Narrow Results

In this paper, we are going to investigate the (3+1)-dimensional nonlinear modified Korteweg–de Vries-Zakharov–Kuznetsov (mKdV–ZK) equation, which governs the behavior of weakly nonlinear ion acoustic waves in magnetized electron–positron plasma. By taking advantage of the newly proposed modified Sardar sub-equation method, we derive a comprehensive set of exact soliton solutions to the mKdV–ZK equation. Additionally, we provide graphical representations of the solutions, including 2D, 3D, and contour plots, to visualize the characteristics and features of these nonlinear wave structures. These solutions encompass a diverse range of wave patterns, including traveling waves, bright solitons, periodic waves, dark–bright solitons, lump-type solitons, and multi-soliton solutions. The obtained solutions provide valuable insights into the nonlinear behaviors and dynamics exhibited by the mKdV–ZK equation. The success of the new modified Sardar sub-equation method in obtaining a diverse range of solutions for the $(3+1)$-dimensional mKdV–ZK equation highlights its potential for applications in the analysis of various nonlinear systems in plasma physics and beyond. Also, the study reviewed the superiority of the modified method compared to the Sardar sub-equation method.

In this paper, we present a novel approach for obtaining optical soliton solutions of the higher-order (2+1)-dimensional Schrödinger equation with a fractional beta time derivative. This model is significant as it accurately simulates complex physical phenomena such as the propagation of optical pulses in non-homogeneous media, which is crucial for advancing technologies in optical communications and information processing. The improved modified extended tanh method is employed to derive various types of solutions, including bright solitons, dark solitons, periodic solutions, and exponential solutions. The obtained solutions are graphically simulated for different values of $\beta$ to showcase the impact of the fractional derivative on the model and its relevance to real-world applications.

This paper proposes a novel extended methodology, establishing the new extended Generalized Exponential Rational Function Method (GERFM), which is known as a “new extended GERFM” for solving the Painleve-integrable $(3+1)$D generalized nonlinear evolution equation. We enhance GERFM by incorporating additional terms that combine functions and derivatives, making it a more generalized and efficient approach for solving nonlinear partial differential equations. Moreover, the unified method is applied to thoroughly delve deeper into the equation’s properties to understand its mathematical characteristics and potential solutions. This approach aims to systematically investigate and elucidate the equation’s inherent features, providing a more complete insight into its behavior and the nature of its solutions. Utilizing these methods, we have derived new exact solutions that involve exponential, trigonometric, hyperbolic, polynomial, and rational functions with additional parameters to the model. To emphasize the physical importance of these solutions, we present three-dimensional (3D) and contour plots, investigating different parameter choices. The graphs illustrate various wave patterns such as irregular periodic solitons, rogue waves, lumps, and the interaction of solitary and lump waves based on different parameter values. This visualization method enriches our comprehension of the solutions and facilitates a thorough exploration of how they could be applied to the generation of prolonged waves with small magnitudes in plasma physics, fluid dynamics, and other media with weak dispersion.

In this work, the dynamical structure for the extended equation is analyzed through unified Riccati equation expansion (UREE) and the Lie isomorphism method for the (3+1)-dimensional Wazwaz–Benjamin–Bona–Mahony (WBBM) equation. This equation represents the unidimensional propagation of short amplitude long waves on the water’s surface in a medium. These employed techniques are the most powerful and effective ways to obtain different sets of new and more generalized exact soliton solutions of the WBBM equation. Furthermore, what distinguishes this study from other studies is that it not only acquires a variety of analytical wave solutions for the studied models but also, demonstrates the interaction phenomena for these results as they propagate over time. Also, shows various meaningful graphs of the processes that provide valuable wisdom for understanding their behavior. The UREE method directly provides various new exact soliton solutions with some novel dynamical properties. We perform a detailed Lie symmetry analysis to governing equation that leaves the system invariant. The Lie group method explores six Lie isomorphism groups to study the WBBM equation. First, we find infinitesimal transformations employing the one-parameter Lie symmetry method. Second, we solve the infinitesimal generator and reduce the order of the equation. Moreover, we illustrate some two-dimensional (2D), three-dimensional (3D), and contour diagrams of the obtained results and compute the exact analytical solution utilizing the used methods. To find novel solutions, the Adomian method is also used, where the Adomian polynomials are utilized to deal with nonlinear terms. Variety of new analytical solutions with different types of dynamical behavior are analyzed by utilizing the computational software like Mathematica. These new analytical exact wave solutions are demonstrated in various dynamical structures of periodic wave soliton, interaction periodic wave and kink wave soliton, lump wave soliton, doubly soliton, multi-wave soliton, kink periodic, parabolic wave, multisoliton, traveling wave, and standing wave-shaped profiles.

