Rogue Waves of the Kundu-Nonlinear Schrodinger Equation

This paper is based on the Darboux transformation of the Kundu-Nonlinear Schr&oumldinger equation. The rogue wave solutions are obtained from periodic seed solutions. After that, the higher order rogue wave solutions of the Kundu-Nonlinear Schr&oumldinger equation are given. Finally, we show that free parameters in eigenfunctions can adjust the patterns of the higher order rogue waves.

On limit fractional Volterra hierarchies

For the limit fractional Volterra(LFV)hierarchy,we construct the n-fold Darboux transformation and the soliton solutions.Furthermore,we extend the LFV hierarchy to the noncommutative LFV(NCLFV)hierarchy,and construct the Darboux transformation expressed by quasi determinant of the noncommutative version.Meanwhile,we establish the relationship between new and old solutions of the NCLFV hierarchy.Finally,the quasi determinant solutions of the NCLFV hierarchy are obtained.

Nonsingular Positon Solutions of a Variable-Coefficient Modified KdV Equation

The determinant representation of three-fold Darboux transformation for a variable-coefficient modified KdV equation is displayed based on the technique used to solve Ablowitz-Kaup-Newell-Segur system. Additionally, the nonsingular positon solutions of the variable-coefficient modified KdV equation are firstly discovered analytically and graphically.

Rogue Waves of the Kundu-DNLS Equation

In this paper, we give the Lax pair and construct the Darboux transformation of the Kundu-DNLS equation. Furthermore, the rogue wave solutions of the Kundu-DNLS equation are derived by using the Taylor expansion of the breather solution. What's more, the triangular and the circular patterns of the third rouge solution are displayed.

Generalizations of the finite nonperiodic Toda lattice and its Darboux transformation

In this paper,we construct Hamiltonian systems for 2 N particles whose force depends on the distances between the particles.We obtain the generalized finite nonperiodic Toda equations via a symmetric group transformation.The solutions of the generalized Toda equations are derived using the tau functions.The relationship between the generalized nonperiodic Toda lattices and Lie algebras is then be discussed and the generalized Kac-van Moerbeke hierarchy is split into generalized Toda lattices,whose integrability and Darboux transformation are studied.

Quantum Torus Lie Algebra in the q-Deformed Kadomtsev-Pe tv iashvili Sys tern

Based on the symmetry of the Q-deformed Kadomtsev-Petviashvili(q-KP)hierarchy,which is a q-deformation of the KP hierarchy,we construct the quantum torus symmetry of the q-KP hierarchy,which further leads to the quantum torus constraint of its tau function.Moreover,we generalize the system to a multi-component q-KP hierarchy that also has the well-known ghost symmetry.

Regular Solution and Lattice Miura Transformation of Bigraded Toda Hierarchy

The authors give finite dimensional exponential solutions of the bigraded Toda hierarchy(BTH).As a specific example of exponential solutions of the BTH,the authors consider a regular solution for the(1,2)-BTH with a 3×3-sized Lax matrix,and discuss some geometric structures of the solution from which the difference between the(1,2)-BTH and the original Toda hierarchy is shown.After this,the authors construct another kind of Lax representation of(N,1)-BTH which does not use the fractional operator of Lax operator.Then the authors introduce the lattice Miura transformation of(AT,1)-BTH which leads to equations depending on one field,and meanwhile some specific examples which contain the Volterra lattice equation(a useful ecological competition model)are given.

Noncommutative Constrained KP Hierarchy and Multi-component Noncommutative Constrained KP Hierarchy

In this paper, the authors define the noncommutative constrained KadomtsevPetviashvili(KP) hierarchy and multi-component noncommutative constrained KP hierarchy. Then they give the recursion operators for the noncommutative constrained KP(NcKP) hierarchy and multi-component noncommutative constrained KP(NmcKP) hierarchy. The authors hope these studies might be useful in the study of D-brane dynamics whose noncommutative coordinates emerge from limits of the M theory and string theory.

Ghost Symmetries and Multi-fold Darboux Transformations of Extended Toda Hierarchy

In this paper,the author constructs ghost symmetries of the extended Toda hierarchy with their spectral representations.After this,two kinds of Darboux transforma-tions in different directions and their mixed Darboux transformations of this hierarchy are constructed.These symmetries and Darboux transformations might be useful in Gromov-Witten theory of CP1.