Riemann–Hilbert approach of the Newell-type long-wave–short-wave equation via the temporal-part spectral analysis

A vector modified Yajima–Oikawa long-wave–short-wave equation is proposed using the zero-curvature presentation. On the basis of the Riccati equations associated with the Lax pair, a method is developed to construct multi-fold classical and generalized Darboux transformations for the vector modified Yajima–Oikawa long-wave–short-wave equation. As applications of the multi-fold classical Darboux transformations and generalized Darboux transformations, various exact solutions for the vector modified long-wave–short-wave equation are obtained, including soliton, breather, and rogue wave solutions.

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A New Integrable Variable-Coefficient 2+1-Dimensional Long Wave-Short Wave Equation and the Generalized Dressing Method

Advances in Mathematical Physics ◽

10.1155/2018/7286574 ◽

2018 ◽

Vol 2018 ◽

pp. 1-7

Author(s):

Ting Su ◽

Hui Hui Dai

Keyword(s):

Wave Equation ◽

Variable Coefficient ◽

Separation Of Variables ◽

Short Wave ◽

Lax Pair ◽

Explicit Solutions ◽

Dressing Method ◽

Long Wave

Based on the generalized dressing method, we propose a new integrable variable-coefficient 2+1-dimensional long wave-short wave equation and derive its Lax pair. Using separation of variables, we have derived the explicit solutions of the equation. With the aid of Matlab, the curves of the solutions are drawn.

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Conservative Difference Scheme of Solitary Wave Solutions of the Generalized Regularized Long-Wave Equation

Indian Journal of Pure and Applied Mathematics ◽

10.1007/s13226-020-0468-7 ◽

2020 ◽

Vol 51 (4) ◽

pp. 1317-1342

Author(s):

Asma Rouatbi ◽

Manel Labidi ◽

Khaled Omrani

Keyword(s):

Wave Equation ◽

Solitary Wave ◽

Difference Scheme ◽

Solitary Wave Solutions ◽

Conservative Difference Scheme ◽

Wave Solutions ◽

Long Wave ◽

Regularized Long Wave Equation ◽

Conservative Difference

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On Galilean Invariant and Energy Preserving BBM-Type Equations

Symmetry ◽

10.3390/sym13050878 ◽

2021 ◽

Vol 13 (5) ◽

pp. 878

Author(s):

Alexei Cheviakov ◽

Denys Dutykh ◽

Aidar Assylbekuly

Keyword(s):

Wave Equation ◽

Conservation Laws ◽

Numerical Simulations ◽

Solitary Waves ◽

Local Symmetry ◽

Symmetry And Conservation Laws ◽

Higher Order ◽

Special Equation ◽

Long Wave ◽

Local Symmetries

We investigate a family of higher-order Benjamin–Bona–Mahony-type equations, which appeared in the course of study towards finding a Galilei-invariant, energy-preserving long wave equation. We perform local symmetry and conservation laws classification for this family of Partial Differential Equations (PDEs). The analysis reveals that this family includes a special equation which admits additional, higher-order local symmetries and conservation laws. We compute its solitary waves and simulate their collisions. The numerical simulations show that their collision is elastic, which is an indication of its S−integrability. This particular PDE turns out to be a rescaled version of the celebrated Camassa–Holm equation, which confirms its integrability.

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