Baxter <Emphasis Type="Italic">Q</Emphasis>-operators for the integrable discrete self-trapping chain

For the integrable discrete self-trapping chain, we construct Baxter Q-operators as the traces of the monodromy of certain M-operators that act in the quantum and auxiliary spaces. With this procedure, we obtain two basic M-operators and derive some functional relations between them such as intertwining relations and Wronskian-type relations between two basic Q-operators.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 2, pp. 310–321, February, 2005.     

Growth theorem of convex mappings on bounded convex circular domains

The growth theorem and the 1/2-covering theorem are obtained for the class of normalized biholomorphic convex mappings on bounded convex circular domains, which extend the corresponding results of Sufridge, Thomas, Liu, Gong, Yu, and Wang. The approach is new, which does not appeal to the automorphisms of the domains; and the domains discussed are rather general on which convex mappings can be studied, since the domain may not have a convex mapping if it is not convex. Project supported by the National Natural Science Foundation of China and the State Education Commission Doctoral Foundation.     

Cauchy problem for integrable discrete equations on quad-graphs

Initial value problems for the integrable discrete equations on quad-graphs are investigated. We give a geometric criterion of when such a problem is well-posed. In the basic example of the discrete KdV equation an effective integration scheme based on the matrix factorization problem is proposed and the interaction of the solutions with the localized defects in the regular square lattice are discussed in details. The examples of kinks and solitons on various quad-graphs, including quasiperiodic tilings, are presented.Dedicated to S. P. Novikov on his 65 birthdayOn leave from Landau Institute for Theoretical Physics, Chernogolovka, Russia.     

A generalization of the coupled integrable dispersionless equations

The paper investigates an extension of the coupled integrable dispersionless equations, which describe the current‐fed string within an external magnetic field. By using the relation among the coupled integrable dispersionless equations, the sine‐Gordon equation and the two‐dimensional Toda lattice equation, we propose a generalized coupled integrable dispersionless system. N‐soliton solutions to the generalized system are presented in the Casorati determinant form with arbitrary parameters. By choosing real or complex parameters in the Casorati determinant, the properties of one‐soliton and two‐soliton solutions are investigated. It is shown that we can obtain solutions in soliton profile and breather profile. Copyright © 2016 John Wiley & Sons, Ltd.     

Explicit solutions to a coupled integrable lattice equation

A coupled integrable lattice equation is derived from a 4 × 4 matrix spectral problem, then with the help of a special Darboux matrix, explicit solutions of the aforementioned equation are given by means of gauge transformation between the Lax pair. Finally, the density profiles of these exact solutions are presented to illustrate these solutions.     

Random mappings with exchangeable in‐degrees

In this article we introduce a new random mapping model, , which maps the set {1,2,…,n} into itself. The random mapping is constructed using a collection of exchangeable random variables which satisfy . In the random digraph, , which represents the mapping , the in‐degree sequence for the vertices is given by the variables , and, in some sense, can be viewed as an analogue of the general independent degree models from random graph theory. We show that the distribution of the number of cyclic points, the number of components, and the size of a typical component can be expressed in terms of expectations of various functions of . We also consider two special examples of which correspond to random mappings with preferential and anti‐preferential attachment, respectively, and determine, for these examples, exact and asymptotic distributions for the statistics mentioned above. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Strongly nonlinear modal equations for nearly integrable PDEs

Summary The purpose of this paper is the derivation of reduced, finite-dimensional dynamical systems that govern the near-integrable modulations ofN-phase, spatially periodic, integrable wavetrains. The small parameter in this perturbation theory is the size of the nonintegrable perturbation in the equation, rather than the amplitude of the solution, which is arbitrary. Therefore, these reduced equations locally approximate strongly nonlinear behavior of the nearly integrable PDE. The derivation we present relies heavily on the integrability of the underlying PDE and applies, in general, to anyN-phase periodic wavetrain. For specific applications, however, a numerical pretest is applied to fix the truncation orderN. We present one example of the reduction philosophy with the damped, driven sine-Gordon system and summarize our present progress toward application of the modulation equations to this numerical study.     

Refined functional equations stemming from cubic,quadratic and additive mappings

Let n ≥ 2 be an integer. In this paper, we investigate the generalized Hyers-Ulam stability problem for the following functional equation f（n-1∑j=1 xj＋2xn）＋f（n-1∑j=1 xj-2xn）＋8 n-1∑j=1f（xj）=2f（n-1∑j=1 xj） ＋4 n-1∑j=1[f（xj＋xn）＋f（xj-xn）] which contains as solutions cubic, quadratic or additive mappings.     

