共查询到20条相似文献,搜索用时 31 毫秒
1.
This paper employs the theory of planar dynamical systems and undetermined coefficient method to study travelling wave solutions of the dissipative (2 + 1)-dimensional AKNS equation. By qualitative analysis, global phase portraits of the dynamic system corresponding to the equation are obtained under different parameter conditions. Furthermore, the relations between the properties of travelling wave solutions and the dissipation coefficient r of the equation are investigated. In addition, the possible bell profile solitary wave solution, kink profile solitary wave solutions and approximate damped oscillatory solutions of the equation are obtained by using undetermined coefficient method. Error estimates indicate that the approximate solutions are meaningful. Based on above studies, a main contribution in this paper is to reveal the dissipation effect on travelling wave solutions of the dissipative (2 + 1)-dimensional AKNS equation. 相似文献
2.
Lei Wang Yi-Tian Gao Xiao-Ling Gai 《Communications in Nonlinear Science & Numerical Simulation》2012,17(7):2833-2844
Whitham-Broer-Kaup (WBK) model is a model for the dispersive long wave in shallow water. With symbolic computation, gauge transformation between the WBK model and a parameter Ablowitz-Kaup-Newell-Segur (AKNS) system is hereby constructed. By selecting seeds, we derive two sorts of multi-soliton solutions for the WBK model via a N-fold Darboux transformation (DT) of the parameter AKNS system, which are expressed in terms of the Vandermonde-like and double Wronskian determinants, respectively. Different from the bilinear way, the double Wronskian solutions can be obtained via the N-fold DT with a linear algebraic system and matrix differential equation solved. A novel inelastic interaction is graphically discussed, in which the soliton complexes are formed after the collision. Our results could be helpful for interpreting certain shallow-water-wave phenomena. 相似文献
3.
Based on a 4 × 4 matrix Lax pair, we propose a negative matrix AKNS system with a Hermitian symmetric space. A Darboux transformation is constructed by setting a restrictive condition and using the loop group method. The restrictive condition can guarantee the evolution relations of the potential matrices. Using this Darboux transformation and different seed solutions and free parameters, we obtain different types of spatial–temporal distribution structures for various explicit solutions of the negative matrix AKNS system with a Hermitian symmetric space, including the rogue wave, Ma breather, the interaction of two Ma breathers, and parabolic-type soliton solutions. 相似文献
4.
The algebraic–geometric solutions of the mixed AKNS equations are investigated through a finite-dimensional Lie–Poisson Hamiltonian system, which is generated by the nonlinearization of the adjoint equation related to the AKNS spectral problem. First, each mixed AKNS equation can be decomposed into two compatible Lie–Poisson Hamiltonian flows. Then the separated variables on the coadjoint orbit are introduced to study these Lie–Poisson Hamiltonian systems. Further, based on the Hamilton–Jacobi theory, the relationship between the action-angle coordinates and the Jacobi-inversion problem is established. In the end, using Riemann–Jacobi inversion, the algebraic–geometric solutions of the first three mixed AKNS equations are obtained. 相似文献
5.
Using an improved direct reduction method, we find the equivalence transformations of (2 + 1)-dimensional AKNS shallow water wave equation with variable coefficients, and obtain the corresponding relationship between explicit solutions of AKNS equation and those of the corresponding reduced equation. In addition, we get some new explicit solutions of AKNS equation by applying Lie symmetry method. 相似文献
6.
《Communications in Nonlinear Science & Numerical Simulation》2011,16(6):2445-2455
The Bäcklund transformations and abundant explicit exact solutions to the AKNS shallow water wave equation are obtained by combining the extended homogeneous balance method with the extended hyperbolic function method. The solutions obtained admit of multiple arbitrary parameters. These solutions include (a) a compound of the rational fractional function and a linear function, (b) a compound of solitary wave solution and a linear function, (c) a compound of the singular travelling wave solutions and a linear function, and (d) a compound of the periodic wave solutions of triangle function and a linear function. In special cases, we can obtain a series of soliton solutions, singular travelling wave solutions, periodic travelling wave solutions, and rational fractional function solution. In addition to re-deriving some known solutions in a systematic way, some brand-new exact solutions are also established. 相似文献
7.
