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In current study, (2+1)-dimensional cubic Klein-Gordon (cKG) equation illustrating dislocation propagation in crystals as well as the behaviour of elementary particles is investigated to establish a variety of new analytic exact solitary wave solutions. Modified exponential expansion method has been implemented to unfold certain wave solutions of considered model. As a result, three sorts of solutions emerge in a fairly systematic manner in the shape of hyperbolic, trigonometric, and rational functions. The kink and periodic wave solitons are acquired and presented geometrically, some 3D plots are simulated and displayed to respond the dynamic behavior of these obtained solutions. In this work we have used symbolic package maxima to obtained our solutions. Our acquired solutions might be most helpful to analyze physical issues that arise from nonlinear complicated dynamical systems.

The extended version of exp(-ψ(κ)) -expansion method is being implemented in a very uniform manner on the (4+1) -dimensional Fokas equation to explore various exact solutions which are new and distinct from literature. Subsequently, hyperbolic, polynomial, trigonometric along with rational function solutions enclosed with arbitrary constants, are produced. Taking some suitable choice of arbitrary constants, the complex dynamics of the obtained solutions are portrayed via 3D plots, which might be used to describe elastic and nonelastic interactions, surface and internal waves in rivers with different physical situations.

The present book of proceedings includes chapters related to the areas of pure, applied and inter-disciplinary mathematics reflecting the potential applications in the domains of sciences and engineering. The main areas include algebra and its applications, analysis and approximation theory, cryptography, computational fluid dynamics, continuum mechanics and vibrations, differential equations and applications, graph theory, fuzzy mathematics and logic, numerical analysis, optimization and its applications, wave propagation, etc. The scientists, engineers, academicians and researchers working in the proposed areas of coding and information theory, computational fluid dynamics, differential equations, fuzzy sets and systems, numerical analysis, optimization, vibrations, etc. looking for new insight and ideas in these areas will be benefitted by the contents of this book. In fact, it will be useful to a large class of readers interested in recent findings related to mathematical sciences and their applications to the diverse domains of knowledge. The book will be a fruitful contribution to all knowledge seekers and researchers in the concerned areas who are looking for new insight and ideas. As it consists of updated research articles on emerging areas of mathematical sciences and their applications, it will be a valued addition to the learning resource centres of various universities, industries, and research and development organizations across the globe.

Background The Zaremaoghaddam model studies internal waves, fluid dynamics, and nonlinear wave equations. In shallow water, internal solitons, or stratified fluids, the procedure may involve modifying or applying nonlinear wave models like Korteweg–de Vries equation, Boussinesq, or Gear–Grimshaw. The Gear–Grimshaw system simulates internal waves in a two-layer stratified fluid, such as an ocean with two density layers, using interconnected nonlinear evolution equations. The Korteweg–de Vries equation is extended to include wave mode interactions. Results The paper recovers solitary waves and shock waves for double-layered fluid flow that is modeled whose basic platform is the Korteweg–de Vries equation. This retrieval is made possible with the usage of the generalized exponential differential rational function approach. Two models for the double-layered flow are taken into consideration, namely the Zaremaoghaddam model and the Gear–Grimshaw model. The parameter restrictions for the existence of such solutions are also enumerated. Conclusions This paper has many implications and opens up many future opportunities. Since double-layered fluid flow never addresses rogue wave features, the paper’s results would be the foundation for studying them. Additionally, viscosity can be considered in the two models in this paper. A practical perspective would result since viscosity is inevitable in any fluid or airflow. Additionally, these models are new. Thus, such models must be investigated by identifying conservation laws and studying soliton perturbation theory.

In the present course of study, we examine a family of Boussinesq equations of distinct structures and dimensions. We investigate the complete integrability of these equations via Painlevé test. Real and complex multiple soliton solutions, for each considered model, are derived by mode of simplified Hirota’s method. Moreover, exponential expansion method has been employed to each equation, resulting into soliton solutions possessing rich spatial structure due to the presence of abundant arbitrary constants.

The present work consists of detailed exploration of modified KdV–Sine-Gordon equation in integrable form, owning to two-component nonlinear channel for modeling laser light propagation. For validating the behavior of this equation in the sense of integrability, we use the Painlevé test. The simplified Hirota’s technique with new complex forms is developed suitably to construct multiple-soliton solutions with complex structure for considered equation. Moreover, Lie symmetry analysis has been implemented for perceiving symmetries of MKdV–SG equation and then culminating the invariant solitary wave solutions. The new findings obviously reveal that simplified Hirota’s technique with complex structure would be highly proficient for fabricating new multiple complex soliton solutions to other nonlinear equations with integrable properties from mathematical physics and dynamical systems community.

In present work, new form of generalized fifth-order nonlinear integrable equation has been investigated by locating movable critical points with aid of Painlevé analysis and it has been found that this equation passes Painlevé test for α = β which implies affirmation toward the complete integrability. Lie symmetry analysis is implemented to obtain the infinitesimals of the group of transformations of underlying equation, which has been further pre-owned to furnish reduced ordinary differential equations. These are then used to establish new abundant exact group-invariant solutions involving various arbitrary constants in a uniform manner.

Lie group formalism is applied to Einstein field equations for perfect fluid distribution and pure radiation fields in the investigation of symmetries and exact solutions. The similarity reductions are obtained by determining the complete sets of point symmetries of these equations. The reduced ordinary differential equations are further studied and some non-trivial exact solutions are successfully furnished.