Robustness of localized waves under temporal modulation in higher-dimensional nonlinear evolution equations

Nonlinear evolution equations (NLEEs) are primarily relevant to nonlinear complex physical systems in a wide range of fields, including ocean physics, plasma physics, chemical physics, optical fibers, fluid dynamics, biology physics, solid-state physics, and marine engineering. This paper investigates the Lie symmetry analysis of a generalized (3+1)-dimensional breaking soliton equation depending on five nonzero real parameters. We derive the Lie infinitesimal generators, one-dimensional optimal system, and geometric vector fields via the Lie symmetry technique. First, using the three stages of symmetry reductions, we converted the generalized breaking soliton (GBS) equation into various nonlinear ordinary differential equations (NLODEs), which have the advantage of yielding a large number of exact closed-form solutions. All established closed-form wave solutions include special functional parameter solutions, as well as hyperbolic trigonometric function solutions, trigonometric function solutions, dark-bright solitons, bell-shaped profiles, periodic oscillating wave profiles, combo solitons, singular solitons, wave-wave interaction profiles, and various dynamical wave structures, which we present for the first time in this research. Eventually, the dynamical analysis of some established solutions is revealed through three-dimensional sketches via numerical simulations. Some of the new solutions are often useful and helpful for studying the nonlinear wave propagation and wave-wave interactions of shallow water waves in many new high-dimensional nonlinear evolution equations.

Extend three-wave method for the (1+2)-dimensional Ito equation

Application of Exp-function method to high-dimensional nonlinear evolution equation

In this paper, by means of the potential systems of the given nonlinear evolution equations, a procedure of symmetry preserving discretization of differential equations is presented. The specific process will be given detailed in section 2. This extended method is effective for discreting the high-order (high-dimensional) nonlinear evolution equations. As examples, the invariant difference models of the mKdV equation and the Boussinesq equation are constructed.

Wave profile analysis of a couple of (3+1)-dimensional nonlinear evolution equations by sine-Gordon expansion approach

Conservation laws and symmetries of radial generalized nonlinear p-Laplacian evolution equations

The factorisation method commonly used in linear supersymmetric quantum mechanics is extended, such that it can be applied to nonlinear quantum mechanical systems. The new method is distinguishable from the linear formalism, as the superpotential is forced to become eigenfunction-dependent. Example solutions are given for the nonlinear Schrödinger equation and its supersymmetric partner equation. This method allows new nonlinear evolution equations to be constructed from the solutions of known nonlinear equations, and has the potential to be a useful tool for mathematicians and physicists working in the field of nonlinear systems, allowing the discovery of previously unknown ‘dualities’ amongst nonlinear evolution equations.

Recursive formulae for the analytic solution of the nonlinear spatial conformable fractional evolution equation

The nonlinear Semilinear Dispersive-Fisher (SDF) model is an important equation describing wave dynamics in dispersive media influenced by nonlinear and diffusive effects. This study enhances both analytical and numerical methods to solve the SDF model and validate the solutions. We utilize the Modified Khater (MKhat) and Unified (UF) methods for analytical solutions, and He’s Variational Iteration (HVI) scheme for numerical approximations. Our investigation connects the SDF model to other nonlinear equations like the Korteweg–de Vries (KdV) and Fisher–Kolmogorov (FK) equations, demonstrating the consistency between analytical and numerical solutions. This research contributes to the understanding and modeling of wave phenomena in dispersive media with nonlinear and diffusive dynamics, offering refined analytical and numerical techniques specific to the SDF model. Key contributions include introducing the MKhat and UF methods for analytical solutions and employing the HVI scheme for numerical approximations, which are less represented in current literature. This work is positioned in mathematical physics, focusing on nonlinear evolution equations.

97/04852 Coal pulsating combustion model: Xu, W. Ranshao Kexue Yu Jishu., 1996, 2, (3), 260–269. (In Chinese)

CANM, a program for numerical solution of a system of nonlinear equations using the continuous analog of Newton's method

98/03268 Reopening of pulverized coal injection in blast furnace after a pause

99/00738 On the combustion of heat pump with solar steam turbine

99/00035 ESR study of the macromolecular structure of copper-porphyrin-labeled coal

Time analyticity for solutions of nonlinear abstract parabolic evolution equations