New Soliton Solutions to the Initial Value Problem for the Two-Component Short Pulse Equation

First, we establish the local well-posedness of the initial value problem for a new three-component Camassa–Holm system with peakons. We then present a precise blowup scenario and several blowup results for strong solutions to that system. Finally, we determine the blowup rate of strong solutions to the system when a blowup occurs. Our results include all earlier results on the Camassa–Holm equation and on a two-component Camassa–Holm system with peakons.

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Two-Component Generalization of a Generalized the Short Pulse Equation

Iraqi Journal of Science ◽

10.24996/ijs.2019.60.8.13 ◽

2019 ◽

pp. 1760-1765

Author(s):

Mohammed Allami

Keyword(s):

Nonlinear Equations ◽

Short Pulse ◽

Three Dimensional ◽

Continuous Group ◽

System Of Nonlinear Equations ◽

Similarity Reduction ◽

Short Pulse Equation ◽

Point Transformations ◽

Two Component ◽

Lie Brackets

In this article, we introduce a two-component generalization for a new generalization type of the short pulse equation was recently found by Hone and his collaborators. The coupled of nonlinear equations is analyzed from the viewpoint of Lieâ€™s method of a continuous group of point transformations. Our results show the symmetries that the system of nonlinear equations can admit, as well as the admitting of the three-dimensional Lie algebra. Moreover, the Lie brackets for the independent vectors field are presented. Similarity reduction for the system is also discussed.

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Riemann-Hilbert problems and soliton solutions for the complex modified short pulse equation

Reports on Mathematical Physics ◽

10.1016/s0034-4877(21)00066-5 ◽

2021 ◽

Vol 88 (2) ◽

pp. 145-159

Author(s):

Xuan Zhou ◽

Engui Fan

Keyword(s):

Short Pulse ◽

Soliton Solutions ◽

Short Pulse Equation

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The Modified Coupled Hirota Equation: Riemann-Hilbert Approach and N-Soliton Solutions

Abstract and Applied Analysis ◽

10.1155/2019/8342876 ◽

2019 ◽

Vol 2019 ◽

pp. 1-10

Author(s):

Siqi Xu

Keyword(s):

Soliton Solution ◽

Initial Value Problem ◽

Soliton Solutions ◽

Compact Form ◽

Initial Value ◽

Hirota Equation ◽

Dynamical Behaviors

The Cauchy initial value problem of the modified coupled Hirota equation is studied in the framework of Riemann-Hilbert approach. The N-soliton solutions are given in a compact form as a ratio of (N+1)×(N+1) determinant and N×N determinant, and the dynamical behaviors of the single-soliton solution are displayed graphically.

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