arXiv subjects

Chuanzhong Li

Publications and source records attributed to Chuanzhong Li.

At least 37 records · Page 2Linked to original sources

arXiv2019

A strongly coupled extended Toda hierarchy and its Virasoro symmetry

As a generalization of the integrable extended Toda hierarchy and a reduction of the extended multicomponent Toda hierarchy, from the point of a commutative subalgebra of $gl(2,\mathbb{C})$, we construct a strongly coupled extended Toda hierarchy(SCETH) which will be proved to possess a Virasoro type additional symmetry by acting on its tau-function. Further we give the multi-fold Darboux transformations of the strongly coupled extended Toda hierarchy.

nlin.SI↗

arXiv2019

Quantum torus symmetries of multicomponent modified KP hierarchy and reductions

In this paper, we construct the multicomponent modified KP hierarchy and its additional symmetries. The additional symmetries constitute an interesting multi-folds quantum torus type Lie algebra. By a reduction, we also construct the constrained multicomponent modified KP hierarchy and its Virasoro type additional symmetries.

nlin.SI↗

arXiv2019

Symmetries and reductions on the noncommutative Kadomtsev-Petviashvili and Gelfand-Dickey hierarchies

In this paper, we construct the additional flows of the noncommutative Kadomtsev-Petviashvili(KP) hierarchy and the additional symmetry flows constitute an infinite dimensional Lie algebra $W_{1+\infty}$. In addition, the generating function of the additional symmetries can also be proved to have a nice form in terms of wave functions and this generating symmetry is used to construct the noncommutative KP hierarchy with self-consistent sources and the constrained noncommutative KP hierarchy. The above results will be further generalized to the noncommutative Gelfand-Dickey hierarchies which contains many interesting noncommutative integrable systems such as the noncommutative KdV hierarchy and noncommutative Boussinesq hierarchy. Meanwhile, we construct two new noncommutative systems including odd noncommutative C type Gelfand-Dickey and even noncommutative C type Gelfand-Dickey hierarchies. Also using the symmetry, we can construct a new noncommutative Gelfand-Dickey hierarchy with self-consistent sources. Basing on the natural differential Lax operator of the noncommutative Gelfand-Dickey hierarchy, the string equations of the noncommutative Gelfand-Dickey hierarchy are also derived.

nlin.SI↗

arXiv2017

Coupled Hall-Littlewood functions, vertex operators and the $q$-boson model

In this paper, we firstly give the definition of the coupled Hall-Littlewood function and its realization in terms of vertex operators. Then we construct the representation of the two-site generalized $q$-boson model in the algebra of coupled Hall-Littlewood functions. Finally, we find that the vertex operators which generate coupled Hall-Littlewood functions can also be used to obtain the partition function of the A-model topological string on the conifold.

math-ph↗

arXiv2017

Universal Character, Phase Model and Topological Strings on $\mathbb C^3$

In this paper, we consider two different subjects: the algebra of universal characters $S_{[λ,μ]}({\bf x},{\bf y})$ (a generalization of Schur functions) and the phase model of strongly correlated bosons. We find that the two-site generalized phase model can be realized in the algebra of universal characters, and the entries in the monodromy matrix of the phase model can be represented by the vertex operators $Γ_i^\pm(z) (i=1,2)$ which generate universal characters. Meanwhile, we find that these vertex operators can also be used to obtain the A-model topological string partition function on $\mathbb C^3$.

math-ph↗

arXiv2017

Rogue waves of the Frobenius nonlinear Schrödinger equation

In this paper, by considering the potential application in two mode nonlinear waves in nonlinear fibers under a certain case, we define a coupled nonlinear Schrödinger equation(called Frobenius NLS equation) including its Lax pair. Afterwards, we construct the Darboux transformations of the Frobenius NLS equation. New solutions can be obtained from known seed solutions by the Dardoux transformations. Soliton solutions are generated from trivial seed solutions. Also we derive breather solutions $q,r$ of the Frobenius NLS equation obtained from periodic seed solutions. Interesting enough, we find the amplitudes of $r$ vary in size in different areas with period-like fluctuations in the background. This is very different from the solution $q$ of the single-component classical nonlinear Schrödinger equation. Then, the rogue waves of the Frobenius NLS equation are given explicitly by a Taylor series expansion about the breather solutions $q,r$. Also the graph of rogue wave solution $r$ shows that the rogue wave has fluctuations around the peak. The reason of this phenomenon should be in the dynamical dependence of $r$ on $q$ which is independent on $r$.

