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The purpose of this paper is to find an exact analytical expression for the periodic solutions of the double-hump Duffing equation and an expression for the period of these solutions.

The double-hump Duffing equation is presented as a Hamiltonian system and a phase portrait of this system has been found. On the ground of analytical calculations performed using Hamiltonian-based technique, the periodic solutions of this system are represented by Jacobi elliptic functions sn, cn and dn.

Expressions for the periodic solutions and their periods of the double-hump Duffing equation have been found. An expression for the solution, in the time domain, corresponding to the heteroclinic trajectory has also been found. An important element in various applications is the relationship obtained between constant Hamiltonian levels and the elliptic modulus of the elliptic functions.

The results obtained in this paper represent a generalization and improvement of the existing ones. They can find various applications, such as analysis of limit cycles in perturbed Duffing equation, analysis of damped and forced Duffing equation, analysis of nonlinear resonance and analysis of coupled Duffing equations.

The purpose of this paper is to propose a Chebyshev collocation method (CCM) for Hallén’s equation of thin wire antennas.

Since the current induced on the thin wire antennas behaves like the square root of the distance from the end, a smoothed current is used to annihilate this end effect. Then the CCM adopts Chebyshev polynomials to approximate the smoothed current from which the actual current can be quickly recovered. To handle the difficulty of the kernel singularity and to realize fast computation, a decomposition is adopted by separating the singularity from the exact kernel. The integrals including the singularity in the linear system can be given in an explicit formula while the others can be evaluated efficiently by the fast cosine transform or the fast Fourier transform.

The CCM convergence rate is fast and this method is more efficient than the other existing methods. Specially, it can attain less than 1 percent relative errors by using 32 basis functions when a/h is bigger than 2×10−5 where h is the half length of wire antenna and a is the radius of antenna. Besides, a new efficient scheme to evaluate the exact kernel has been proposed by comparing with most of the literature methods.

Since the kernel evaluation is vital to the solution of Hallén’s and Pocklington’s equations, the proposed scheme to evaluate the exact kernel may be helpful in improving the efficiency of existing methods in the study of wire antennas. Due to the good convergence and efficiency, the CCM may be a competitive method in the analysis of radiation properties of thin wire antennas. Several numerical experiments are presented to validate the proposed method.

Fractional order nonlinear evolution equations (FNLEEs) pertaining to conformable fractional derivative are considered to be revealed for well-furnished analytic solutions due to their importance in the nature of real world. In this article, the autors suggest a productive technique, called the rational fractional (DξαG/G)-expansion method, to unravel the nonlinear space-time fractional potential Kadomtsev–Petviashvili (PKP) equation, the nonlinear space-time fractional Sharma–Tasso–Olver (STO) equation and the nonlinear space-time fractional Kolmogorov–Petrovskii–Piskunov (KPP) equation. A fractional complex transformation technique is used to convert the considered equations into the fractional order ordinary differential equation. Then the method is employed to make available their solutions. The constructed solutions in terms of trigonometric function, hyperbolic function and rational function are claimed to be fresh and further general in closed form. These solutions might play important roles to depict the complex physical phenomena arise in physics, mathematical physics and engineering.

The rational fractional (DξαG/G)-expansion method shows high performance and might be used as a strong tool to unravel any other FNLEEs. This method is of the form U(ξ)=∑i=0nai(DξαG/G)i/∑i=0nbi(DξαG/G)i.

Achieved fresh and further abundant closed form traveling wave solutions to analyze the inner mechanisms of complex phenomenon in nature world which will bear a significant role in the of research and will be recorded in the literature.

The rational fractional (DξαG/G)-expansion method shows high performance and might be used as a strong tool to unravel any other FNLEEs. This method is newly established and productive.

The purpose of this paper is to discuss the homoclinic breathe-wave solutions and the singular periodic solutions for (2 + 1)-dimensional generalized shallow water wave (GSWW) equation.

The Hirota bilinear method, the Lie symmetry method and the non-Lie symmetry method are applied to the (2 + 1)D GSWW equation.

A reduced (1 + 1)D potential KdV equation can be derived, and its soliton solutions are also presented.

As a typical nonlinear evolution equation, some dynamical behaviors are also discussed.

