Symmetries of optimal system, various closed-form solutions, and propagation of different wave profiles for the Boussinesq–Burgers system in ocean waves

Abstract

Water waves, a common natural phenomenon, have been influential in various fields, such as energy development, offshore engineering, mechanical engineering, and hydraulic engineering. To describe the shallow water waves near an ocean coast or in a lake, we use the (1 + 1)-dimensions Boussinesq–Burgers system. By means of Lie symmetry analysis, symmetry groups and infinitesimal generators are obtained for the (1 + 1)-dimension Boussinesq–Burgers system. For the sake of finding the invariant solutions of the Boussinesq–Burgers system, the optimal one-dimensional subalgebra system is computed. Furthermore, using similarity reduction and the generalized Kudryashov method, we attain the abundant wave solutions of the Boussinesq–Burgers system presented in this research paper. Additionally, the exact solutions, which illustrate the effectiveness of the proposed method, also reveal the physical interpretation of the nonlinear models. To demonstrate the significance of interaction phenomena, dynamical behaviors of some attained solutions are depicted geometrically and theoretically through suitable parameter values. Consequently, kink, singular, periodic, solitary wave solutions, and their elastic nature have been shown to validate these solutions with physical phenomena. With the aid of the obtained results, the researchers could gain an understanding of the different modes of shallow water waves nearby an ocean beach. The computational work ascertained that the imposed methods are sturdy, precise, modest, and widely applicable.

Diverse analytical wave solutions and dynamical behaviors of the new (2 + 1)-dimensional Sakovich equation emerging in fluid dynamics

“…Therefore, the exact closed-form solutions of these equations play a fundamental role in unraveling the dynamic and help to understand/comprehend the mechanism of the existent state. A variety of novel approaches have been effectively utilized, developed, and improved by collections of assorted researchers for retrieving the exact solutions of NLEEs where the most important goes back to Lax pair [1], bifurcation method [2,3], extended mapping method [4], the tanh method [5], extended multiple Riccati equations expansion method [6], inverse scattering method [7], extended Jacobian elliptic function expansion method [8,9], Lie symmetry analysis [10][11][12][13][14][15][16], generalized Kudryashov method [17], etc. Although there is no particular universal technique that is applicable to all NLPDEs.…”

“…As a result of the ceaseless research on the analogous topics, for the first time, we reveal new and further general exact solitary wave solutions for the (2+1)-dimensional nonlinear Sakovich equation ( 5) by applying two renewed techniques named the Lie symmetry analysis and the extended Jacobian elliptic function expansion method. The Lie symmetry (LS) is theorized to be one of the significant approaches to determining competent solutions of NLEEs of type [15,16]. The main purpose of the symmetry approach is to reduce the dimensions of non linear partial differential equations (NLPDEs) to an ordinary differential equations (ODEs), which works on the principle of invariance under various symmetries.…”

Explicit Soliton Structure Formation for the Riemann Wave Equation and a Sensitive Demonstration

“…Many schemes and approaches, such as the Kudryashov method [28], sine-Gordon expansion scheme [29], bilinear neural network technique [30], and a simple extended equation method [31], have been developed to secure exact analytical solutions for partial nonlinear differential equations to find soliton solutions, [32] F-expansion technique [33], unified auxiliary equation technique (m + G G ) expansion strategy [34], Hirota bilinear technique [35], extended exponential function method [36], generalized exponential rational function method [37], variational iteration method [38], and several others [39][40][41][42][43][44][45][46].…”

Analytical Soliton Solutions of the Coupled Radhakrishnan‐Kundu‐Lakshmanan Equation via Three Techniques

“…The theory and investigation of soliton solutions is one of the important research fields relating to nonlinear partial differential equations ascending in telecommunication engineering, optics, mathematical physics, and other domains of nonlinear sciences. Therefore, diverse academics and researchers developed a number of numerical and analytical techniques, namely, the ðm + 1/G ′ Þ-expansion technique [1], the truncated M-fractional derivative scheme [2], the q-homotopy analysis technique [3], Atangana-Baleanu operator scheme [4], the improved Bernoulli subequation function process [5], the sine-Gordon expansion approach [6], the Haar wavelet technique [7], the biframelet systems process [8], the Lie symmetry technique [9], the generalized exponential rational function mode [10], the Painlevé analysis [11], the extended subequation method [12], the improved ðG′/GÞ-expansion scheme [13], the Hirota simplified method [14], the onedimensional subalgebra system [15], Painlevé analysis and multi-soliton solutions technique [16], the one-parameter Lie group of transformations approach [17], etc.…”