Exact closed-form solutions and dynamics of solitons for a $$(2+1)$$-dimensional universal hierarchy equation via Lie approach

Abstract

The dynamics of localised solitary wave solutions play an essential role in the fields of mathematical sciences such as optical physics, plasma physics, nonlinear dynamics and many others. The prime objective of this study is to obtain localised solitary wave solutions and exact closed-form solutions of the (2 + 1)-dimensional universal hierarchy equation (UHE) using the Lie symmetry approach. Besides, the Lie infinitesimals, all the vector fields, commutation relations of Lie algebra and symmetry reductions are derived via the Lie transformation method. Meanwhile, the universal hierarchy equation is reduced into nonlinear ODEs through two stages of symmetry reductions. The closed-form invariant solutions are attained under some parametric conditions imposed on infinitesimal generators. Because of the presence of arbitrary independent functional parameters and other constants, these group-invariant solutions are explicitly displayed in terms of arbitrary functions that are more relevant, beneficial and useful for explaining nonlinear complex physical phenomena. Furthermore, the dynamical structures of the obtained exact solutions are illustrated for suitable values of arbitrary constants through 3D-plots based on numerical simulation. Some of these localised solitary waves are double solitons, periodic lump solitons, dark solitons, five-solitons, hemispherical solitons and lump-type solitons.

Abundant Bounded and Unbounded Solitary, Periodic, Rogue‐Type Wave Solutions and Analysis of Parametric Effect on the Solutions to Nonlinear Klein–Gordon Model

“…ese solutions always provide different types of solitary waves. Solution ( 14) is complex if α 2 k 2 < ω 2 and it can be expressed as the trigonometric function in (15) and ( 16). e profile in Figures 5(a For the condition α 2 k 2 > ω 2 , solution ( 14) can be expressed in the hyperbolic form in (18), (19), and (20).…”

“…Many scientific experimental models are employed in nonlinear differential form from the phenomena of nonlinear fiber optics, high-amplitude waves, fluids, plasma, solid state particle motions, etc. Surveying literature, we realized ideas that many scientists worked to disclose innovative, efficient techniques for explaining internal behaviors of NLDEs with constant coefficients that are significant to elucidate different intricate problems such as a discrete algebraic framework [1], IRM-CG method [2], transformed rational function scheme [3], fractional residual method [4], new multistage technique [5], new analytical technique [6], extended tanh approach [7], Hirota-bilinear approach [8][9][10], multi exp-expansion method [11,12], Jacobi elliptic expansion method [13,14], Lie approach [15], Lie symmetry analysis techniques [16], generalized Kudryashov scheme [17,18], generalized exponential rational function scheme [19], MSE method [20][21][22], and many more. Such or similar schemes are also used to solve the model with variable coefficients to visualize various new nonlinear dynamics [23][24][25].…”

New Fractal Soliton Solutions and Sensitivity Visualization for Double-Chain DNA Model

“…Many physicists and mathematicians have worked hard to develop further precise alternatives to NLPDEs for a better understanding of these processes. Therefore, exact solutions of NLPDEs are essential for exploring physical explanations and qualitative aspects of different mechanisms [8][9][10][11][12][13][14][15][16][17][18]. These solutions demonstrate the dynamics of several nonlinear complex models symbolically and physically.…”