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Kelei Tian

Publications and source records attributed to Kelei Tian.

17 recordsLinked to original sources
arXiv2025

On a matrix constrained CKP hierarchy

The algebraic structures of integrable hierarchies play an important role in the study of soliton equations. In this paper, we use splitting theory to give a matrix representation of a constrained CKP hierarchy, which can be considered as a generalization of the $\hat{A}_{2n}^{(2)}$-KdV hierarchy and the constrained KP hierarchy. An equivalent construction in terms of the pseudo-differential operator is discussed. Darboux transformations, scaling transformation and tau functions $\ln τ_f$ for this constrained hierarchy are studied. Moreover, we present formulas for the Virasoro vector fields on $\ln τ_f$ for the $\hat{A}_{2 n}^{(2)}$-KdV hierarchy.

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arXiv2025

The (3+1)-dimensional dispersionless integrable hierarchy and nonlinear Riemann-Hilbert problem associated with the Doubrov-Ferapontov modified heavenly equation

According to the classification of integrable complex Monge-Ampere equations by Doubrov and Ferapontov, the modified heavenly equation is a typical (3+1)-dimensional dispersionless and canonical integrable equation.In this paper we use the eigenfunctions of the Doubrov-Ferapontov modified heavenly equation to obtain a related hierarchy. Next we construct the Lax-Sato equations with Hamiltonian vector fields and Zakharov-Shabat type equations which are equivalent to the hierarchy. The nonlinear Riemann-Hilbert problem is also applied to study the solution of Doubrov-Ferapontov modified heavenly equation.

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arXiv2025

On one typical Einstein-Weyl equation: inverse spectral transform for theCauchy problem, longtime behaviour of the solutions and implicit solutions

In this paper, we study one typical Einstein-Weyl equation. It arises from Ferapontov and Kruglikov's investigation on the integrability of several dispersionless partial differential equations and the geometry of their formal linearizations. First, by using Manakov-Santini IST (inverse spectral transform) method, we investigate its Cauchy problem including the direct problem from the initial data, the time evolution of the scattering and spectral data, and the inverse problem. Second, based on the nonlinear RH (Riemann-Hilbert) dressing, the longtime behaviour of the solutions is constructed. In addition, some implicit solutions are presented.

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arXiv2024

The integrable hierarchy and the nonlinear Riemann-Hilbert problem associated with one typical Einstein-Weyl physico-geometric dispersionless system

From a specific series of exchange conditions for a one-parameter Hamiltonian vector field, we establish an integrable hierarchy using Lax pairs derived from the dispersionless partial differential equation. An exterior differential form of the integrable hierarchy is introduced, further confirming the existence of the tau function. Subsequently, we present the twistor structure of the hierarchy. By constructing the nonlinear Riemann Hilbert problem for the equation, the structure of the solution to the equation is better understood.

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arXiv2024

On the c-k constrained KP and BKP hierarchies: the Fermionic pictures, solutions and additional symmetries

In this paper, we study two generalized constrained integrable hierarchies, which are called the $c$-$k$ constrained KP and BKP hierarchies. The Fermionic picture of the $c$-$k$ constrained KP hierarchy is given. We give some solutions for the $c$-$k$ constrained KP hierarchy by using the free Fermion operators and define its additional symmetries. Its additional flows form a subalgebra of the Virasoro algebra. Furthermore, the additional flows acting on eigenfunctions $q_{i}(t)$ and adjoint eigenfunctions $r_{i}(t)$ of the $c$-$k$ constrained KP hierarchy are presented. Next, we define the $c$-$k$ constrained BKP hierarchy and obtain its bilinear identity and solutions. The algebra formed by the additional symmetric flow of the $c$-$k$ constrained BKP hierarchy that we defined is still a subalgebra of the Virasoro algebra and it is a subalgebra of the algebra formed by the additional flows of the $c$-$k$ constrained KP hierarchy.

