Invariance analysis, optimal system, closed-form solutions and dynamical wave structures of a (2+1)-dimensional dissipative long wave system

Abstract

By employing the one-parameter Lie group of transformations method, abundant exact invariant solutions are obtained for a (2+1)-dimensional dissipative long wave (DLW) system, which describes the water wave model of hydrodynamics with wide channels or open seas of finite depth and can also be used to illustrate nonlinear wave propagation in the dissipative medium. Initially, we derived the Lie symmetries, geometric vector fields, and commutative table, and adjoint relations of the examined vectors for the system. A one-dimensional optimal system of symmetry subalgebras is well-structured. It leads to the reduction of independent variables of the considered DLW system and leaves the system invariant. The (2+1)-dimensional DLW system is reduced into nonlinear ordinary differential equations (NLODEs) for the various subalgebras obtained through two phases of similarity reductions. Some novel closed-form solutions are obtained, including arbitrary independent functional parameters, and have never been reported in the literature. Moreover, we depict the dynamical behaviors of obtained closed-form invariant solutions like multi-wave solitons, single solitons, breather-waves, rogue waves, bell-shaped structures, Kink-waves, and Lump-waves, wavewave interactions, and annihilation of Kink-wave profiles through three-dimensional graphics. Eventually, a comparison with other results is discussed.

Lie group of invariance technique for analyzing propagation of strong shock wave in a rotating non‐ideal gas with azimuthal magnetic field

“…Some valuable results and related topics are studied in previous studies 36–42 . One can see the work based on the Lie group of transformation method for solitary waves and shock waves in previous studies 43–48 . According to the authors' knowledge, the problem under consideration has not been solved yet by the similarity method, which makes this work different from previous works.…”

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.…”

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.…”