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Yunbo Zeng

Publications and source records attributed to Yunbo Zeng.

At least 37 records · Page 2Linked to original sources
arXiv2007

A new multi-component CKP hierarchy

We construct a new multi-component CKP hierarchy based on the eigenfunction symmetry reduction. It contains two types of CKP equation with self-consistent sources which Lax representations are presented. Also it admits reductions to $k$-constrained CKP hierarchy and to a (1+1)-dimensional soliton hierarchy with self-consistent source, which include two types of Kaup-Kuperschmidt equation with self-consistent sources and of bi-directional Kaup-Kuperschmidt equation with self-consistent sources.

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arXiv2006

Restricted Flows and the Soliton Equation with Self-Consistent Sources

The KdV equation is used as an example to illustrate the relation between the restricted flows and the soliton equation with self-consistent sources. Inspired by the results on the Backlund transformation for the restricted flows (by V.B. Kuznetsov et al.), we constructed two types of Darboux transformations for the KdV equation with self-consistent sources (KdVES). These Darboux transformations are used to get some explicit solutions of the KdVES, which include soliton, rational, positon, and negaton solutions.

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arXiv2006

Bäcklund transformations for the constrained dispersionless hierarchies and dispersionless hierarchies with self-consistent sources

The Bäcklund transformations between the constrained dispersionless KP hierarchy (cdKPH) and the constrained dispersionless mKP hieararchy (cdmKPH) and between the dispersionless KP hieararchy with self-consistent sources (dKPHSCS) and the dispersionless mKP hieararchy with self-consistent sources (dmKPHSCS) are constructed. The auto-Bäcklund transformations for the cdmKPH and for the dmKPHSCS are also formulated.

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arXiv2006

Integrable dispersionless KdV hierarchy with sources

An integrable dispersionless KdV hierarchy with sources (dKdVHWS) is derived. Lax pair equations and bi-Hamiltonian formulation for dKdVHWS are formulated. Hodograph solution for the dispersionless KdV equation with sources (dKdVWS) is obtained via hodograph transformation. Furthermore, the dispersionless Gelfand-Dickey hierarchy with sources (dGDHWS) is presented.

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arXiv2005

The constrained dispersionless mKP hierarchy and the dispersionless mKP hierarchy with self-consistent sources

We first show that the quasiclassical limit of the squared eigenfunction symmetry constraint of the Sato operator for the mKP hierarchy leads to a reduction of the Sato function for the dispersionless mKP hierarchy. The constrained dispersionless mKP hierarchy (cdmKPH) is obtained and it is shown that the (2+1)-dimensional dispersionless mKP hierarchy is decomposed to two (1+1)-dimensional hierarchies of hydrodynamical type. The dispersionless mKP hierarchy with self-consistent sources (dmKPHSCS) together with its associated conservation equations are also constructed. Some solutions of dmKPESCS are obtained by hodograph reduction method.

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arXiv2005

Bäcklund transformations for the KP and mKP hierarchies with self-consistent sources

Using gauge transformations for the corresponding generating pseudo-differential operators $L^n$ in terms of eigenfunctions and adjoint eigenfunctions, we construct several types of auto-Bäcklund transformations for the KP hierarchy with self-consistent sources (KPHSCS) and mKP hierarchy with self-consistent sources (mKPHSCS) respectively. The Bäcklund transformations from the KPHSCS to mKPHSCS are also constructed in this way.

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arXiv2005

The quasiclassical limit of the symmetry constraint of the KP hierarchy and the dispersionless KP hierarchy with self-consistent sources

For the first time we show that the quasiclassical limit of the symmetry constraint of the KP hierarchy leads to the generalized Zakharov reduction of the dispersionless KP (dKP) hierarchy which has been proved to be result of symmetry constraint of the dKP hierarchy recently. By either regarding the constrained dKP hierarchy as its stationary case or taking the dispersionless limit of the KP hierarchy with self-consistent sources directly, we construct a new integrable dispersionless hierarchy, i.e., the dKP hierarchy with self-consistent sources and find its associated conservation equations (or equations of Hamilton-Jacobi type). Some solutions of the dKP equation with self-consistent sources are also obtained by hodograph transformations.

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arXiv2005

On the Toda Lattice Equation with Self-Consistent Sources

The Toda lattice hierarchy with self-consistent sources and their Lax representation are derived. We construct a forward Darboux transformation (FDT) with arbitrary functions of time and a generalized forward Darboux transformation (GFDT) for Toda lattice with self-consistent sources (TLSCS), which can serve as a non-auto-Backlund transformation between TLSCS with different degrees of sources. With the help of such DT, we can construct many type of solutions to TLSCS, such as rational solution, solitons, positons, negetons, and soliton-positons, soliton-negatons, positon-negatons etc., and study properties and interactions of these solutions.

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arXiv2004

A new constrained mKP hierarchy and the generalized Darboux transformation for the mKP equation with self-consistent sources

The mKP equation with self-consistent sources (mKPESCS) is treated in the framework of the constrained mKP hierarchy. We introduce a new constrained mKP hierarchy which may be viewed as the stationary hierarchy of the mKP hierarchy with self-consistent sources. This offers a natural way to obtain the Lax representation for the mKPESCS. Based on the conjugate Lax pairs, we construct the generalized Darboux transformation with arbitrary functions in time $t$ for the mKPESCS which, in contrast with the Darboux transformation for the mKP equation, provides a non-auto-Bäcklund transformation between two mKPESCSs with different degrees. The formula for $n$-times repeated generalized Darboux transformation is proposed and enables us to find the rational solutions (including the lump solutions), soliton solutions and the solutions of breather type of the mKPESCS.

