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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 paper studies the optical soliton solutions of the Biswas–Arshed equation with the help of two different techniques, such as the generalised exponential rational function (GERF) technique and Kudryashov's simplest equation technique. The GERF technique extracts distinct families of exact solitary wave solutions involving trigonometric function solutions, hyperbolic function solutions, rational function solutions, etc. After that, we apply Kudryashov's simplest equation method in the context of Bernoulli and Riccati equations to attain different kinds of families of exact soliton solutions. All the acquired solutions of the equation have numerous applications in many branches of nonlinear sciences such as plasma physics, superconductivity, nonlinear optics, biophysics, star formation, quantum mechanics, etc. and many more connected fields of nonlinear wave sciences. The exact solitary wave solutions obtained by GERF technique and Kudryashov's simplest equation technique are in more generalised form as they contained several arbitrary parameters. Subsequently, to understand the behaviour of deduced solutions, we graphically discuss the real part, imaginary part and modulus of these solutions by suitable choice of involved arbitrary parameters. [ABSTRACT FROM AUTHOR]

In the present article, our main aim is to construct abundant exact solutions for the (2 + 1) -dimensional Kadomtsev–Petviashvili-Benjamin–Bona–Mahony (KP-BBM) equation by using two powerful techniques, the Lie symmetry method and the generalised exponential rational function (GERF) method with the help of symbolic computations via Mathematica. Firstly, we have derived infinitesimals, geometric vector fields, commutation relations and optimal system. Therefore, the KP-BBM equation is reduced into several nonlinear ODEs under two stages of symmetry reductions. Furthermore, abundant solutions are obtained in different shapes of single solitons, solitary wave solutions, quasiperiodic wave solitons, elastic multisolitons, dark solitons and bright solitons, which are more relevant, meaningful and useful to describe physical phenomena due to the existence of free parameters and constants. All these generated exact soliton solutions are new and completely different from the previous findings. Moreover, the dynamical behaviour of the obtained exact closed-form solutions is analysed graphically by their 3D, 2D-wave profiles and the corresponding density plots by using the mathematical software, which will be comprehensively used to explain complex physical phenomena in the fields of nonlinear physics, plasma physics, optical physics, mathematical physics, nonlinear dynamics, etc. [ABSTRACT FROM AUTHOR]

This paper is devoted to obtaining some new types of exact solutions of the (2+1)-dimensional Date–Jimbo–Kashiwara–Miwa (DJKM) equation utilizing the Lie symmetry method. All the Lie symmetries, infinitesimal generators, and possible geometric vector fields have been obtained by using the invariance condition of the group-theoretic method. Meanwhile, the Lie symmetry reductions and explicit exact solutions are obtained by a one-dimensional (1D) optimal system. All the obtained exact solutions are absolutely new and completely different from the earlier established results in the literature. Moreover, the dynamical behavior of obtained solitons like doubly solitons, dark solitons, kink wave, curved shaped multi-solitons, parabolic waves, solitary waves, and annihilation of elastic multi-soliton profiles is depicted graphically via interesting 3D-shapes. That will be widely used to provide many more attractive complex physical phenomena in the fields of plasma physics, statistical physics, fiber optics, fluid dynamics, condensed matter physics, and so on. Finally, we have verified all the achieved soliton solutions through symbolic computations with Mathematica. [ABSTRACT FROM AUTHOR]