A Chain of Type Ⅱ and Its Exact Solutions

The exact solutions of a chain of type Ⅱ are investigated.The chain of type E is first transformed to an integrable differential-difference equation,which has the Kaup-Newell spectral problem as its continuous spatial spectral problem and a Darboux transformation of the Kaup-Newell equation as its discrete temporal spectral problem.Then,with these spectral problems,a Darboux transformation of the transformed equation is constructed.Finally,as an application of the Darboux transformation,an exact solution of the transformed equation and thus the chain of type Ⅱ are presented.     

Symmetry and asymmetry rogue waves in two-component coupled nonlinear Schrdinger equations

By means of the modified Darboux transformation we obtain some types of rogue waves in two-coupled nonlinear Schr ¨odinger equations.Our results show that the two components admits the symmetry and asymmetry rogue wave solutions,which arises from the joint action of self-phase,cross-phase modulation,and coherent coupling term.We also obtain the analytical transformation from the initial seed solution to unique rogue waves with the bountiful pair structure.In a special case,the asymmetry rogue wave can own the spatial and temporal symmetry gradually,which is controlled by one parameter.It is worth pointing out that the rogue wave of two components can share the temporal inversion symmetry.     

Darboux Transformation with a Double Spectral Parameter for the Myrzakulov-I Equation

The Myrzakulov-I equation is a 2＋l-dimensional generalization of the Heisenberg ferromagnetic equa- tion and has a non-isospectral Lax pair. The ex- plicit solutions to the Myrzakulov-I equation have been discussed by many researchers. Darboux transformation is one of the useful methods to ob- tain explicit solutions to the nonlinear partial differ- ential equation. The Darboux transformation of de- gree 1 for this equation has been constructed and exact global ＇one-soliton＇ solutions are derived.     

Soliton,Breather and Rogue Wave Solutions for the Nonlinear Schrdinger Equation Coupled to a Multiple Self-Induced Transparency System

We derive an N-fold Darboux transformation for the nonlinear Schrdinger equation coupled to a multiple selfinduced transparency system, which is applicable to optical fiber communications in the erbium-doped medium.The N-soliton, N-breather and N th-order rogue wave solutions in the compact determinant representations are derived using the Darboux transformation and limit technique. Dynamics of such solutions from the first-to second-order ones are shown.     

N-soliton solutions of an integrable equation studied by Qiao

In this paper, we studied N-soliton solutions of a new integrable equation studied by Qiao [J. Math. Phys. 48 082701 (2007)]. Firstly, we employed the Darboux matrix method to construct a Darboux transformation for the modified Korteweg-de Vries equation. Then we use the Darboux transformation and a transformation, introduced by Sakovich [J. Math. Phys. 52 023509 (2011)], to derive N-soliton solutions of the new integrable equation from the seed solution. In particular, the multiple soliton solutions are explicitly obtained and shown through some figures.     

Darboux Transformation Method for Solving the Sime—Gordon Equation in a Laboratory Reference

The sine-Gordon equation is solved in a laboratory reference using the method of Darboux transformation.Using the Liouville theorem,explicit expressions of the single soliton solution and the breather solution are derived from the Darboux matrix in the case of a null spectral parameter.     

Darboux Transformation and Variable Separation Approach： the Nizhnik-Novikov-Veselov Equation

Darboux transformation (DT) is developed to systematically find variable separation solutions for the Nizhnik-Novikov-Veselov equation. Starting from a seed solution with some arbitrary functions, the one-step DT yields the variable separable solutions, which can be obtained from the truncated Painleve analysis, and the two-step DT leads to some new variable separable solutions, which are the generalization of the known results obtained by using a guess ansatz to solve the generalized trilinear equation.     

Solutions of Two Kinds of Non-Isospectral Generalized Nonlinear Schrodinger Equation Related to Bose-Einstein Condensates

Two non-isospectral generalized nonlinear Schrodinger （ONLS） equations, which are two important models of nonlinear excitations of matter waves in Bose-Einstein condensates, are studied. Two novel transformations are constructed such that these two GNLS equations are transformed to the well-known nonlinear Schr6dinger （NLS） equation, which is an isospectral equation. Therefore, once one solution of the NLS equation is provided, we can immediately obtain one solution for two ONLS equations by these transformations. Thus it is unnecessary to solve these two non-isospectral GNLS equations directly. Soliton solutions and periodic solutions are obtained for them by two transformations from the corresponding solutions of the NLS equation, which are generated by Darboux transformation.     

Dynamics of Bright Solitons in Bose-Einstein Condensates with Complicated Potential

We present analytical solutions of the one-dimensional nonlinear Schrodinger equations of Bose-Einstein condensates in an expulsive parabolic background with a complex potential and gravitational field, by performing the Darboux transformation from a trivial seed solution. It is shown that under a safe range of parameter, the shape of bright soliton can be controlled well by adjusting the experimental parameter of the ratio of axial oscillation to radial oscillation and feeding condensates from a thermal cloud. The gravitational field can change the contrail of the bright soliton trains without changing their peak and width.     

Nonsingular Travelling Complexiton Solutions to a Coupled Korteweg--de Vries Equation

A class of novel nonsingular travelling complexiton solutions to a coupled Korteweg-de Vries （KdV） equation is presented via the first step Darboux transformation of the complex KdV equation with nonzero seed solution. Furthermore, the properties of the nonsingular solutions are discussed.     

Rogue Wave Solutions for Nonlinear Schrdinger Equation with Variable Coefficients in Nonlinear Optical Systems

In this paper, the generalized Darboux transformation is constructed to variable coefficient nonlinear Schrdinger(NLS) equation. The N-th order rogue wave solution of this variable coefficient NLS equation is obtained by determinant expression form. In particular, we present rogue waves from first to third-order through some figures and analyze their dynamics.     