nlin.SI↗

arXiv2017

Quantum Torus symmetries of the CKP and multi-component CKP hierarchies

In this paper, we construct a series of additional flows of the CKP hierarchy and the multi-component CKP hierarchy and these flows constitute a N-folds direct product of the positive half of the quantum Torus symmetry. Comparing to the $W_{\infty}$ infinite dimensional Lie symmetry, this quantum Torus symmetry has a nice algebraic structure with double indices.

nlin.SI↗

arXiv2017

Quantum torus algebras and B(C) type Toda systems

In this paper, we construct a new even constrained B(C) type Toda hierarchy and derive its B(C) type Block type additional symmetry. Also we generalize the B(C) type Toda hierarchy to the $N$-component B(C) type Toda hierarchy which is proved to have symmetries of a coupled $\bigotimes^NQT_+ $ algebra ( $N$-folds direct product of the positive half of the quantum torus algebra $QT$).

nlin.SI↗

arXiv2017

Bäcklund transformations of $Z_n$-Sine-Gordon systems

In this paper, from the algebraic reductions from the Lie algebra $gl(n,\mathbb C)$ to its commutative subalgebra $Z_n$, we construct the general $Z_n$-Sine-Gordon and $Z_n$-Sinh-Gordon systems which contain many multi-component Sine-Gordon type and Sinh-Gordon type equations. Meanwhile, we give the Bäcklund transformations of the $Z_n$-Sine-Gordon and $Z_n$-Sinh-Gordon equations which can generate new solutions from seed solutions. To see the $Z_n$-systems clearly, we consider the $Z_2$-Sine-Gordon and $Z_3$-Sine-Gordon equations explicitly including their Bäcklund transformations, the nonlinear superposition formula and Lax pairs.

nlin.SI↗

arXiv2017

Dispersionless and multicomponent BKP hierarchies with quantum torus symmetries

In this article, we will construct the additional perturbative quantum torus symmetry of the dispersionless BKP hierarchy basing on the $W_{\infty}$ infinite dimensional Lie symmetry. These results show that the complete quantum torus symmetry is broken from the BKP hierarchy to its dispersionless hierarchy. Further a series of additional flows of the multicomponent BKP hierarchy will be defined and these flows constitute an $N$-folds direct product of the positive half of the quantum torus symmetries.

nlin.SI↗

arXiv2017

Bosonic symmetries of the extended fermionic $(2N,2M)$-Toda hierarchy

In this paper, we construct the additional symmetries of the fermionic $(2N,2M)$-Toda hierarchy basing on the generalization of the $N{=}(1|1)$ supersymmetric two dimensional Toda lattice hierarchy. These additional flows constitute a $w_{\infty}\times w_{\infty}$ Lie algebra. As a Bosonic reduction of the $N{=}(1|1)$ supersymmetric two dimensional Toda lattice hierarchy and the fermionic $(2N,2M)$-Toda hierarchy, we define a new extended fermionic $(2N,2M)$-Toda hierarchy which admits a Bosonic Block type superconformal structure.

nlin.SI↗

arXiv2016

The extended $Z_N$-Toda hierarchy

The extended flow equations of a new $Z_N$-Toda hierarchy which takes values in a commutative subalgebra $Z_N$ of $gl(N,\mathbb C)$ is constructed. Meanwhile we give the Hirota bilinear equations and tau function of this new extended $Z_N$-Toda hierarchy(EZTH). Because of logarithm terms, some extended Vertex operators are constructed in generalized Hirota bilinear equations which might be useful in topological field theory and Gromov-Witten theory. Meanwhile the Darboux transformation and bi-hamiltonian structure of this hierarchy are given. From hamiltonian tau symmetry, we give another different tau function of this hierarchy with some unknown mysterious connections with the one defined from the point of Sato theory.

nlin.SI↗

arXiv2016

Virasoro symmetry of the constrained multi-component KP hierarchy and its integrable discretization

