A New Class of Periodic Solutions to (2+1)-Dimensional KdV Equations

Huang, Wenhua; Liu, Yulu; Zhang, Jie-Fang; Lai, Xian-Jing

doi:10.1088/6102/44/3/401

Cited by 6 publications

(2 citation statements)

References 30 publications

(14 reference statements)

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“…Ring types of solutions, periodic solutions and localized coherent solutions of Eqs. (25) and (26) can be found in [37][38][39][40][41].…”

Section: Application Of the Methodsmentioning

confidence: 97%

A generalized auxiliary equation method and its application to (2+1) -dimensional Korteweg–de Vries equations

Zhang

2007

Computers & Mathematics with Applications

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A generalized auxiliary equation method is proposed for constructing more general exact solutions of nonlinear partial differential equations. With the aid of symbolic computation, we choose the (2 + 1)-dimensional Korteweg-de Vries equations to illustrate the validity and advantages of this method. As a result, many new and more general exact non-travelling wave and coefficient function solutions are obtained, which include soliton-like solutions, triangular-like solutions, single and combined non-degenerate Jacobi elliptic wave function-like solutions and Weierstrass elliptic doubly-like periodic solutions.

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“…Ring types of solutions, periodic solutions and localized coherent solutions of Eqs. (25) and (26) can be found in [37][38][39][40][41].…”

Section: Application Of the Methodsmentioning

confidence: 97%

A generalized auxiliary equation method and its application to (2+1) -dimensional Korteweg–de Vries equations

Zhang

2007

Computers & Mathematics with Applications

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“…The ring type of solutions, periodic solutions and localized coherent solutions of Eqs. (5) and (6) can be found in [14][15][16][17][18]. In this section, we will obtain many new and more general solutions by using our method presented in Section 2.…”

Section: Application To the (2 + 1)-dimensional Kdv Equationsmentioning

confidence: 99%

An improved generalized F-expansion method and its application to the (2+1)-dimensional KdV equations

Zhang

Xia

2008

Communications in Nonlinear Science and Numerical Simulation

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Superposition principle and composite solutions to coupled nonlinear Schrödinger equations

Sakkaf

Khawaja²

2020

Math Methods in App Sciences

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We show that the superposition principle applies to coupled nonlinear Schrödinger equations with cubic nonlinearity where exact solutions may be obtained as a linear combination of other exact solutions. This is possible due to the cancelation of cross terms in the nonlinear coupling. First, we show that a composite solution, which is a linear combination of the two components of a seed solution, is another solution to the same coupled nonlinear Schrödinger equation. Then, we show that a linear combination of two composite solutions is also a solution to the same equation. With emphasis on the case of Manakov system of two-coupled nonlinear Schrödinger equations, the superposition is shown to be equivalent to a rotation operator in a two-dimensional function space with components of the seed solution being its coordinates. Repeated application of the rotation operator, starting with a specific seed solution, generates a series of composite solutions, which may be represented by a generalized solution that defines a family of composite solutions. Applying the rotation operator to almost all known exact seed solutions of the Manakov system, we obtain for each seed solution the corresponding family of composite solutions. Composite solutions turn out, in general, to possess interesting features that do not exist in the seed solution. Using symmetry reductions, we show that the method applies also to systems of N-coupled nonlinear Schrödinger equations. Specific examples for the three-coupled nonlinear Schrödinger equation are given.

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A New Class of Periodic Solutions to (2+1)-Dimensional KdV Equations

Cited by 6 publications

References 30 publications

A generalized auxiliary equation method and its application to (2+1) -dimensional Korteweg–de Vries equations

A generalized auxiliary equation method and its application to (2+1) -dimensional Korteweg–de Vries equations

An improved generalized F-expansion method and its application to the (2+1)-dimensional KdV equations

Superposition principle and composite solutions to coupled nonlinear Schrödinger equations

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