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The main goal of this research work is to explore novel soliton solutions and dynamics of solitonic structures to asymmetric Nizhnik–Novikov–Veselov (ANNV) equations by two methods, namely, the generalized exponential rational function (GERF) method and the modified extended tanh expansion (METE) method. These techniques are the most effective and reliable tools for solving nonlinear equations with partial derivatives. The used methods provide a wide range of soliton solutions that can motivate applied scientists and researchers for these specific structures. These acquired solutions are in the form of exponential, trigonometric and hyperbolic functions including, tan , sin , sec , cot , csc , sinh , tanh , sech , cosh , coth , csch and of their combinations. It has also been observed that the obtained soliton solutions of the ANNV equations are bell-shape, anti-bell-shape, periodic solution, multisoliton and different types of soliton solutions. To illustrate the physical features and dynamics characteristics of some obtained solutions, three-dimensional (3D) and two-dimensional (2D) figures are exhibited through the Mathematica software. The methods applied in this paper are suitable to study the soliton solutions for the ANNV equation without producing the complexities in some other known analytical method. Finally, the employed methods can be earmarked easily for solving various classes of nonlinear evolution equations appearing in mathematical physics, fluid dynamics, plasma physics, nonlinear waves and nonlinear sciences. [ABSTRACT FROM AUTHOR]

This research studies the Kudryashov's equation by using the generalised exponential rational function (GERF) technique. This technique allows us to use the traveling wave reduction for Kudryashov's equation which provides the system of equations associated with the real and imaginary parts of the profile pulse for a complex function. First and foremost, we reduced these systems of equations into nonlinear ordinary differential equations (ODE) under the travelling wave transformations. The critical steps of the GERF technique are then applied to the resulting nonlinear ODE. With the assistance of the mentioned technique, we obtained different kinds of families of optical soliton solutions for Kudryashov's equation. The real part, imaginary part and modulus of some of the obtained solutions are investigated/explored by the three-dimensional visual representations under the appropriate choice of the involved arbitrary parameters with suitable range space, demonstrating that the solutions represent a series of solitons, bright solitons, dark solitons, singular solitons, the interaction of solitons with solitary wave profiles, the interaction of kink waves with solitons and the interaction of kink waves with solitons. [ABSTRACT FROM AUTHOR]

This investigation focuses on two novel Kadomtsev–Petviashvili (KP) equations with time-dependent variable coefficients that describe the nonlinear wave propagation of small-amplitude surface waves in narrow channels or large straits with slowly varying width and depth and non-vanishing vorticity. These two variable coefficients, Kadomtsev–Petviashvili (VCKP) equations in (2+1)-dimensions, are the main extensions of the KP equation. Applying the Lie symmetry technique, we carry out infinitesimal generators, potential vector fields, and various similarity reductions of the considered VCKP equations. These VCKP equations are converted into nonlinear ODEs via two similarity reductions. The closed-form analytic solutions are achieved, including in the shape of distinct complex wave structures of solitons, dark and bright soliton shapes, double W-shaped soliton shapes, multi-peakon shapes, curved-shaped multi-wave solitons, and novel solitary wave solitons. All the obtained solutions are verified and validated by using back substitution to the original equation through Wolfram Mathematica. We analyze the dynamical behaviors of these obtained solutions with some three-dimensional graphics via numerical simulation. The obtained variable coefficient solutions are more relevant and useful for understanding the dynamical structures of nonlinear KP equations and shallow water wave models. [ABSTRACT FROM AUTHOR]