Coexistence of algebraic and nonalgebraic limit cycles in Kukles systems

A class of Kukles differential systems of degree five having an invariant conic is examined. We show the coexistence of small amplitude limit cycles, large amplitude limit cycles, under perturbations of the coefficients of the systems. Financed partially by: USM Grant No. 12.06.28 and 12.06.27.     

一类代数系统及其应用

本文利用已有的loop代数$\widetilde{A}_{1}$构造出代数系统$X$,然后建立了一个新的等谱问题得到著名的Volterra lattice可积系,最后通过构造出的$X$的扩展代数系统$\widetilde{X}$得到已有的可积系的可积耦合系统.     

Discontinuous wave equations and a topological degree for some classes of multi-valued mappings

The Leray-Schauder degree is extended to certain multi-valued mappings on separable Hilbert spaces with applications to the existence of weak periodic solutions of discontinuous semilinear wave equations with fixed ends.     

Integrable discretization and numerical simulations of the generalized coupled integrable dispersionless equations

In this paper, we study the generalized coupled integrable dispersionless (GCID) equations and construct two integrable discrete analogues including a semi-discrete system and a full-discrete one. The results are based on the relations among the GCID equations, the sine-Gordon equation and the two-dimensional Toda lattice equation. We also present the N-soliton solutions to the semi-discrete and fully discrete systems in the form of Casorati determinant. In the continuous limit, we show that the fully discrete GCID equations converge to the semi-discrete GCID equations, then further to the continuous GCID equations. By using the integrable semi-discrete system, we design two numerical schemes to the GCID equations and carry out several numerical experiments with solitons and breather solutions.     

Classification of integrable Vlasov-type equations

The classification of integrable Vlasov-type equations reduces to a functional equation for a generating function. We find a general solution of this functional equation in terms of hypergeometric functions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 2, pp. 249–260, February, 2008.     

Generalization of preinvex and B-vex fuzzy mappings

A class of fuzzy mappings, called B-preinvex, and strictly B-preinvex fuzzy mappings, is introduced by relaxing the definition of preinvexity of a fuzzy mapping. Similarly, based on a notion of differentiability of fuzzy mappings different from the usual one, the class of pseudo-B-vex, B-invex, and pseudo-B-invex fuzzy mappings is defined as a generalization of pseudo-convex, invex, and pseudo-invex fuzzy mappings. We prove that a strictly B-vex, or B-preinvex fuzzy mapping has at most one global minimum point, and that the class of B-vex fuzzy mappings forms a subset of the class of quasi-convex fuzzy mappings. B-vex (resp. B-preinvex) fuzzy mappings satisfy most of the basic properties of convex (resp. preinvex) fuzzy mappings. In addition characterizations and sufficient optimality conditions are obtained for B-vex and B-preinvex fuzzy mappings.     

A family of Liouville integrable lattice equations and its conservation laws

A new discrete spectral problem of matrix three-by-three with three potentials is introduced and by using the discrete trace identity a hierarchy of Liouville lattice equations is obtained. The infinite number of conservation laws of the hierarchy of Hamiltonian system are presented.     

On an integrable case of Kozlov-Treshchev Birkhoff integrable potentials

We establish, using a new approach, the integrability of a particular case in the Kozlov-Treshchev classification of Birkhoff integrable Hamiltonian systems. The technique used is a modification of the so called quadratic Lax pair for D _n Toda lattice combined with a method used by M. Ranada in proving the integrability of the Sklyanin case.      

The convenient setting for quasianalytic Denjoy-Carleman differentiable mappings

For quasianalytic Denjoy-Carleman differentiable function classes CQ where the weight sequence Q=(Qk) is log-convex, stable under derivations, of moderate growth and also an L-intersection (see (1.6)), we prove the following: The category of CQ-mappings is cartesian closed in the sense that CQ(E,CQ(F,G))≅CQ(E×F,G) for convenient vector spaces. Applications to manifolds of mappings are given: The group of CQ-diffeomorphisms is a regular CQ-Lie group but not better.     

Convergence and finite determination of formal CR mappings

It is shown that a formal mapping between two real-analytic hypersurfaces in complex space is convergent provided that neither hypersurface contains a nontrivial holomorphic variety. For higher codimensional generic submanifolds, convergence is proved e.g. under the assumption that the source is of finite type, the target does not contain a nontrivial holomorphic variety, and the mapping is finite. Finite determination (by jets of a predetermined order) of formal mappings between smooth generic submanifolds is also established.

     

On the uniqueness problem of harmonic quasiconformal mappings

In this paper, we give an affirmative answer to Sheretov's problem on the uniqueness of harmonic mappings and improve the unique minimal mapping theorem of Reich and Strebel. Meanwhile, we also solve a problem posed by Reich and obtain the uniqueness theorem on related weight functions.