Darboux transformation for two‐level system

We develop the Darboux procedure for the case of the two‐level system. In particular, it is demonstrated that one can construct the Darboux intertwining operator that does not violate the specific structure of the equations of the two‐level system, transforming only one real potential into another real potential. We apply the obtained Darboux transformation to known exact solutions of the two‐level system. Thus, we find three classes of new solutions for the two‐level system and the corresponding new potentials that allow such solutions.     

The determinant representation of an N-fold Darboux transformation for the short pulse equation

We present an explicit representation of an N-fold Darboux transformation T?_N for the short pulse equation, by the determinants of the eigenfunctions of its Lax pair. In the course of the derivation of T?_N, we show that the quasi-determinant is avoidable, and it is contrast to a recent paper (J. Phys. Soc. Jpn. 81 (2012), 094008) by using this relatively new tool which was introduced to study noncommutative mathematical objectives. T?_N produces new solutions u^[N] and x^[N] which are expressed by ratios of two corresponding determinants. We also obtain the soliton solutions, which have a variable trajectory, of the short pulse equation from new “seed” solutions.     

Broer-Kaup系统的达布变换和新的精确解

利用Broer-Kaup系统Lax对的两个基本解，构造了该系统一个新的达布变换，并利用该变换求出了Broer-Kaup系统一组新的双向孤子解. 关键词： Lax对 达布变换 孤子解     

A hierarchy of lattice soliton equations and its Darboux transformation

A new discrete spectral problem is introduced and the corresponding hierarchy of the lattice soliton equations are derived by means of the trace identity. We find a new Darboux transformation of the lattice soliton equation, through which the explicit solutions are shown.     

Darboux transformation and dark rogue wave states arising from two-wave resonance interaction

Exact rogue wave solutions of the long wave–short wave resonance equation are obtained via Darboux transformations. Compared to the real long-wave field which always features a single hump structure, the short-wave field can be manifested as bright rogue wave, intermediate rogue wave, or dark rogue wave, depending on the subtle nonlinearity driven by the long-wave field.     

The Darboux Transformation for NLS-MB Equations

The Darboux transformations for the nonlinear Schrodinger equation and Maxwell-Bloch equations areconstructed. The one-soliton solution and periodic solution are obtained from the different “seeds“.     

Factorization method and new potentials from the inverted oscillator

In this article we will apply the first- and second-order supersymmetric quantum mechanics to obtain new exactly-solvable real potentials departing from the inverted oscillator potential. This system has some special properties; in particular, only very specific second-order transformations produce non-singular real potentials. It will be shown that these transformations turn out to be the so-called complex ones. Moreover, we will study the factorization method applied to the inverted oscillator and the algebraic structure of the new Hamiltonians.     

Darboux transformation and exact multisolitons for a matrix coupled dispersionless system

We present a matrix coupled dispersionless(CD) system. A Lax pair for the matrix CD system is proposed and Darboux transformation is constructed on the solutions of the matrix CD system and the associated Lax pair. We express an N soliton formula for the solutions of the matrix CD system in terms of quasideterminants. By using properties of the quasideterminants, we obtain some exact solutions, including bright and dark-type solitons, rogue wave and breather solutions of the matrix CD system. Furthermore, it has been shown that the solutions of the matrix CD system are expressed in terms of solutions to the usual CD system, sine-Gordon equation and Maxwell-Bloch system.     

Darboux transformation and explicit solutions for the derivative versions of Toda equation

The derivative versions of Toda equation are produced from the Lax triad composed of the Kaup-Newell and negative Kaup-Newell spectral problems and a new discrete eigenvalue problem. With the help of decomposition and the Darboux transformation, explicit solutions of these derivative Toda equations are obtained.     

The Darboux Transformation for NLS-MB Equations

The Darboux transformations for the nonlinear Schrödinger equation and Maxwell-Bloch equations are constructed. The one-soliton solution and periodic solution are obtained from the different “seeds”.     

Multi-soliton solutions of a two-component Camassa–Holm system:Darboux transformation approach

We propose and develop another approach to constructing multi-soliton solutions of an integrable two-component Camassa–Holm(CH2)system.With the help of a reciprocal transformation and a gauge transformation,we relate the CH2 system to a negative flow of the Broer–Kaup or twoboson hierarchy.The solutions of this negative flow are given in terms of Wronskians via Darboux transformation.Then the multi-soliton solutions of the CH2 system are recovered in parametric form by inverting the reciprocal transformation and the gauge transformation.     

Darboux transformation and soliton solutions of the semi-discrete massive Thirring model

A one-fold Darboux transformation between solutions of the semi-discrete massive Thirring model is derived using the Lax pair and the dressing method. This transformation is used to find the exact expressions for soliton solutions on zero and nonzero backgrounds. It is shown that the discrete solitons have the same properties as their continuous counterparts.     

Darboux transformation and soliton solutions of a nonlocal Hirota equation

Starting from local coupled Hirota equations,we provide a reverse space-time nonlocal Hirota equation by the symmetry reduction method known as the Ablowitz–Kaup–Newell–Segur scattering problem.The Lax integrability of the nonlocal Hirota equation is also guaranteed by existence of the Lax pair.By Lax pair,an n-fold Darboux transformation is constructed for the nonlocal Hirota equation by which some types of exact solutions are found.The solutions with specific properties are distinct from those of the local Hirota equation.In order to further describe the properties and the dynamic features of the solutions explicitly,several kinds of graphs are depicted.     

Mapping of Shape Invariant Potentials by the Point Canonical Transformation

In this paper by using the method of point canonical transformation we find that the Coulomb and Kratzer potentials can be mapped to the Morse potential. Then we show that the Pöschl-Teller potential type I belongs to the same subclass of shape invariant potentials as Hulthén potential. Also we show that the shape-invariant algebra for Coulomb, Kratzer, and Morse potentials is SU(1,1), while the shape-invariant algebra for Pöschl-Teller type I and Hulthén is SU(2).     

Darboux transformation and elementary exact solutions of the Schrödinger equation

Darboux transformation is applied to three classical potentials, namely the harmonic oscillator, effective Coulomb and Morse potentials to generate exactly solvable potentials of elementary form. For every potential, the isospectral families of potentials are constructed. For almost all potentials, a set of normalized discrete spectrum wave functions is given.     

Darboux transformation and novel solutions for the long wave-short wave model

Firstly, we establish the relation between the loop group method and gauge transformation method for 3×3 spectral problem. Some novel solutions of long wave-short wave model are obtained by Darboux transformation method. Besides, we give the analysis and classification of solution in detail.     

Wronskian differential formula for confluent supersymmetric quantum mechanics

A Wronskian differential formula, useful for applying the confluent second-order SUSY transformations to arbitrary potentials, will be obtained. This expression involves a parametric derivative with respect to the factorization energy which, in many cases, is simpler for calculations than the previously found integral equation. This alternative mechanism will be applied to the free particle and the single-gap Lamé potential.     

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 are governed by femtosecond pulse propagation through inhomogeneous doped fibre.The determinant representation of Darboux transformation is used to derive soliton solutions,positon solutions to the IH-MB equations.     

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.