Breathers and solitons for the coupled nonlinear Schr?dinger system in three-spine α-helical protein

We mainly investigate the variable-coefficient 3-coupled nonlinear Schr?dinger(NLS) system, which describes soliton dynamics in the three-spine α-helical protein with inhomogeneous effect. The variable-coefficient NLS equation is transformed into the constant coefficient NLS equation by similarity transformation firstly. The Hirota method is used to solve the constant coefficient NLS equation, and then we get the one-and two-breather solutions of the variable-coefficient NLS equation. The results show that, in the background of plane waves and periodic waves, the breather can be transformed into some forms of combined soliton solutions. The influence of different parameters on the soliton solution and the collision between two solitons are discussed by some graphs in detail. Our results are helpful to study the soliton dynamics inα-helical protein.     

Explicit N—Fold Darboux Transformations and Soliton Solutions for Nonlinear Derivative Schrodinger Equations

An explicit N-fold Darboux transformation for a coupled of derivative nonlinear Schrodinger equations is constructed with the help of a gauge transformation of spectral problems.As a reduction,the Darboux transformation for well-known Gerdjikov-Ivanov equation is further obtained,from which a general form of N-soliton solutions for Gerdjikov-Ivanov equation is given.     

Localized waves of the coupled cubic–quintic nonlinear Schrdinger equations in nonlinear optics

We investigate some novel localized waves on the plane wave background in the coupled cubic–quintic nonlinear Schr o¨dinger(CCQNLS) equations through the generalized Darboux transformation(DT). A special vector solution of the Lax pair of the CCQNLS system is elaborately constructed, based on the vector solution, various types of higherorder localized wave solutions of the CCQNLS system are constructed via the generalized DT. These abundant and novel localized waves constructed in the CCQNLS system include higher-order rogue waves, higher-order rogues interacting with multi-soliton or multi-breather separately. The first-and second-order semi-rational localized waves including several free parameters are mainly discussed:(i) the semi-rational solutions degenerate to the first-and second-order vector rogue wave solutions;(ii) hybrid solutions between a first-order rogue wave and a dark or bright soliton, a second-order rogue wave and two dark or bright solitons;(iii) hybrid solutions between a first-order rogue wave and a breather, a second-order rogue wave and two breathers. Some interesting and appealing dynamic properties of these types of localized waves are demonstrated, for example, these nonlinear waves merge with each other markedly by increasing the absolute value of α.These results further uncover some striking dynamic structures in the CCQNLS system.     

Darboux Transformation and Exact Solutions of the Myrzakulov-I Equation

The Myrzakulov-I equation is a 2＋1-dimensional generalization of the Heisenberg ferromagnetic equation and has a non-isospectral Lax pair. The Darboux transformation with non-constant spectral parameter is constructed and an extra constraint on the spectral parameter for the existence of the Darboux transformation is derived. Explicit expressions of the solutions of the Myrzakulov-I equation are presented.     

Explicit solutions from residual symmetry of the Boussinesq equation

The Bcklund transformation related symmetry is nonlocal, which is hard to be applied in constructing solutions for nonlinear equations. In this paper, the residual symmetry of the Boussinesq equation is localized to Lie point symmetry by introducing multiple new variables. By applying the general Lie point method, two main results are obtained: a new type of Backlund transformation is derived, from which new solutions can be generated from old ones; the similarity reduction solutions as well as corresponding reduction equations are found. The localization procedure provides an effective way to investigate interaction solutions between nonlinear waves and solitons.     

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.     

Darboux Transformation and Explicit Solutions for Drinfel＇d-Sokolov-Wilson Equation

A generalized Drinfel＇d Sokolov-Wilson （DSW） equation and its Lax pair are proposed. A Daorboux transformation for the generalized DSW equation is constructed with the help of the gauge transformation between spectral problems, from which a Darboux transformation for the DSW equation is obtained through a reduction technique. As an application of the Darboux transformations, we give some explicit solutions of the generalized DSW equation and DEW equation such as rational solutions, soliton solutions, periodic solutions.     

Deformed soliton,breather,and rogue wave solutions of an inhomogeneous nonlinear Schrdinger equation

We use the 1-fold Darboux transformation (DT) of an inhomogeneous nonlinear Schro¨dinger equation (INLSE) to construct the deformed-soliton, breather, and rogue wave solutions explicitly. Furthermore, the obtained first-order deformed rogue wave solution, which is derived from the deformed breather solution through the Taylor expansion, is different from the known rogue wave solution of the nonlinear Schro¨dinger equation (NLSE). The effect of inhomogeneity is fully reflected in the variable height of the deformed soliton and the curved background of the deformed breather and rogue wave. By suitably adjusting the physical parameter, we show that a desired shape of the rogue wave can be generated. In particular, the newly constructed rogue wave can be reduced to the corresponding rogue wave of the nonlinear Schro¨dinger equation under a suitable parametric condition.     

Solutions to the Complex Korteweg-de Vries Equation: Blow-up Solutions and Non-Singular Solutions

In the paper two kinds of solutions are derived for the complex Korteweg-de Vries equation, includ- ing blow-up solutions and non-singular solutions. We derive blow-up solutions from known 1-soliton solution and a double-pole solution. There is a complex Miura transformation between the complex Korteweg-de Vries equation and a modified Kortcweg-de Vries equation. Using the transformation, solitons, breathers and rational solutions to the com- plex Korteweg-de Vries equation are obtained from those of the modified Korteweg-de Vries equation. Dynamics of the obtained solutions are illustrated.