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arXiv2004

The Solutions of the NLS Equations with Self-Consistent Sources

We construct the generalized Darboux transformation with arbitrary functions in time $t$ for the AKNS equation with self-consistent sources (AKNSESCS) which, in contrast with the Darboux transformation for the AKNS equation, provides a non-auto-Bäcklund transformation between two AKNSESCSs with different degrees of sources. The formula for N-times repeated generalized Darboux transformation is proposed. By reduction the generalized Darboux transformation with arbitrary functions in time $t$ for the Nonlinear Schrödinger equation with self-consistent sources (NLSESCS) is obtained and enables us to find the dark soliton, bright soliton and positon solutions for NLS$^{+}$ESCS and NLS$^{-}$ESCS. The properties of these solution are analyzed.

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arXiv2004

Generalized Darboux transformations for the KP equation with self-consistent sources

The KP equation with self-consistent sources (KPESCS) is treated in the framework of the constrained KP equation. This offers a natural way to obtain the Lax representation for the KPESCS. Based on the conjugate Lax pairs, we construct the generalized binary Darboux transformation with arbitrary functions in time $t$ for the KPESCS which, in contrast with the binary Darboux transformation of the KP equation, provides a non-auto-Bäcklund transformation between two KPESCSs with different degrees. The formula for N-times repeated generalized binary Darboux transformation is proposed and enables us to find the N-soliton solution and lump solution as well as some other solutions of the KPESCS.

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arXiv2003

Negaton and Positon solutions of the soliton equation with self-consistent sources

The KdV equation with self-consistent sources (KdVES) is used as a model to illustrate the method. A generalized binary Darboux transformation (GBDT) with an arbitrary time-dependent function for the KdVES as well as the formula for $N$-times repeated GBDT are presented. This GBDT provides non-auto-Bäcklund transformation between two KdV equations with different degrees of sources and enable us to construct more general solutions with $N$ arbitrary $t$-dependent functions. By taking the special $t$-function, we obtain multisoliton, multipositon, multinegaton, multisoliton-positon, multinegaton-positon and multisoliton-negaton solutions of KdVES. Some properties of these solutions are discussed.

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arXiv2002

Integration of the mKdV hierarchy with integral type of source

We investigate the mKdV hierarchy with integral type of source (mKdVHWS), which consist of the reduced AKNS eigenvalue problem with $r=q$ and the mKdV hierarchy with extra term of the integration of square eigenfunction. First we propose a method to find the explicit evolution equation for eigenfunction of the auxiliary linear problems of the mKdVHWS. Then we determine the evolution equations of scattering data corresponding to the mKdVHWS which allow us to solve the equation in the mKdVHWS by inverse scattering transformation.

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arXiv2002

Higher Order Potential Expansion for the Continuous Limits of the Toda Hierarchy

A method for introducing the higher order terms in the potential expansion to study the continuous limits of the Toda hierarchy is proposed in this paper. The method ensures that the higher order terms are differential polynomials of the lower ones and can be continued to be performed indefinitly. By introducing the higher order terms, the fewer equations in the Toda hierarchy are needed in the so-called recombination method to recover the KdV hierarchy. It is shown that the Lax pairs, the Poisson tensors, and the Hamiltonians of the Toda hierarchy tend towards the corresponding ones of the KdV hierarchy in continuous limit.

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arXiv2001

Bäcklund transformations for high-order constrained flows of the AKNS hierarchy: canonicity and spectrality property

New infinite number of one- and two-point Bäcklund transformations (BTs) with explicit expressions are constructed for the high-order constrained flows of the AKNS hierarchy. It is shown that these BTs are canonical transformations including Bäcklund parameter $η$ and a spectrality property holds with respect to $η$ and the 'conjugated' variable $μ$ for which the point $(η, μ)$ belongs to the spectral curve. Also the formulas of m-times repeated Darboux transformations for the high-order constrained flows of the AKNS hierarchy are presented.

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arXiv2001

Canonical explicit Bäcklund transformations with spectrality for constrained flows of soliton hierarchies

It is shown that explicit Bäcklund transformations (BTs) for the high-order constrained flows of soliton hierarchy can be constructed via their Darboux transformations and Lax representation, and these BTs are canonical transformations including Bäcklund parameter $η$ and possess a spectrality property with respect to $η$ and the 'conjugated' variable $μ$ for which the pair $(η, μ)$ lies on the spectral curve. As model we present the canonical explicit BTs with the spectrality for high-order constrained flows of the Kaup-Newell hierarchy and the KdV hierarchy.

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arXiv2001

Binary Constrained Flows and Separation of Variables for Soliton Equations

In contrast to mono-constrained flows with N degrees of freedom, binary constrained flows of soliton equations, admitting 2x2 Lax matrices, have 2N degrees of freedom. By means of the existing method, Lax matrices only yield the first N pairs of canonical separated variables. An approach for constructing the second N pairs of canonical separated variables with additional N separated equations is introduced. The Jacobi inversion problems for binary constrained flows are then established. Finally, the separability of binary constrained flows together with the factorization of soliton equations by the spatial and temporal binary constrained flows leads to the Jacobi inversion problems for soliton equations.

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