Nonpropagating solitons of the variable coefficient and nonisospectral Korteweg–de Vries equation

Chan, Wai Ming; Li, Kam-Shun

doi:10.1063/1.528533

Cited by 80 publications

(39 citation statements)

References 4 publications

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“…It is easy to see that the nonisospectral parameter = (2 ) /2 in [33] is a special case of (12). Here in (12) is equivalent to in [33].…”

Section: Introductionmentioning

confidence: 99%

“…Since the initialvalue problem of the Korteweg-de Vries (KdV) equation was exactly solved by the IST method [1], finding soliton solutions of nonlinear PDEs has become extremely active and some effective methods were proposed such as Hirota's bilinear method [2], Painlevé expansion [3], homogeneous balance method [4], and function expansion methods [5][6][7][8][9][10]. Among these methods, the IST [1] is a systematic method which has achieved considerable development and received a wide range of applications like those in [11][12][13][14][15][16][17][18][19][20][21] since it is put forward by Gardner, Greene, Kruskal, and Miura in 1967. One of the advantages of the IST is that it can solve a whole hierarchy of nonlinear PDEs associated with a certain spectral problem.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Lax Integrability and Soliton Solutions for a Nonisospectral Integrodifferential System

Zhang

Hong

2017

Complexity

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Searching for integrable systems and constructing their exact solutions are of both theoretical and practical value. In this paper, Ablowitz-Kaup-Newell-Segur (AKNS) spectral problem and its time evolution equation are first generalized by embedding a new spectral parameter. Based on the generalized AKNS spectral problem and its time evolution equation, Lax integrability of a nonisospectral integrodifferential system is then verified. Furthermore, exact solutions of the nonisospectral integrodifferential system are formulated through the inverse scattering transform (IST) method. Finally, in the case of reflectionless potentials, the obtained exact solutions are reduced to -soliton solutions. When = 1 and = 2, the characteristics of soliton dynamics of one-soliton solutions and two-soliton solutions are analyzed with the help of figures.

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“…It is easy to see that the nonisospectral parameter = (2 ) /2 in [33] is a special case of (12). Here in (12) is equivalent to in [33].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Lax Integrability and Soliton Solutions for a Nonisospectral Integrodifferential System

Zhang

Hong

2017

Complexity

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“…We discuss Equation (1) in Section 2 and its symmetry transformations and in Section 3, we derive solutions of some equations of type (1) from solutions of the KdV and mKdV equation including a few illustrating examples taken from physical models [11,12].…”

Section: Introductionmentioning

confidence: 99%

A new symmetry approach to solve space–time‐dependent KdV systems

Ruan

Chen

2003

Math Methods in App Sciences

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SUMMARYIn this paper, a new method to solve space-time-dependent non-linear equations is proposed. After considering the variable coe cient of a non-linear equation as a new dependent variable, some special types of space-time-dependent equations can be solved from corresponding space-time-independent equations by using the general classical Lie approach. The rich soliton solutions of space-time-dependent KdV equation and mKdV equation are given with the help of the approach.

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“…Equation (8) can be reduced to other more physical forms [21][22][23][24][25][26] which has been discussed in Ref. [16].…”

Section: Exact Solutions To the Gvkdv Equationmentioning

confidence: 99%

New Exact Analytical Solutions for the General KdV Equation with Variable Coefficients

Hong

2014

MCA

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Abstract-In this paper, a general algebraic method based on the generalized Jacobi elliptic functions expansion method, the improved general mapping deformation method and the extended auxiliary function method with computerized symbolic computation is proposed to construct more new exact solutions of a generalized KdV equation with variable coefficients. As a result, eight families of new generalized Jacobi elliptic function wave solutions and Weierstrass elliptic function solutions of the equation are obtained by using this method, some of these solutions are degenerated to soliton-like solutions, trigonometric function solutions in the limit cases when the modulus of the Jacobi elliptic functions 1 m  or 0, which shows that the general method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differential equations arising in mathematical physics.

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Nonpropagating solitons of the variable coefficient and nonisospectral Korteweg–de Vries equation

Cited by 80 publications

References 4 publications

Lax Integrability and Soliton Solutions for a Nonisospectral Integrodifferential System

Lax Integrability and Soliton Solutions for a Nonisospectral Integrodifferential System

A new symmetry approach to solve space–time‐dependent KdV systems

New Exact Analytical Solutions for the General KdV Equation with Variable Coefficients

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