<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>The equation of the surface waves in deep water, given here by (1), is extended to (2+1) dimensions, which is a novel equation.. It is shown that the surface waves equation is self- free source. So, it has a class of infinite solutions. Here many types of self-similar and semi-self similar solutions are obtained. The self-similar waves show various geometric structures. Among them, wave crest in the form of coupled lumps and soliton wave moving along the characteristic curve in the plane. It is entrained by troughs with cavities. The semi-self similar waves exhibit multi lumps or periodic waves with troughs and multi-periodic waves. The study of bifurcation shows that the trajectories are open, so that the traveling wave solutions are unstable. The time-dependent steepness-function is defined here and it is found that it attains a maximum value and then it decreases with time. The results found are interesting in ocean engineering and sciences. The extended unified method is used, here, to find the exact solutions, which was proposed recently.</p>
</abstract>

The equation of the surface waves in deep water, given here by (1), is extended to (2+1) dimensions, which is a novel equation.. It is shown that the surface waves equation is self- free source. So, it has a class of infinite solutions. Here many types of self-similar and semi-self similar solutions are obtained. The self-similar waves show various geometric structures. Among them, wave crest in the form of coupled lumps and soliton wave moving along the characteristic curve in the plane. It is entrained by troughs with cavities. The semi-self similar waves exhibit multi lumps or periodic waves with troughs and multi-periodic waves. The study of bifurcation shows that the trajectories are open, so that the traveling wave solutions are unstable. The time-dependent steepness-function is defined here and it is found that it attains a maximum value and then it decreases with time. The results found are interesting in ocean engineering and sciences. The extended unified method is used, here, to find the exact solutions, which was proposed recently.

Similarity Solutions of the Surface Waves Equation in (2+1) Dimensions and Bifurcation

Mathematics Department, Faculty of Sciences

Applied Mathematics and Nonlinear Sciences

This work is licensed under the Creative Commons Attribution 4.0 International License.

The equation of the surface waves in deep water, given here by (1), is extended to (2+1) dimensions, which is a novel equation.. It is shown that the surface...

Similarity Solutions of the Surface Waves Equation in (2+1) Dimensions and Bifurcation

Published Online: Oct 14, 2022

Page range: 419 - 430

Received: Aug 01, 2020

Accepted: Jan 31, 2021

DOI: https://doi.org/10.2478/amns.2022.1.00102

Keywords
extension to (2+1) dimensions, surface waves, self similar solutions, extended unified method

© 2023 Hamdy I. Abdel-Gawad et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

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Similarity Solutions of the Surface Waves Equation in (2+1) Dimensions and Bifurcation

Published Online: Oct 14, 2022

Page range: 419 - 430

Received: Aug 01, 2020

Accepted: Jan 31, 2021

DOI: https://doi.org/10.2478/amns.2022.1.00102

Keywordsextension to (2+1) dimensions, surface waves, self similar solutions, extended unified method

© 2023 Hamdy I. Abdel-Gawad et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Keywords
extension to (2+1) dimensions, surface waves, self similar solutions, extended unified method