersionless KP hierarchy in a particular small phase space. We discuss also the dispersionless KP hierarchy in large phase spaces by reformulating the hierarchy, and show that the $W$ potential takes a universal form, which does not depend on a specific form of the solution in a large space.

Abstract:
The dispersionless differential Fay identity is shown to be equivalent to a kernel expansion providing a universal algebraic characterization and solution of the dispersionless Hirota equations. Some calculations based on D-bar data of the action are also indicated.

Abstract:
We study soliton solutions to the DKP equation which is defined by the Hirota bilinear form, \[ {\begin{array}{llll} (-4D_xD_t+D_x^4+3D_y^2) \tau_n\cdot\tau_n=24\tau_{n-1}\tau_{n+1}, (2D_t+D_x^3\mp 3D_xD_y) \tau_{n\pm 1}\cdot\tau_n=0 \end{array} \quad n=1,2,.... \] where $\tau_0=1$. The $\tau$-functions $\tau_n$ are given by the pfaffians of certain skew-symmetric matrix. We identify one-soliton solution as an element of the Weyl group of D-type, and discuss a general structure of the interaction patterns among the solitons. Soliton solutions are characterized by $4N\times 4N$ skew-symmetric constant matrix which we call the $B$-matrices. We then find that one can have $M$-soliton solutions with $M$ being any number from $N$ to $2N-1$ for some of the $4N\times 4N$ $B$-matrices having only $2N$ nonzero entries in the upper triangular part (the number of solitons obtained from those $B$-matrices was previously expected to be just $N$).

Abstract:
The paper concerns the topology of an isospectral real smooth manifold for certain Jacobi element associated with real split semisimple Lie algebra. The manifold is identified as a compact, connected completion of the disconnected Cartan subgroup of the corresponding Lie group $\tilde G$ which is a disjoint union of the split Cartan subgroups associated to semisimple portions of Levi factors of all standard parabolic subgroups of $\tilde G$. The manifold is also related to the compactified level sets of a generalized Toda lattice equation defined on the semisimple Lie algebra, which is diffeomorphic to a toric variety in the flag manifold ${\tilde G}/B$ with Borel subgroup $B$ of $\tilde G$. We then give a cellular decomposition and the associated chain complex of the manifold by introducing colored-signed Dynkin diagrams which parametrize the cells in the decomposition.

Abstract:
This paper concerns the topology of isospectral real manifolds of certain Jacobi elements associated with real split semisimple Lie algebras. The manifolds are related to the compactified level sets of the generalized (nonperiodic) Toda lattice equations defined on the semisimple Lie algebras. We then give a cellular decomposition and the associated chain complex of the manifold by introducing colored Dynkin diagrams which parametrize the cells in the decomposition. We also discuss the Morse chain complex of the manifold.

Abstract:
This paper begins with an observation that the isospectral leaves of the signed Toda lattice as well as the Toda flow itself may be constructed from the Tomei manifolds by cutting and pasting along certain chamber walls inside a polytope. It is also observed through examples that although there is some freedom in this procedure of cutting and pasting the manifold and the flow, the choices that can be made are not arbitrary. We proceed to describe a procedure that begins with an action of the Weyl group on a set of signs; it uses the Convexity Theorem in \cite{BFR:90} and combines the resulting polytope with the chosen Weyl group action to paste together a compact manifold. This manifold which is obtained carries an action of the Weyl group and a Toda lattice flow which is related to this action. This construction gives rise to a large family of compact manifolds which is parametrized by twisted sign actions of the Weyl group. For example, the trivial action gives rise to Tomei manifolds and the standard action of the Weyl group on the connected components of a split Cartan subgroup of a split semisimple real Lie group gives rise to the isospectral leaves of the signed Toda lattice. This clarifies the connection between the polytope in the Convexity Theorem and the topology of the compact smooth manifolds arising from the isospectral leaves of a Toda flow. Furthermore, this allows us to give a uniform treatment to two very different cases that have been studied extensively in the literature producing new cases to look at. Finally we describe the unstable manifolds of the Toda flow for these more general manifolds and determine which of these give rise to cycles.

Abstract:
It is known that a large class of integrable hydrodynamic type systems can be constructed through the Lauricella function, a generalization of the classical Gauss hypergeometric function. In this paper, we construct novel class of integrable hydrodynamic type systems which govern the dynamics of critical points of confluent Lauricella type functions defined on finite dimensional Grassmannian Gr(2,n), the set of 2xn matrices of rank two. Those confluent functions satisfy certain degenerate Euler-Poisson-Darboux equations. It is also shown that in general, hydrodynamic type system associated to the confluent Lauricella function is given by an integrable and non-diagonalizable quasi-linear system of a Jordan matrix form. The cases of Grassmannian Gr(2,5) for two component systems and Gr(2,6) for three component systems are considered in details.