Additional symmetry of the modified extended Toda hierarchy

In this paper, one new integrable modified extended Toda hierarchy(METH) is constructed with the help of two logarithmic Lax operators. With this modification, the interpolated spatial flow is added to make all flows complete. To show more integrable properties of the METH, the bi-Hamiltonian structure and tau symmetry of the METH will be given. The additional symmetry flows of this new hierarchy are presented. These flows form an infinite dimensional Lie algebra of Block type.Comment: 13 page

Darboux transformation and positons of the inhomogeneous Hirota and the Maxwell-Bloch equation

In this paper, we derive Darboux transformation of the inhomogeneous Hirota and the Maxwell-Bloch(IH-MB) equations which is governed by femtosecond pulse propagation through inhomogeneous doped fibre. The determinant representation of Darboux transformation is used to derive soliton solutions, positon solutions of the IH-MB equations.Comment: accepted by SCIENCE CHINA Physics, Mechanics & Astronomy. arXiv admin note: substantial text overlap with arXiv:1205.119

The wronskian solution of the constrained discrete KP hierarchy

From the constrained discrete KP (cdKP) hierarchy, the Ablowitz-Ladik lattice has been derived. By means of the gauge transformation, the Wronskian solution of the Ablowitz-Ladik lattice have been given. The $u_1$ of the cdKP hierarchy is a Y-type soliton solution for odd times of the gauge transformation, but it becomes a dark-bright soliton solution for even times of the gauge transformation. The role of the discrete variable $n$ in the profile of the $u_1$ is discussed.Comment: 19 pages,13 figure

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.Comment: 20 Pages, Theoretical and Mathematical Physics, 187(2016), 871-88

Solutions of the (2+1)-dimensional KP, SK and KK equations generated by gauge transformations from non-zero seeds

By using gauge transformations, we manage to obtain new solutions of (2+1)-dimensional Kadomtsev-Petviashvili(KP), Kaup-Kuperschmidt(KK) and Sawada-Kotera(SK) equations from non-zero seeds. For each of the preceding equations, a Galilean type transformation between these solutions $u_2$ and the previously known solutions $u_2^{\prime}$ generated from zero seed is given. We present several explicit formulas of the single-soliton solutions for $u_2$ and $u_2^{\prime}$, and further point out the two main differences of them under the same value of parameters, i.e., height and location of peak line, which are demonstrated visibly in three figures.Comment: 18 pages, 6 figures, to appear in Journal of Nonlinear Mathematical Physic

Dispersionless bigraded Toda Hierarchy and its additional symmetry

In this paper, we firstly give the definition of dipersionless bigraded Toda hierarchy (dBTH) and introduce some Sato theory on dBTH. Then we define Orlov-Schulman's \M_L, \M_R operator and give the additional Block symmetry of dBTH. Meanwhile we give tau function of dBTH and some some related dipersionless bilinear equations.Comment: 31 pages, Accepted by Reviews in Mathematical Physic

Regular solution and lattice miura transformation of bigraded Toda Hierarchy

In this paper, we give finite dimensional exponential solutions of the bigraded Toda Hierarchy(BTH). As an specific example of exponential solutions of the BTH, we consider a regular solution for the $(1,2)$-BTH with $3\times 3$-sized Lax matrix, and discuss some geometric structure of the solution from which the difference between $(1,2)$-BTH and original Toda hierarchy is shown. After this, we construct another kind of Lax representation of $(N,1)$-bigraded Toda hierarchy($(N,1)$-BTH) which does not use the fractional operator of Lax operator. Then we introduce lattice Miura transformation of $(N,1)$-BTH which leads to equations depending on one field, meanwhile we give some specific examples which contains Volterra lattice equation(an useful ecological competition model).Comment: Accepted by Chinese Annals of Mathematics, Series

The extended ZNZ_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.Comment: 22 Pages, Theoretical and Mathematical Physics, 185(2015), 1614-163

Designable integrability of the variable coefficient nonlinear Schr\"odinger equation

The designable integrability(DI) of the variable coefficient nonlinear Schr\"odinger equation (VCNLSE) is first introduced by construction of an explicit transformation which maps VCNLSE to the usual nonlinear Schr\"odinger equation(NLSE). One novel feature of VCNLSE with DI is that its coefficients can be designed artificially and analytically by using transformation. A special example between nonautonomous NLSE and NLSE is given here. Further, the optical super-lattice potentials (or periodic potentials) and multi-well potentials are designed, which are two kinds of important potential in Bose-Einstein condensation(BEC) and nonlinear optical systems. There are two interesting features of the soliton of the VCNLSE indicated by the analytic and exact formula. Specifically, its the profile is variable and its trajectory is not a straight line when it evolves with time $t$.Comment: 11 pages, 5 figures, accepted by Studies in Applied Mathematic

Quantum Torus symmetry of the KP, KdV and BKP hierarchies

In this paper, we construct the quantum Torus symmetry of the KP hierarchy and further derive the quantum torus constraint on the tau function of the KP hierarchy. That means we give a nice representation of the quantum Torus Lie algebra in the KP system by acting on its tau function. Comparing to the $W_{\infty}$ symmetry, this quantum Torus symmetry has a nice algebraic structure with double indices. Further by reduction, we also construct the quantum Torus symmetries of the KdV and BKP hierarchies and further derive the quantum Torus constraints on their tau functions. These quantum Torus constraints might have applications in the quantum field theory, supersymmetric gauge theory and so on.Comment: published in Lett. Math. Phys. online ahead of print 15 August 201