On the extension of solutions of the real to complex KdV equation and a mechanism for the construction of rogue waves

The formation of rogue waves in shallow water is presented in this Rapid Communication by providing the three lowest-order exact rational solutions to the Korteweg-de Vries (KdV) equation. They have been obtained from the modified KdV equation by using the complex Miura transformation. It is found that the amplitude amplification factor of such waves formed in shallow water is much larger than the amplitude amplification factor of those occurring in deep water. These solutions clearly demonstrate a potential hazard for coastal areas. They can also provide a solid mathematical basis for the existence of abnormally large-amplitude waves in other branches of nonlinear physics such as optics, unidirectional crystal growth, and in quantum mechanics.

show abstract

“…However, overall, little has been done on this subject, and our approach here is quite different from that of Ref. [16].…”

Section: Introductionmentioning

confidence: 94%

“…The first attempts to find rogue wave solutions of the complex KdV equation were made in Ref. [16]. However, overall, little has been done on this subject, and our approach here is quite different from that of Ref.…”

Section: Introductionmentioning

confidence: 99%

Shallow-water rogue waves: An approach based on complex solutions of the Korteweg–de Vries equation

2019

View full text Add to dashboard Cite

show abstract

“…The unified, extended and generalized unified methods have been proposed in [42][43][44][45][46][47][48][49]. The unified method asserts that , the solutions of a nonlinear evolution equation can be written in the forms of polynomial or rational functions in auxiliary functions, that satisfy appropriate auxiliary equations.…”

Section: Solutions Of the Fractal And Fractional Time-derivative Bnementioning

confidence: 99%

Analytic Solutions of Fractal and Fractional Time Derivative-Burgers–Nagumo Equation

Abdel‐Gawad

Tantawy

Abdel-Aziz

et al. 2021

Int. J. Appl. Comput. Math

Self Cite

View full text Add to dashboard Cite

The Nagumo equation describes a reaction-diffusion system in biology. Here, it is coupled to Burgers equation, via including convection, which is the Burgers-Nagumo equation (BNE). The first objective of this work is to present a theorem to reduce, approximately, the different versions of the fractional time derivatives (FTD) to an ordinary derivative (OD) with time dependent coefficients (non autonomous). The second objective is to find the exact solutions of the fractal and FTD-BNE is reduced to BNE with time dependent coefficient. Further similarity transformations are introduced. The unified and extended unified method are used to find the exact traveling waves solutions. Also, self-similar solutions are obtained. The novelties in this work are (i) reducing, via an analytic approximation, the different versions of FTD to non autonomous OD. (ii) Traveling and self-similar waves solutions of the FTD-BNE are derived. (iii) The effect of the order of fractional and fractal derivatives, on waves structure, are investigated. It is found that significant fractal effects hold for smaller order derivatives. While significant fractional effects hold for higher-order derivatives. It is found that, the solutions obtained show solitary wave, wrinkle soliton, solitons with double kinks or with spikes and undulated wave. Further It is shown that wrinkle soliton, with double kink configuration holds for smaller fractal order. While in the case of fractional derivative, this holds for higher orders. We mention that the results found here are completely new. Symbolic computations are carried by using Mathematica.

show abstract

“…We solve Equation by using the unified method . For polynomial solutions, we write

U false(z false) = true \sum_{j = 0}^{j = n} a_{j} g^{j} false(z false), {false(g^{'} false(z false) false)}^{p} = true \sum_{j = 0}^{j = k} c_{j} g^{j} false(z false),

where

p = 1

, the auxiliary equation solves to elementary fiction or implicit functions.…”

Section: Applicationsmentioning

confidence: 99%

Fractional KdV and Boussenisq‐Burger's equations, reduction to PDE and stability approaches

2020

Self Cite

View full text Add to dashboard Cite

The generalized fractional Caputo‐operator is discussed. The reduction of partial differential equations retarded or advanced with delays is obtained. The applications to the space‐time‐fractional KdV and time‐fractional Boussenisq‐Burger's equation are carried. Semi‐self‐similar SS wave solutions are obtained. That is, there exists a soliton wave propagates along a specific characteristic curve in the xt‐ plane. This may be due to the effects associated with the distributed time delay. The effects of space fractional derivative on a system are attributed to the transition states. Thus, anomalous wave transport is produced mainly near the origin.

show abstract

On the extension of solutions of the real to complex KdV equation and a mechanism for the construction of rogue waves

Cited by 25 publications

References 28 publications

Shallow-water rogue waves: An approach based on complex solutions of the Korteweg–de Vries equation

Shallow-water rogue waves: An approach based on complex solutions of the Korteweg–de Vries equation

Analytic Solutions of Fractal and Fractional Time Derivative-Burgers–Nagumo Equation

Fractional KdV and Boussenisq‐Burger's equations, reduction to PDE and stability approaches

Contact Info

Product

Resources

About