Multi-soliton, Rogue Wave and Periodic Wave Solutions of Generalized ($$2+1$$) Dimensional Boussinesq Equation

Vinodh, D.; Asokan, R.

doi:10.1007/s40819-020-0768-y

“…Among the aforementioned methods, researchers have proved that the HBM is widely popular due to its simplicity and directness, which is particularly useful to derive -soliton solutions to any nonlinear integrable models. Recently, with the use of the long-wave limit approach in coordination with constraint parameters, the localized waves including solitons, lumps, breathers, rogue waves, and their interaction solutions from -soliton solutions have been found through a variety of nonlinear models, which are mostly published in 2018-2020 [1,10,[39][40][41][42][43][44][45][46][47][48]. Among the cited refs.…”

Section: Introductionmentioning

confidence: 99%

“…Among the cited refs. [1,10,[39][40][41][42][43][44][45][46][47][48], no significant studies have been found for analyzing the wave propagation dynamics of the localized waves and their interaction structures in shallow water through some relevant models. Although, in the past, Wazwaz [49] mentioned that the study of wave propagation of long waves in shallow water has become a dynamic research field in nonlinear science.…”

Section: Introductionmentioning

confidence: 99%

Novel localized waves and interaction solutions for a dimensionally reduced (2 + 1)-dimensional Boussinesq equation from N-soliton solutions

Kumar

¹

,

Nuruzzaman

²

,

Paul

³

et al. 2022

Nonlinear Dyn

16

0

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The Boussinesq equation (BqE) has been of considerable interest in coastal and ocean engineering models for simulating surface water waves in shallow seas and harbors, tsunami wave propagation, wave over-topping, inundation, and near-shore wave process in which nonlinearity and dispersion effects are taken into consideration. The study deals with the dynamics of localized waves and their interaction solutions to a dimensionally reduced (2+1)dimensional BqE from N-soliton solutions with the use of Hirota's bilinear method (HBM).Taking the long-wave limit approach in coordination with some constraint parameters in the N-soliton solutions, the localized waves (i.e., soliton, breather, lump, and rogue waves) and their interaction solutions are constructed. The interaction solutions can be obtained among localized waves, such as (i) one breather or one lump from the two solitons, (ii) one stripe and one breather, and one stripe and one lump from the three solitons, and (iii) two stripes and one breather, one lump and one periodic breather, two stripes and one lump, two breathers, and two lumps from the four solitons. It is to be found that all interactions among the solitons are elastic.The energy, phase shift, shape, and propagation direction of these localized waves and their interaction solutions can be influenced and controlled by the involved constraint parameters.The dynamical characteristics of these localized waves and their interaction solutions are demonstrated through some 3D and density graphs. The outcomes achieved in this study can be used to illustrate the wave interaction phenomena in shallow water.

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“…Over the past few years, many methods are being developed to expose new traveling solitary wave solutions for the nonlinear partial differential equation (PDE) representing the different areas of science and engineering. [1][2][3][4] Some of the analytical methods such as extended (G ′ /G)-expansion method, Darboux transformation, Pfaffian technique, sech-tanh method, sine-cosine method, Painlevé analysis, 5 Hirota bilinear method, [6][7][8][9][10][11][12][13][14][15] extended generalized Darboux transformation method, 16,17 Bäcklund transformation, and simplified Hirota's method [18][19][20][21][22][23][24] are used to solve different models involving nonlinear PDE. There is no specified method to solve all types nonlinear PDE.…”

Section: Introductionmentioning

confidence: 99%

New various multisoliton kink‐type solutions of the (1 + 1)‐dimensional Mikhailov–Novikov–Wang equation

Ray

¹

,

Singh

²

2021

Math Methods in App Sciences

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show abstract

“…Among the aforementioned methods, researchers have proved that the HBM is widely popular due to its simplicity and directness, which is particularly useful to derive -soliton solutions to any nonlinear integrable models. Recently, with the use of the long-wave limit approach in coordination with constraint parameters, the localized waves including solitons, lumps, breathers, rogue waves, and their interaction solutions from -soliton solutions have been found through a variety of nonlinear models, which are mostly published in 2018-2020 [1,10,[39][40][41][42][43][44][45][46][47][48]. Among the cited refs.…”

Section: Introductionmentioning

confidence: 99%

“…Among the cited refs. [1,10,[39][40][41][42][43][44][45][46][47][48], no significant studies have been found for analyzing the wave propagation dynamics of the localized waves and their interaction structures in shallow water through some relevant models. Although, in the past, Wazwaz [49] mentioned that the study of wave propagation of long waves in shallow water has become a dynamic research field in nonlinear science.…”

Section: Introductionmentioning

confidence: 99%

Novel Localized Waves and their Interaction Solutions for a Dimensionally Reduced (2+1)-dimensional Boussinesq Equation from N-soliton Solutions

Kumar

¹

,

Nuruzzaman

²

,

Gc

³

et al. 2021

Preprint

0

View full text Add to dashboard Cite

The Boussinesq equation (BqE) has been of considerable interest in coastal and ocean engineering models for simulating surface water waves in shallow seas and harbors, tsunami wave propagation, wave over-topping, inundation, and near-shore wave process in which nonlinearity and dispersion effects are taken into consideration. The study deals with the dynamics of localized waves and their interaction solutions to a dimensionally reduced (2 + 1)-dimensional BqE from N-soliton solutions with the use of Hirota’s bilinear method (HBM). Taking the long-wave limit approach in coordination with some constraint parameters in the N-soliton solutions, the localized waves (i.e., soliton, breather, lump, and rogue waves) and their interaction solutions are constructed. The interaction solutions can be obtained among localized waves, such as (i) one breather or one lump from the two solitons, (ii) one stripe and one breather, and one stripe and one lump from the three solitons, and (iii) two stripes and one breather, one lump and one periodic breather, two stripes and one lump, two breathers, and two lumps from the four solitons. It is to be found that all interactions among the solitons are elastic. The energy, phase shift, shape, and propagation direction of these localized waves and their interaction solutions can be influenced and controlled by the involved constraint parameters. The dynamical characteristics of these localized waves and their interaction solutions are demonstrated through some 3D and density graphs. The outcomes achieved in this study can be used to illustrate the wave interaction phenomena in shallow water.

show abstract

Multi-soliton, Rogue Wave and Periodic Wave Solutions of Generalized ($$2+1$$) Dimensional Boussinesq Equation

Cited by 13 publications

References 65 publications

Novel localized waves and interaction solutions for a dimensionally reduced (2 + 1)-dimensional Boussinesq equation from N-soliton solutions

Novel localized waves and interaction solutions for a dimensionally reduced (2 + 1)-dimensional Boussinesq equation from N-soliton solutions

New various multisoliton kink‐type solutions of the (1 + 1)‐dimensional Mikhailov–Novikov–Wang equation

Novel Localized Waves and their Interaction Solutions for a Dimensionally Reduced (2+1)-dimensional Boussinesq Equation from N-soliton Solutions

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