相关论文: The algebro-geometric solutions for Degasperis-Pro…
The quaternionic KP hierarchy is the integrable hierarchy of p.d.e obtained by replacing the complex numbers with the quaternions, mutatis mutandis, in the standard construction of the KP hierarchy equations and solutions; it is equivalent…
We provide a detailed derivation of all complex-valued algebro-geometric finite-band solutions of the Ablowitz-Ladik hierarchy. In addition, we survey a recursive construction of the Ablowitz-Ladik hierarchy and its zero-curvature and Lax…
In this paper we show some new exact solutions for the generalized modified Degasperis$-$Procesi equation (mDP equation)
The class of differential equations describing pseudospherical surfaces enjoys important integrability properties which manifest themselves by the existence of infinite hierarchies of conservation laws (both local and non-local) and the…
In this paper we discuss an integrable hierarchy of compatible Lax equations that is obtained by a wider deformation of a commutative algebra in the loop space of ${\rm sl}_{2}$ than that in the AKNS-case and whose Lax equations are based…
We derive the Lax operator for a very large family of classical minimal surface solutions in $AdS_3$ describing Wilson loops in $\mathcal{N}=4$ SYM theory. These solutions, constructed by Ishizeki, Kruczenski and Ziama, are associated with…
Hessian operators arising in inverse problems governed by partial differential equations (PDEs) play a critical role in delivering efficient, dimension-independent convergence for both Newton solution of deterministic inverse problems, as…
We give the solution to the complete noncommutative Kadomtsev--Petviashvili (KP) hierarchy. We achieve this via direct linearisation which involves the Gelfand--Levitan--Marchenko (GLM) equation. This is a linear integral equation in which…
We have constructed series of the spectral parameter dependent solutions to the Yang-Baxter equations defined on the tensor product of reducible representations with symmetry of quantum algebra. These series are produced as descendant…
This thesis explores algorithmic applications and limitations of convex relaxation hierarchies for approximating some discrete and continuous optimization problems. - We show a dichotomy of approximability of constraint satisfaction…
This paper deals with the category of nonlinear evolution equations (NLEEs) associated with the spectral problem and provides an approach for constructing their algebraic structure and $r$-matrix. First we introduce the category of NLEEs,…
We construct integrals of motion for multidimensional classical systems from ladder operators of one-dimensional systems. This method can be used to obtain new systems with higher order integrals. We show how these integrals generate a…
We put the Adler-Gelfand-Dickey approach to classical W-algebras in the framework of Poisson vertex algebras. We show how to recover the bi-Poisson structure of the KP hierarchy, together with its generalizations and reduction to the N-th…
The discrete Lax operators with the spectral parameter on an algebraic curve are defined. A hierarchy of commuting flows on the space of such operators is constructed. It is shown that these flows are linearized by the spectral transform…
Additional non-isospectral symmetries are formulated for the constrained Kadomtsev-Petviashvili (\cKP) integrable hierarchies. The problem of compatibility of additional symmetries with the underlying constraints is solved explicitly for…
In the application of the Deift-Zhou steepest descent method to the Riemann-Hilbert problem for orthogonal polynomials, a model Riemann-Hilbert problem that appears in the multi-cut case is solved with the use of hyperelliptic theta…
We construct a convergent family of outer approximations for the problem of optimizing polynomial functions over convex bodies subject to polynomial constraints. This is achieved by generalizing the polarization hierarchy, which has…
We define and study a combinatorial Hopf algebra dRec with basis elements indexed by diagonal rectangulations of a square. This Hopf algebra provides an intrinsic combinatorial realization of the Hopf algebra tBax of twisted Baxter…
We constructed the three nonequivalent gradings in the algebra $D_4 \simeq so(8)$. The first one is the standard one obtained with the Coxeter automorphism $C_1=S_{\alpha_2} S_{\alpha_1}S_{\alpha_3}S_{\alpha_4}$ using its dihedral…
A completely integrable nonlinear partial differential equation (PDE) can be associated with a system of linear PDEs in an auxiliary function whose compatibility requires that the original PDE is satisfied. This associated system is called…