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arXiv2015

Ghost symmetry of the discrete KP hierarchy

In this paper, with the help of the $S$ function and ghost symmetry for the discrete KP hierarchy which is a semi-discrete version of the KP hierarchy, the ghost flow on its eigenfunction(adjoint eigenfunction) and the spectral representation of its Baker-Akhiezer function and adjoint Baker-Akhiezer function are derived. From these observations above, some important distinctions between the discrete KP hierarchy and KP hierarchy are shown. Also we give the ghost flow on the tau function and another kind of proof of the ASvM formula of the discrete KP hierarchy.

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arXiv2014

Symmetric $q$-deformed KP hierarch

Based on the analytic property of the symmetric $q$-exponent $e_q(x)$, a new symmetric $q$-deformed Kadomtsev-Petviashvili ($q$-KP) hierarchy associated with the symmetric $q$-derivative operator $\partial_q$ is constructed. Furthermore, the symmetric $q$-CKP hierarchy and symmetric $q$-BKP hierarchy are defined. Here we also investigate the additional symmetries of the symmetric $q$-KP hierarchy.

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arXiv2011

The Additional Symmetries for the BTL and CTL Hierarchies

The Toda lattice (TL) hierarchy was first introduced by K.Ueno and K.Takasaki in \cite{uenotaksasai} to generalize the Toda lattice equations\cite{toda}. Along the work of E. Date, M. Jimbo, M. Kashiwara and T. Miwa \cite{DJKM} on the KP hierarchy, K.Ueno and K.Takasaki in \cite{uenotaksasai} develop the theory for the TL hierarchy: its algebraic structure, the linearization, the bilinear identity, $τ$ function and so on. Also the analogues of the B and C types for the TL hierarchy, i.e. the BTL and CTL hierarchies, are considered in \cite{uenotaksasai}, which are corresponding to infinite dimensional Lie algebras $\textmd{o}(\infty)$ and $\textmd{sp}(\infty)$ respectively. In this paper, we will focus on the study of the additional symmetries for the BTL and CTL hierarchies.

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arXiv2010

Additional symmetries of constrained CKP and BKP hierarchies

The additional symmetries of the constrained CKP (cCKP) and BKP (cBKP) hierarchies are given by their actions on the Lax operators, and their actions on the eigenfunction and adjoint eigenfunction $\{Φ_i,Ψ_i \}$ are presented explicitly. Furthermore, we show that acting on the space of the wave operator, $\partial_k^*$ forms new centerless $W^{cC}_{1+\infty}$ and $W^{cB}_{1+\infty}$-subalgebra of centerless $W_{1+\infty}$ respectively. In order to define above symmetry flows $\partial_k^*$ of the cCKP and cBKP hierarchies, two vital operators $Y_k$ are introduced to revise the additional symmetry flows of the CKP and BKP hierarchies.

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arXiv2009

String Equations of the q-KP Hierarchy

Based on the Lax operator $L$ and Orlov-Shulman's $M$ operator, the string equations of the $q$-KP hierarchy are established from special additional symmetry flows, and the negative Virasoro constraint generators \{$L_{-n}, n\geq1$\} of the $2-$reduced $q$-KP hierarchy are also obtained.

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arXiv2008

Negative Generators of the Virasoro Constraints for the BKP Hierarchy

We give a straightforward derivation of the string equation and Virasoro constraints on the $τ$ function of the BKP hierarchy by means of some special additional symmetry flows. The explicit forms of the actions of these additional symmetry flows on the wave function and then the negative Virasoro generators $L_{-k}$ are given, where $k$ is a positive integer.

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arXiv2007

Additional Symmetry of CKP hierarchy

Based on the Orlov and Shulman's M operator, the additional symmetries and the string equation of the CKP hierarchy are established, and then the higher order constraints on $L^l$ are obtained. In addition, the generating function and some properties are also given. In particular, the additional symmetry flows form a new infinite dimensional algebra $W^C_{1+\infty}$, which is a subalgebra of $W_{1+\infty}$.

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