Lump and mixed rogue-soliton solutions to the 2+1 dimensional Ablowitz-Kaup-Newell-Segur equation
下载免费PDF全文
![点击此处可从《Journal of Applied Analysis & Computation》网站下载免费的PDF全文](/ch/ext_images/free.gif)
In this paper, the 2+1 dimensional Ablowitz-Kaup-Newell-Segur (AKNS) equation which obtained from the potential Boiti-Leon-Manna-Pempi nelli (pBLMP) equation, is introduced. Through the bilinear method and ansatz technique, the rational solutions consisting of rogue wave and lump soliton solutions are constructed, where we discuss the condition of guaranteeing the positiveness and analyticity of the lump solutions. The collection of a quadratic function with an exponential function describing rational-exponential solutions is proved, the interaction consisting of one lump and one soliton with fission and fusion phenomena. The second kind of interaction comprises the line rogue wave and soliton solution, which is inelastic. With the usage of the extended homoclinic test approach, the homoclinic breather-wave solution is derived. The characteristics of these various solutions are exhibited and illustrated graphically. 相似文献
8.
Wen‐Xiu Ma 《Mathematical Methods in the Applied Sciences》2019,42(4):1099-1113
An arbitrary order matrix spectral problem is introduced and its associated multicomponent AKNS integrable hierarchy is constructed. Based on this matrix spectral problem, a kind of Riemann‐Hilbert problems is formulated for a multicomponent mKdV system in the resulting AKNS integrable hierarchy. Through special corresponding Riemann‐Hilbert problems with an identity jump matrix, soliton solutions to the presented multicomponent mKdV system are explicitly worked out. A specific reduction of the multicomponent mKdV system is made, together with its reduced Lax pair and soliton solutions. 相似文献
9.
《Communications in Nonlinear Science & Numerical Simulation》2012,17(6):2319-2332
A new generalized AKNS hierarchy is presented starting from a 4 × 4 matrix spectral problem with four potentials. Its generalized bi-Hamiltonian structure is also investigated by using the trace identity. Moreover, the special coupled nonlinear equation, the coupled KdV equation, the KdV equation, the coupled mKdV equation and the mKdV equation are produced from the generalized AKNS hierarchy. Most importantly, a Darboux transformation for the generalized AKNS hierarchy is established with the aid of the gauge transformation between the corresponding 4 × 4 matrix spectral problem, by which multiple soliton solutions of the generalized AKNS hierarchy are obtained. As a reduction, a Darboux transformation of the mKdV equation and its new analytical positon, negaton and complexiton solutions are given. 相似文献
10.
马文秀 《数学物理学报(B辑英文版)》2019,(2)
Based on a 4 x 4 matrix spectral problem, an AKNS soliton hierarchy with six potentials is generated. Associated with this spectral problem, a kind of Riemann-Hilbert problems is formulated for a six-component system of mKdV equations in the resulting AKNS hierarchy. Soliton solutions to the considered system of coupled mKdV equations are computed, through a reduced Riemann-Hilbert problem where an identity jump matrix is taken. 相似文献
11.
G.‐F. Yu 《Studies in Applied Mathematics》2015,135(2):117-138
The paper investigates the integrable discretization of a negative order AKNS equation. Two semidiscrete and one fully discrete versions of the system are given via Hirota's bilinear method. Soliton solutions for the derived discrete systems are also presented. Dynamics of one‐soliton and two‐soliton solutions of spacial‐discrete system are characterized. 相似文献
12.
G. F. Helminck 《Theoretical and Mathematical Physics》2017,192(3):1324-1336
We present an integrable hierarchy that includes both the AKNS hierarchy and its strict version. We split the loop space g of gl2 into Lie subalgebras g≥0 and g<0 of all loops with respectively only positive and only strictly negative powers of the loop parameter. We choose a commutative Lie subalgebra C in the whole loop space s of sl2 and represent it as C = C≥0⊕C<0. We deform the Lie subalgebras C≥0 and C<0 by the respective groups corresponding to g<0 and g≥0. Further, we require that the evolution equations of the deformed generators of C≥0 and C<0 have a Lax form determined by the original splitting. We prove that this system of Lax equations is compatible and that the equations are equivalent to a set of zero-curvature relations for the projections of certain products of generators. We also define suitable loop modules and a set of equations in these modules, called the linearization of the system, from which the Lax equations of the hierarchy can be obtained. We give a useful characterization of special elements occurring in the linearization, the so-called wave matrices. We propose a way to construct a rather wide class of solutions of the combined AKNS hierarchy. 相似文献
13.