In this paper, we construct the Virasoro type additional symmetries of a kind of constrained multi-component KP hierarchy and give the Virasoro flow equation on eigenfunctions and adjoint eigenfunctions. It can also be seen that the algebraic structure of the Virasoro symmetry is kept after discretization from the constrained multi-component KP hierarchy to the discrete constrained multi-component KP hierarchy.

nlin.SI↗

arXiv2016

Addition formulae of discrete KP, q-KP and two-component BKP systems

In this paper, we constructed the addition formulae for several integrable hierarchies, including the discrete KP, the q-deformed KP, the two-component BKP and the D type Drinfeld-Sokolov hierarchies. With the help of the Hirota bilinear equations and $τ$ functions of different kinds of KP hierarchies, we prove that these addition formulae are equivalent to these hierarchies. These studies show that the addition formula in the research of the integrable systems has good universality.

nlin.SI↗

arXiv2016

Gauge transformation and symmetries of the commutative multi-component BKP hierarchy

In this paper, we defined a new multi-component BKP hierarchy which takes values in a commutative subalgebra of $gl(N,\mathbb C)$. After this, we give the gauge transformation of this commutative multi-component BKP (CMBKP) hierarchy. Meanwhile we construct a new constrained CMBKP hierarchy which contains some new integrable systems including coupled KdV equations under a certain reduction. After this, the quantum torus symmetry and quantum torus constraint on the tau function of the commutative multi-component BKP hierarchy will be constructed.

nlin.SI↗

arXiv2016

Constrained lattice-field hierarchies and Toda system with Block symmetry

In this paper, we construct the additional $W$-symmetry and ghost symmetry of two-lattice field integrable hierarchies. Using the symmetry constraint, we construct constrained two-lattice integrable systems which contain several new integrable difference equations. Under a further reduction, the constrained two-lattice integrable systems can be combined into one single integrable system, namely the well-known one dimensional original Toda hierarchy. We prove that the one dimensional original Toda hierarchy has a nice Block Lie symmetry.

nlin.SI↗

arXiv2016

Multi-fold Darboux transformations of the extended bigraded Toda hierarchy

With the extended logarithmic flow equations and some extended Vertex operators in generalized Hirota bilinear equations, extended bigraded Toda hierarchy(EBTH) was proved to govern the Gromov-Witten theory of orbiford $c_{NM}$ in literature. The generating function of these Gromov-Witten invariants is one special solution of the EBTH. In this paper, the multi-fold Darboux transformations and their determinant representations of the EBTH are given with two different gauge transformation operators. The two Darboux transformations in different directions are used to generate new solutions from known solutions which include soliton solutions of $(N,N)$-EBTH, i.e. the EBTH when $N=M$. From the generation of new solutions, one can find the big difference between the EBTH and the extended Toda hierarchy(ETH). Meanwhile we plotted the soliton graphs of the $(N,N)$-EBTH from which some approximation analysis will be given. From the analysis on velocities of soliton solutions, the difference between the extended flows and other flows are shown. The two different Darboux transformations constructed by us might be useful in Gromov-Witten theory of orbiford $c_{NM}$.

nlin.SI↗

arXiv2015

Rogue waves in a resonant erbium-doped fiber system with higher-order effects

We mainly investigate a coupled system of the generalized nonlinear Schrödinger equation and the Maxwell-Bloch equations which describes the wave propagation in an erbium-doped nonlinear fiber with higher-order effects including the forth-order dispersion and quintic non-Kerr nonlinearity. We derive the one-fold Darbox transformation of this system and construct the determinant representation of the $n$-fold Darboux transformation. Then the determinant representation of the $n$th new solutions $(E^{[n]},\, p^{[n]},\, η^{[n]})$ which were generated from the known seed solutions $(E, \, p, \, η)$ is established through the $n$-fold Darboux transformation. The solutions $(E^{[n]},\, p^{[n]},\, η^{[n]})$ provide the bright and dark breather solutions of this system. Furthermore, we construct the determinant representation of the $n$th-order bright and dark rogue waves by Taylor expansions and also discuss the hybrid solutions which are the nonlinear superposition of the rogue wave and breather solutions.

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