The main objectives of this work are to deduce the Lax pair for Modified Toda lattice hierarchy and find whether there exists a transformation between (2+1)-dimensional derivative equation equation. By new spectral problem, we obtain hierarchy, furthermore give Hirota’s form solution equation, thus can also be attained correspondingly.

The five integrable vortex equations, recently studied by Manton, are generalized to include magnetic impurities of the Tong–Wong type. Under certain conditions, these generalizations remain integrable. We further set up a gauge theory with product group, two complex scalar fields and general charge matrix. second species vortices, when frozen, interpreted as impurity for all equations. then gi...

For a family of Poisson algebras, parametrized by r ∈ Z, and an associated Lie algebraic splitting, we consider the factorization of given canonical transformations. In this context we rederive the recently found r-th dispersionless modified KP hierachies and r-th dispersionless Dym hierarchies, giving a new Miura map among them. We also found a new integrable hierarchy which we call the r-th d...

We consider the C 2 Toda theory in the reduced WZNW framework. Analysing the classical representation space of the symmetry algebra (which is the corresponding C 2 W algebra) we determine its classical highest weight representations. We quantise the model promoting only the relevant quantities to operators. Using the quantised equation of motion we determine the selection rules for the u field ...

We prove that finite Morse index solutions to the Allen-Cahn equation in R2 have finitely many ends and linear energy growth. The main tool is a curvature decay estimate on level sets of these finite Morse index solutions, which in turn is reduced to a problem on the uniform second order regularity of clustering interfaces for the singularly perturbed Allen-Cahn equation in Rn. Using an indirec...

The non-perturbative corrections to the universal hypermultiplet moduli space metric in the type-IIA superstring compactification on a Calabi-Yau threefold are investigated in the presence of 4d, N=2 supergravity. These corrections come from multiple wrapping of the BPS (Euclidean) D2-branes around certain (BPS) Calabi-Yau 3-cycles, and they are known as the D-instantons. The exact universal hy...

For a family of Poisson algebras, parametrized by r ∈ Z, and an associated Lie algebraic splitting, we consider the factorization of given canonical transformations. In this context we rederive the recently found r-th dispersionless modified KP hierachies and r-th dispersionless Dym hierarchies, giving a new Miura map among them. We also found a new integrable hierarchy which we call the r-th d...

By using the relativistic top theory, we derive a relativistic top deviation equation. This equation turns out to be a generalization of the geodesic deviation equation for a pair of nearby point particles. In fact, we show that when the spin angular momentum tensor associated to the top vanishes, such a relativistic top deviation equation reduces to the geodesic deviation equation for spinless...

We develop the approach to the problem of integrable discretization based on the notion of r–matrix hierarchies. One of its basic features is the coincidence of Lax matrices of discretized systems with the Lax matrices of the underlying continuous time systems. A common feature of the discretizations obtained in this approach is non–locality. We demonstrate how to overcome this drawback. Namely...

By using the relativistic top theory, we derive a relativistic top deviation equation. This equation turns out to be a generalization of the geodesic deviation equation for a pair of nearby point particles. In fact, we show that when the spin angular momentum tensor associated to the top vanishes, such a relativistic top deviation equation reduces to the geodesic deviation equation for spinless...