Exact similarity and traveling wave solutions to an integrable evolution equation for surface waves in deep water

Meng, Qing; He, Bin; Liu, Wei

doi:10.1007/s11071-018-4093-5

Cited by 5 publications

(2 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The Lie symmetry group applied to the surface wave equation was used to find exact similarity and traveling wave solution in [20]. Solutions of the periodic waves via nonlinear Schrodinger NLS equation was carried in [21].…”

Section: Introductionmentioning

confidence: 99%

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

Abdel‐Gawad

Belić

2022

Applied Mathematics and Nonlinear Sciences

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%

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

Abdel‐Gawad

Belić

2022

Applied Mathematics and Nonlinear Sciences

View full text Add to dashboard Cite

show abstract

“…In the current study, the Lie symmetry analysis technique [8][9][10][11][12][13][14][15][16][17][18][19] is using to analyze the considered model. This method applies to all kinds of nonlinear PDEs.…”

Section: Introductionmentioning

confidence: 99%

Some exact explicit solutions and conservation laws of Chaffee-Infante equation by Lie symmetry analysis

et al. 2021

View full text Add to dashboard Cite

In this work, the tanh method is employed to compute some traveling wave patterns of the nonlinear third-order (2+1) dimensional Chaffee-Infante (CI) equation. The tanh technique is successfully used to get the traveling wave solutions of a considered model in the form of some hyperbolic functions. The Lie symmetry technique is used to analyze the Chaffee-Infante (CI) equation and compute the Infinitesimal generators under the invariance criteria of Lie groups. Then we construct the commutator table, adjoint representation table, and we have represented symmetry groups for each Infinitesimal generator. The optimal system and similarity reduction method is used to obtain some analytical solutions of the considered model. With the help of the similarity reduction method, we have converted the nonlinear partial differential equation into nonlinear ordinary differential equations (ODEs). Moreover, we have shown graphically obtained wave solutions by using the different values of involving parameters. Conserved quantities of nonlinear CI equation are obtained by the multiplier approach.

show abstract

Quasi-periodic, chaotic and travelling wave structures of modified Gardner equation

Jhangeer

Hussain

Junaid-U-Rehman

et al. 2021

Chaos, Solitons & Fractals

View full text Add to dashboard Cite

Exact similarity and traveling wave solutions to an integrable evolution equation for surface waves in deep water

Cited by 5 publications

References 18 publications

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

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

Some exact explicit solutions and conservation laws of Chaffee-Infante equation by Lie symmetry analysis

Quasi-periodic, chaotic and travelling wave structures of modified Gardner equation

Contact Info

Product

Resources

About