On the transverse instabilities of solitary waves

Allen, Michael A.; Rowlands, G.

doi:10.1016/s0375-9601(97)00618-x

Cited by 37 publications

(22 citation statements)

References 4 publications

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“…We verify that the three-soliton condition (11) can be written as a determinant relation (17). Noting the even property of the polynomial…”

Section: Appendix: a Proof Of The Three-soliton Determinant Conditionmentioning

confidence: 56%

“…This implies that the three-soliton determinant condition (17) holds for the generalized KP and Boussinesq bilinear equation (13).…”

Section: Discussionmentioning

confidence: 96%

“…In analyzing the existence of the three soliton solution for the generalized KP and Boussinesq equations (1), the difficulty is to compute the three-soliton condition (11), and our success is to rewrite the three-soliton condition (11) as a determinant relation (17), which is put in the appendix. An approach of Darboux transformations [15] could be used to generate multi-soliton solutions directly from the three-soliton solution.…”

Section: Discussionmentioning

confidence: 99%

“…Step 2: Now, apply the three-soliton condition (17) to the transformed equation (26), and then we see from the arbitrariness character of the parameters l i, j that there is at most one non-zero constant, let us say b 2 , among the coefficients b i , 2 ≤ i ≤ M.…”

Section: Uniqueness Propertymentioning

confidence: 99%

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Uniqueness of the Kadomtsev-Petviashvili and Boussinesq Equations

W.-X.¹,

Pekcan²

2011

Z. Naturforsch. A

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The Kadomtsev-Petviashvili and Boussinesq equations (u xxx − 6uu x ) x − u tx ± u yy = 0, (u xxx − 6uu x ) x + u xx ± u tt = 0, are completely integrable, and in particular, they possess the three-soliton solution. This article aims to expose a uniqueness property of the Kadomtsev-Petviashvili (KP) and Boussinesq equations in the integrability theory. It is shown that the Kadomtsev-Petviashvili and Boussinesq equations and their dimensional reductions are the only integrable equations among a class of generalized Kadomtsev-Petviashvili and Boussinesq equations (uwhere the a i j 's are arbitrary constants and M is an arbitrary natural number, if the existence of the three-soliton solution is required.

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“…We verify that the three-soliton condition (11) can be written as a determinant relation (17). Noting the even property of the polynomial…”

Section: Appendix: a Proof Of The Three-soliton Determinant Conditionmentioning

confidence: 56%

“…This implies that the three-soliton determinant condition (17) holds for the generalized KP and Boussinesq bilinear equation (13).…”

Section: Discussionmentioning

confidence: 96%

Section: Discussionmentioning

confidence: 99%

Section: Uniqueness Propertymentioning

confidence: 99%

See 2 more Smart Citations

Uniqueness of the Kadomtsev-Petviashvili and Boussinesq Equations

W.-X.¹,

Pekcan²

2011

Z. Naturforsch. A

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“…As a check, the soliton case, further limited to perpendicular instabilities, was recovered as found by Laedke and Spatschek (1982). Since then several authors have looked at the propagation and some at stability in various configurations (Infeld and Frycz, 1987& 1989, Allen and Rowlands, 1993, 1995& 1997 Here we present a small K stability analysis of a nonlinear wave propagating at an angle to the magnetic field. Its shape is treated exactly and a cubic is obtained for the frequency of the perturbation (or growth rate).…”

Section: Introductionmentioning

confidence: 99%