These results are very useful for investigating some localized geometry structures of dynamical behaviors and enriching dynamical features of solutions for the higher dimensional systems.

The purpose of this paper is to propose a numerical scheme based on forward finite difference, quasi-linearisation process and polynomial differential quadrature method to find the numerical solutions of nonlinear Klein-Gordon equation with Dirichlet and Neumann boundary condition.

In first step, time derivative is discretised by forward difference method. Then, quasi-linearisation process is used to tackle the non-linearity in the equation. Finally, fully discretisation by differential quadrature method (DQM) leads to a system of linear equations which is solved by Gauss-elimination method.

The accuracy of the proposed method is demonstrated by several test examples. The numerical results are found to be in good agreement with the exact solutions and the numerical solutions exist in literature. The proposed scheme can be expended for multidimensional problems.

The main advantage of the present scheme is that the scheme gives very accurate and similar results to the exact solutions by choosing less number of grid points. Secondly, the scheme gives better accuracy than (Dehghan and Shokri, 2009; Pekmen and Tezer-Sezgin, 2012) by choosing less number of grid points and big time step length. Also, the scheme can be extended for multidimensional problems.

The purpose of this paper is to use He's Exp‐function method (EFM) to construct solitary and soliton solutions of the nonlinear evolution equation.

This technique is straightforward and simple to use and is a powerful method to overcome some difficulties in the nonlinear problems.

This method is developed for searching exact traveling wave solutions of the nonlinear partial differential equations. The EFM presents a wider applicability for handling nonlinear wave equations.

The paper shows that EFM, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear evolution equations. Application of EFM to Fitzhugh‐Nagumo equation illustrates its effectiveness.

The purpose of this paper is to apply the exp‐function method to construct exact solutions of nonlinear wave equations. The proposed technique is tested on the (2+1) and (3+1) dimensional extended shallow water wave equations. These equations play a very important role in mathematical physics and engineering sciences.

In this paper, the authors apply the exp‐function method to construct exact solutions of nonlinear wave equations.

In total, four forms of the extended shallow water wave equation have been studied, from the point of view of its exact solutions using computational method. Exp‐function method was employed to achieve the goal set for this work. The applied method will be used in further works to establish more entirely new solutions for other kinds of nonlinear wave equations. Finally, it is worthwhile to mention that the proposed method is straightforward, concise, and it is a promising and powerful new method for other nonlinear wave equations in mathematical physics.

The algorithm suggested in the paper is quite efficient and is practically well suited for use in these problems. The method is straightforward and concise, and its applications are promising.

The purpose of this study is to investigate the nonlinear Schrödinger equation (NLS) incorporating spatiotemporal dispersion and other dispersive effects. The goal is to derive various soliton solutions, including bright, dark, singular, periodic and exponential solitons, to enhance the understanding of soliton propagation dynamics in nonlinear metamaterials (MMs) and contribute new findings to the field of nonlinear optics.

The research uses a range of powerful mathematical approaches to solve the NLS. The proposed methodologies are applied systematically to derive a variety of optical soliton solutions, each demonstrating unique optical behaviors and characteristics. The approach ensures that both the theoretical framework and practical implications of the solutions are thoroughly explored.

The study successfully derives several types of soliton solutions using the aforementioned mathematical approaches. Key findings include bright optical envelope solitons, dark optical envelope solitons, periodic solutions, singular solutions and exponential solutions. These results offer new insights into the behavior of ultrashort solitons in nonlinear MMs, potentially aiding further research and applications in nonlinear wave studies.

This study makes an original contribution to nonlinear optics by deriving new soliton solutions for the NLS with spatiotemporal dispersion. The diversity of solutions, including bright, dark, periodic, singular and exponential solitons, adds substantial value to the existing body of knowledge. The use of distinct and reliable methodologies to obtain these solutions underscores the novelty and potential applications of the research in advancing optical technologies. The originality lies in the novel approaches used to obtain these diverse soliton solutions and their potential impact on the study and application of nonlinear waves in MMs.

Under this heading are published regularly abstracts of all Reports and Memoranda of the Aeronautical Research Committee, Reports and Technical Notes of the U.S. National Advisory Committee for Aeronautics, and publications of other similar research bodies as issued