scholarly journals Mixed spectral AKNS hierarchy from linear isospectral problem and its exact solutions

Open Physics ◽  
2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Sheng Zhang ◽  
Xu-Dong Gao
Keyword(s):  
Exact Solutions ◽  
Inverse Scattering ◽  
Variable Coefficient ◽  
Soliton Solutions ◽  
Variable Coefficients ◽  
Necessary Condition ◽  
Scattering Problems ◽  
Systematic Analysis ◽  
Akns Hierarchy ◽  
Isospectral Problem

AbstractIn this paper, the AKNS isospectral problem and its corresponding time evolution are generalized by embedding three coefficient functions. Starting from the generalizedAKNS isospectral problem, a mixed spectralAKNS hierarchy with variable coefficients is derived. Thanks to the selectivity of these coefficient functions, the mixed spectral AKNS hierarchy contains not only isospectral equations but also nonisospectral equations. Based on a systematic analysis of the related direct and inverse scattering problems, exact solutions of the mixed spectral AKNS hierarchy are obtained through the inverse scattering transformation. In the case of reflectionless potentials, the obtained exact solutions are reduced to n-soliton solutions. This paper shows that the AKNS spectral problem being nonisospectral is not a necessary condition to construct a nonisospectral AKNS hierarchy and that the inverse scattering transformation can be used for solving some other variable-coefficient mixed hierarchies of isospectral equations and nonisospectral equations.

scholarly journals Inverse scattering transform for a new non-isospectral integrable non-linear AKNS model

2017 ◽  
Vol 21 (suppl. 1) ◽  
pp. 153-160 ◽  
Author(s):  
Xudong Gao ◽  
Sheng Zhang
Keyword(s):  
Evolution Equation ◽  
Exact Solutions ◽  
Inverse Scattering ◽  
Spectral Parameter ◽  
Soliton Solutions ◽  
Transformation Method ◽  
Linear Partial Differential Equations ◽  
Scattering Transform ◽  
Non Linear ◽  
Isospectral Problem

Constructing integrable systems and solving non-linear partial differential equations are important and interesting in non-linear science. In this paper, Ablowitz-Kaup-Newell-Segur (AKNS)?s linear isospectral problem and its accompanied time evolution equation are first generalized by embedding a new non-isospectral parameter whose varying with time obeys an arbitrary smooth enough function of the spectral parameter. Based on the generalized AKNS linear problem and its evolution equation, a new non-isospectral Lax integrable non-linear AKNS model is then derived. Furthermore, exact solutions of the derived AKNS model is obtained by extending the inverse scattering transformation method with new time-varying spectral parameter. In the case of reflectinless potentials, explicit n-soliton solutions are finally formulated through the obtained exact solutions.

Integrability and Multi-Soliton Solutions for a Variable-Coefficient Coupled Gross–Pitaevskii System for Atomic MatterWaves

2012 ◽  
Vol 67 (10-11) ◽  
pp. 525-533
Author(s):  
Zhi-Qiang Lin ◽  
Bo Tian ◽  
Ming Wang ◽  
Xing Lu
Keyword(s):  
Variable Coefficient ◽  
Soliton Solutions ◽  
Variable Coefficients ◽  
Einstein Condensate ◽  
Matter Wave ◽  
Bilinear Method ◽  
Matter Waves ◽  
Bose Einstein Condensate ◽  
Bose Einstein ◽  
Hirota Bilinear

Under investigation in this paper is a variable-coefficient coupled Gross-Pitaevskii (GP) system, which is associated with the studies on atomic matter waves. Through the Painlev´e analysis, we obtain the constraint on the variable coefficients, under which the system is integrable. The bilinear form and multi-soliton solutions are derived with the Hirota bilinear method and symbolic computation. We found that: (i) in the elastic collisions, an external potential can change the propagation of the soliton, and thus the density of the matter wave in the two-species Bose-Einstein condensate (BEC); (ii) in the shape-changing collision, the solitons can exchange energy among different species, leading to the change of soliton amplitudes.We also present the collisions among three solitons of atomic matter waves.

Riemann–Hilbert approach and multi-soliton solutions of a variable-coefficient fifth-order nonlinear Schrödinger equation with N distinct arbitrary-order poles

2021 ◽  
pp. 2150194
Author(s):  
Zhi-Qiang Li ◽  
Shou-Fu Tian ◽  
Tian-Tian Zhang ◽  
Jin-Jie Yang
Keyword(s):  
Schrödinger Equation ◽  
Inverse Scattering ◽  
Nonlinear Schrödinger Equation ◽  
Variable Coefficient ◽  
Schrodinger Equation ◽  
Arbitrary Order ◽  
Soliton Solutions ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Fifth Order

Based on inverse scattering transformation, a variable-coefficient fifth-order nonlinear Schrödinger equation is studied through the Riemann–Hilbert (RH) approach with zero boundary conditions at infinity, and its multi-soliton solutions with [Formula: see text] distinct arbitrary-order poles are successfully derived. By deriving the eigenfunction and scattering matrix, and revealing their properties, a RH problem (RHP) is constructed based on inverse scattering transformation. Via solving the RHP, the formulae of multi-soliton solutions are displayed when the reflection coefficient possesses [Formula: see text] distinct arbitrary-order poles. Finally, some appropriate parameters are selected to analyze the interaction of multi-soliton solutions graphically.