Xiangpeng Xin Lihua Zhang Yarong Xi Hanze Liu 《Journal of Applied Analysis & Computation》2020,10(6):2669-2681
In this paper, nonlocal symmetries of variable coefficient Ablowitz-Kaup-Newell-Segur(AKNS) system are studied for the first time. In order to construct some new analytic solutions, a new variable is introduced, which can transform nonlocal symmetries into Lie point symmetries. Furthermore, using the Lie point symmetries of closed system, we give out two types of symmetry reductions and some analytic solutions. For some interesting solutions, such as interaction solutions among solitons and other complicated waves, we give corresponding images to describe their dynamic behavior. 相似文献
14.
Jie Ji 《Nonlinear Analysis: Theory, Methods & Applications》2009,71(9):4034-4046
In this paper, the bilinear form of a negative order non-isospectral AKNS equation is given. The soliton solutions are obtained through Hirota’s direct method and the Wronskian technique, respectively. 相似文献
15.
《Studies in Applied Mathematics》2018,141(1):113-141
In this paper, nonlocal reductions of the Ablowitz–Kaup–Newell–Suger (AKNS) hierarchy are collected, including the nonlocal nonlinear Schrödinger hierarchy, nonlocal modified Korteweg‐de Vries hierarchy, and nonlocal versions of the sine‐Gordon equation in nonpotential form. A reduction technique for solutions is employed, by which exact solutions in double Wronskian form are obtained for these reduced equations from those double Wronskian solutions of the AKNS hierarchy. As examples of dynamics, we illustrate new interaction of two‐soliton solutions of the reverse‐t nonlinear Schrödinger equation. Although as a single soliton, it is stationary that two solitons travel along completely symmetric trajectories in plane and their amplitudes are affected by phase parameters. Asymptotic analysis is given as demonstration. The approach and relation described in this paper are systematic and general and can be used to other nonlocal equations. 相似文献
16.
In the paper, we continue to consider symmetries related to the Ablowitz–Ladik hierarchy. We derive symmetries for the integrable discrete nonlinear Schrödinger hierarchy and discrete AKNS hierarchy. The integrable discrete nonlinear Schrödinger hierarchy is in scalar form and its two sets of symmetries are shown to form a Lie algebra. We also present discrete AKNS isospectral flows, non‐isospectral flows and their recursion operator. In continuous limit these flows go to the continuous AKNS flows and the recursion operator goes to the square of the AKNS recursion operator. These discrete AKNS flows form a Lie algebra that plays a key role in constructing symmetries and their algebraic structures for both the integrable discrete nonlinear Schrödinger hierarchy and discrete AKNS hierarchy. Structures of the obtained algebras are different structures from those in continuous cases which usually are centerless Kac–Moody–Virasoro type. These algebra deformations are explained through continuous limit and degree in terms of lattice spacing parameter h. 相似文献
17.
V. S. Oganesyan 《Theoretical and Mathematical Physics》2018,196(1):983-995
We consider the AKNS hierarchy and find the necessary and sufficient conditions for functions p and q to become solutions of the AKNS hierarchy. Using the functions p and q, we construct finite-gap Schrödinger potentials. 相似文献
18.
M.A. Kaashoek 《Journal of Functional Analysis》2005,228(1):207-233
A discrete analog of a skew self-adjoint canonical (Zakharov-Shabat or AKNS) system with a pseudo-exponential potential is introduced. For the corresponding Weyl function the direct and inverse problem are solved explicitly in terms of three parameter matrices. As an application explicit solutions are obtained for the discrete integrable nonlinear equation corresponding to the isotropic Heisenberg magnet model. State-space techniques from mathematical system theory play an important role in the proofs. 相似文献
19.
Ruguang ZHOU 《数学年刊B辑(英文版)》2012,33(2):191-206
It is well-known that every member of the KdV hierarchy of equations can be obtained from the AKNS hierarchy of equations by imposing a simple reduction. The author finds that the reduction conditions ... 相似文献
20.
New double Wronskian solutions of the AKNS equation 总被引:2,自引:0,他引:2
Soliton solutions, rational solutions, Matveev solutions, complexitons and interaction solutions of the AKNS equation are derived through a matrix method for constructing double Wronskian entries. The latter three solutions are novel. Moreover, rational solutions of the nonlinear Schrodinger equation are obtained by reduction. 相似文献