A Study of (2+1)-Dimensional Konopelchenko-Dubrovsky (KD) System: Closed-Form Solutions, Solitary Waves, Bifurcation Analysis and Quasi-Periodic Solution

Nonlinear evolution equations (NLEEs) are extensively used to establish the elementary propositions of natural circumstances. In this work, we study the Konopelchenko-Dubrovsky (KD) equation which depicts non-linear waves in mathematical physics with weak dispersion. The considered model is investigated using the combination of generalized exponential rational function (GERF) method and dynamical system method. The GERF method is utilized to generate closedform invariant solutions to the (2+1)-dimensional KD model in terms of trigonometric, hyperbolic, and exponential forms with the assistance of symbolic computations. Moreover, three-dimensional graphics are displayed to depict the behavior of obtained solitary wave solutions. The model is observed to have single and multiple soliton profiles, kink-wave profiles, and periodic oscillating nonlinear waves. These generated solutions have never been published in the literature. All the newly generated soliton solutions are checked by putting them back into the associated system with the soft computation via Wolfram Mathematica. Moreover, the system is converted into a planer dynamical system using a certain transformation and the analysis of bifurcation is examined. Furthermore, the quasi-periodic solution is investigated numerically for the perturbed system by inserting definite periodic forces into the considered model. With regard to the parameter of the perturbed model, two-dimensional and three-dimensional phase portraits are plotted.

Fractal Characteristic Evolution of Coastal Settlement Land Use: A Case of Xiamen, China

Coastal settlements in urban areas show certain degrees of spatial complexity. Understanding the evolution law of fractal settlements is practically important for marine engineering and urban planning. In this paper, we investigate the fractal evolution of coastal settlement land use based on fractal theory. The fractal dimensions of the land uses for three typically coastal settlements in Xiamen city, China, are obtained to quantify their spatial complexity. The results reveal the fractal characteristics and regional differences of the coastal settlements. Furthermore, nonlinear modeling is applied to describe the fractal dimension evolution of the coastal settlement land uses from 2000 to 2018. Three settlements in rapid urbanization show different nonlinear evolution equations of the fractal dimension due to their different land uses. This study might provide a theoretical basis for understanding the fractal characteristic evolution of coastal settlements in urban areas and show its potential application in urban geography.

Analytical solutions of the fifth-order time fractional nonlinear evolution equations by the unified method

This key purpose of this study is to investigate soliton solution of the fifth-order Sawada–Kotera and Caudrey–Dodd–Gibbon equations in the sense of time fractional local [Formula: see text]-derivatives. This important goal is achieved by employing the unified method. As a result, a number of dark and rational soliton solutions to the nonlinear model are retrieved. Some of the achieved solutions are illustrated graphically in order to fully understand their physical behavior. The results demonstrate that the presented approach is more effective in solving issues in mathematical physics and other fields.

A Study on Controllability of a Class of Impulsive Fractional Nonlinear Evolution Equations with Delay in Banach Spaces

Under a new generalized definition of exact controllability we introduced and with a appropriately constructed time delay term in a special complete space to overcome the delay-induced-difficulty, we establish the sufficient conditions of the exact controllability for a class of impulsive fractional nonlinear evolution equations with delay by using the resolvent operator theory and the theory of nonlinear functional analysis. Nonlinearity in the system is only supposed to be continuous rather than Lipschitz continuous by contrast. The results obtained in the present work are generalizations and continuations of the recent results on this issue. Further, an example is presented to show the effectiveness of the new results.

Kink solutions of two generalized fifth-order nonlinear evolution equations

In this paper, two generalized fifth-order nonlinear evolution equations are introduced and investigated: One is (1+1)-dimensional, the other is (2+1)-dimensional. The Hereman–Nuseir method is used to derive the multiple kink solutions and singular kink solutions, and the conditions for the cases of complete integrability of these two equations. Meanwhile, it is found that these equations have completely different dispersion relations and physical structures. The corresponding graphs with specific parameters are given to show the effectiveness and validness of the obtained results.

Solitary wave solutions along with Painleve analysis for the Ablowitz–Kaup–Newell–Segur water waves equation

This study focuses on the Ablowitz–Kaup–Newell–Segur (AKNS) water waves equation. Painleve test (P-test) will be implemented to check the integrability of AKNS equation and an extended modified auxiliary equation mapping (EMAEM) architectonic is implemented to get a new set of traveling wave solutions like periodic and doubly periodic, bell type, kink, singular kink, anti-kink, trigonometric, singular, rational, combined soliton like solutions and hyperbolic solutions. Furthermore, it is analyzed that the implemented algorithm is efficient and accurate for solving nonlinear evolution equations (NLEEs). Finally, graphical simulations (2D, 3D and contours) are also provided to illustrate the detailed behavior of the solution and effectiveness of the proposed method.

Dynamics of exact closed-form solutions to the Schamel Burgers and Schamel equations with constant coefficients using a novel analytical approach

In this paper, we investigate two different constant-coefficient nonlinear evolution equations, namely the Schamel Burgers equation and the Schamel equation. These models also have a great deal of potential for studying ion-acoustic waves in plasma physics and fluid dynamics. The primary goal of this paper is to establish closed-form solutions and dynamics of analytical solutions to the Schamel Burgers and the Schamel equations, which are special examples of the well-known Schamel–Korteweg-de Vries (S-KdV) equation. We derive completely novel solutions to the considered models using a variety of computation programmes and a newly proposed extended generalized [Formula: see text] expansion approach. The newly formed solutions, which include hyperbolic and trigonometric functions as well as rational function solutions, have been produced. The annihilation of three-dimensional shock waves, periodic waves, single soliton, singular soliton, and combo soliton, multisoliton as well as their three-dimensional and contour plots are used to show the dynamical representations of the acquired solutions. These results demonstrate that the proposed extended technique is efficient, reliable and simple.

Analytical soliton solutions to the generalized (3+1)-dimensional shallow water wave equation

In this paper, the soliton solutions and dynamical wave structures for the generalized (3+1)-dimensional shallow water wave (SWW) equation, which is an important physical property in ocean engineering and hydrodynamics, are presented. The generalized exponential rational function (GERF) method is used to investigate the closed-form wave solutions of the generalized SWW equation, which is used to describe the evolutionary dynamics of SWW. We successfully archive a variety of soliton solutions such as exponential solutions, kink wave solutions, non-topological solutions, periodic singular solutions, and topological solutions. These newly established results are also important for understanding the wave-propagation and dynamics of exact solutions of the equation, which is of great significance in physical oceanography and chemical oceanography. Eventually, it is shown that the proposed GERF technique is effective, robust, and straightforward and is also used to solve other types of higher-dimensional nonlinear evolution equations. In our work, we have used Mathematica extensively for such complicated algebraic calculations.

Unified method applied to the new Hamiltonian amplitude equation: Wave solutions and stability analysis

The new Hamiltonian amplitude (nHA) equation deals with some of the disabilities of the modulation wave-train. The main task of this paper is to extract the analytical wave solutions of the nHA equation. Based on the unified scheme, analytical wave solutions are attained in terms of hyperbolic and trigonometric function solutions. In order to prompt the underlying wave propagation characteristics, three-dimensional (3D), two-dimensional (2D) are illustrated from the solutions obtained with the help of computational packages Mathematica and also made comparisons between wave profiles for various values. The proposed method can also be used for many other nonlinear evolution equations.