Bilinear forms and soliton solutions for a (2 + 1)-dimensional variable-coefficient nonlinear Schrödinger equation in an optical fiber

2020 ◽  
Vol 34 (30) ◽  
pp. 2050336
Author(s):  
Dong Wang ◽  
Yi-Tian Gao ◽  
Jing-Jing Su ◽  
Cui-Cui Ding
Keyword(s):  
Schrödinger Equation ◽  
Optical Fiber ◽  
Nonlinear Schrödinger Equation ◽  
Variable Coefficient ◽  
Schrodinger Equation ◽  
Soliton Solutions ◽  
Variable Coefficients ◽  
Nonlinear Schrodinger Equation ◽  
Bilinear Forms ◽  
Dimensional Variable

In this paper, under investigation is a (2 + 1)-dimensional variable-coefficient nonlinear Schrödinger equation, which is introduced to the study of an optical fiber, where [Formula: see text] is the temporal variable, variable coefficients [Formula: see text] and [Formula: see text] are related to the group velocity dispersion, [Formula: see text] and [Formula: see text] represent the Kerr nonlinearity and linear term, respectively. Via the Hirota bilinear method, bilinear forms are obtained, and bright one-, two-, three- and N-soliton solutions as well as dark one- and two-soliton solutions are derived, where [Formula: see text] is a positive integer. Velocities and amplitudes of the bright/dark one solitons are obtained via the characteristic-line equations. With the graphical analysis, we investigate the influence of the variable coefficients on the propagation and interaction of the solitons. It is found that [Formula: see text] can only affect the phase shifts of the solitons, while [Formula: see text], [Formula: see text] and [Formula: see text] determine the amplitudes and velocities of the bright/dark solitons.

scholarly journals Exact Solutions for a Third-Order KdV Equation with Variable Coefficients and Forcing Term

2009 ◽  
Vol 2009 ◽  
pp. 1-13 ◽  
Author(s):  
Alvaro H. Salas S ◽  
Cesar A. Gómez S
Keyword(s):  
Exact Solutions ◽  
Periodic Solutions ◽  
Riccati Equation ◽  
Function Method ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Variable Coefficients ◽  
Third Order ◽  
Forcing Term ◽  
Projective Riccati Equation Method

The general projective Riccati equation method and the Exp-function method are used to construct generalized soliton solutions and periodic solutions to special KdV equation with variable coefficients and forcing term.

Exact Soliton Solutions of the Variable Coefficient KdV Equation with Forced Term

2014 ◽  
Vol 548-549 ◽  
pp. 1196-1200
Author(s):  
Yong Mei Bao ◽  
Siqintana Bao
Keyword(s):  
Variable Coefficient ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Nonlinear Ordinary Differential Equation ◽  
Variable Coefficients ◽  
Nonlinear Evolution Equations ◽  
Forced Term ◽  
Exact Soliton Solutions

In order to construct exact soliton solutions of nonlinear evolution equations with variable coefficients. By using a transformation, the variable coefficient KdV equation with forced Term is reduced to nonlinear ordinary differential equation (NLODE), after that, a number of exact solitons solutions of variable coefficient KdV equation with forced Term are obtained by using the equation shorted in NLODE. As it showed above, this kind of method can be applied in solving a large number of nonlinear evolution equations.

EXACT SOLUTIONS OF NONLINEAR SU(N) PRINCIPAL σ-MODELS

1988 ◽  
Vol 03 (05) ◽  
pp. 1147-1154
Author(s):  
TIBOR KISS-TOTH
Keyword(s):  
Exact Solutions ◽  
Inverse Scattering ◽  
Soliton Solutions ◽  
Static Solution ◽  
Finite Energy ◽  
Single Step ◽  
Inverse Scattering Method ◽  
Scattering Method ◽  
Axially Symmetric ◽  
Symmetric Static Solution

The superpotential for n-step soliton solution is derived in the case of an arbitrary dimensional projector for axially symmetric, static solution of nonlinear principal SU (N) σ-models. This was done by using an inverse scattering method developed by Belinski and Zakharov. Finite energy solutions are constructed for all SU (N) one soliton solutions generated by a single step.

scholarly journals Bell Polynomials Approach Applied to (2 + 1)-Dimensional Variable-Coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Wen-guang Cheng ◽  
Biao Li ◽  
Yong Chen
Keyword(s):  
Variable Coefficient ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Variable Coefficients ◽  
Bell Polynomials ◽  
Bilinear Method ◽  
Bilinear Bäcklund Transformation ◽  
Dimensional Variable ◽  
Binary Bell Polynomials ◽  
Hirota Bilinear

The bilinear form, bilinear Bäcklund transformation, and Lax pair of a (2 + 1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation are derived through Bell polynomials. The integrable constraint conditions on variable coefficients can be naturally obtained in the procedure of applying the Bell polynomials approach. Moreover, theN-soliton solutions of the equation are constructed with the help of the Hirota bilinear method. Finally, the infinite conservation laws of this equation are obtained by decoupling binary Bell polynomials. All conserved densities and fluxes are illustrated with explicit recursion formulae.

scholarly journals Exact solutions of time fractional heat-like and wave-like equations with variable coefficients

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 689-693 ◽  
Author(s):  
Sheng Zhang ◽  
Ran Zhu ◽  
Luyao Zhang
Keyword(s):  
Boundary Conditions ◽  
Exact Solutions ◽  
Variable Coefficient ◽  
Variable Coefficients ◽  
Separation Method ◽  
Variable Separation ◽  
Science And Engineering ◽  
Special Solutions ◽  
Initial And Boundary Conditions ◽  
Variable Separation Method

In this paper, a variable-coefficient time fractional heat-like and wave-like equation with initial and boundary conditions is solved by the use of variable separation method and the properties of Mittag-Leffler function. As a result, exact solutions are obtained, from which some known special solutions are recovered. It is shown that the variable separation method can also be used to solve some others time fractional heat-like and wave-like equation in science and engineering.

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