In this paper, we use a newly formed generalized exponential differential rational function (GEDRF) method to show the dynamic patterns of closed-form analytical solutions to the (4+1)-dimensional Davey–Stewartson–Kadomtsev–Petviashvili (DSKP) equation. To help comprehend these solutions, we have also provided extensive three-dimensional and contour illustrations that show their distinctive characteristics such as lump forms, soliton forms, and interactions between solitons and newly formed waves. These visual representations enhance the comprehension of the solutions behaviors and properties. The derived generalized soliton solutions have extensive applications across various domains: in oceanography for modeling wave dynamics, in physics for studying wave propagation in different media, in engineering for the development of wave-related technologies, and in nonlinear dynamics for theoretical advancements. We used MATHEMATICA to show the multiple dynamic formations of soliton solutions, such as multi-peakons, multi-lumps, and interactions between them, which have been generated with various parameter values using numerical simulations and symbolic computations. Solitons can be found in a wide range of fields, including plasma physics, oceanography, and optical fibers. These findings highlight the versatility and significance of the DSKP equation within various scientific and related technology areas.

This study examines the dynamical equation of the modified Korteweg–de Vries–Zakharov–Kuznetsov (mKdV-ZK), which is used to describe wave propagation in a dispersive and nonlinear medium. This equation is an extension of the well-known KdV-ZK equation, which has been extensively studied in the literature. In this study, we examine the solitary wave solutions of the (3+1)-dimensional mKdV-ZK equation using two analytical techniques: the generalized $\mathrm{\text{exp}}(-\varphi (\xi ))$-expansion approach and the two-variable $(G^{\prime }/G,1/G)$-expansion techniques. As a result, novel soliton solutions in a variety of forms, including Kink- and anti-Kink-type breather waves, dark and bright solitons, Kink soliton and multi-peak solitons, etc. are attained. The solitary wave solutions (which represent the electrostatic field potential), quantum statistical pressures, electric fields and magnetic fields are accomplished with the use of software. These solutions have numerous applications in various areas of physics and other sciences. These results also have applications in electromagnetic wave propagation, nonlinear optics, and plasma physics. Graphical representations of these results have also been presented. These results demonstrate the effectiveness of the two-variable expansion strategy, which will also be useful in solving many other nonlinear models that arise in mathematical physics and several other applied sciences fields. This work contributes to the advancement of novel wave manipulation and control methods, the construction of improved photonic devices for sensing and communications, and plasma confinement in fusion devices, among other uses.

In this paper, the $(4+1)$-dimensional Fokas model is investigated analytically with extensive applications in nonlinear wave theory including the evolution of a three-dimensional wave packet in water with a finite depth, and surface waves and internal waves in straits or channels of varied depth and width. Applying the Hirota bilinear approach, the lump wave solitons and interaction of lump with periodic waves, the interactions of lump wave along single, double-kink solitons as well as the interactions of lump, periodic along double-kink solitons of the governing model are developed. Additionally, by using the extended tanh function technique, certain new traveling wave solitons are established. The physical nature of several solitons is illustrated by drawing the 3D, contours and 2D plots. Consequently, a set of periodic, bright, dark, rational as well as elliptic function solitons are constructed. The employed techniques seem to be more effective and well organized to build some interesting analytical wave solitons to several appealing nonlinear models.

We study $h$-almost conformal $\omega$-Ricci–Bourguignon solitons and $h$-almost gradient conformal $\omega$-Ricci–Bourguignon solitons in spacetimes of general relativity. At first, we provide an example of $h$-almost conformal $\omega$-Ricci–Bourguignon soliton. Next, we prove that if a generalized Robertson–Walker spacetime admits an $h$-almost conformal $\omega$-Ricci–Bourguignon soliton or an $h$-almost gradient conformal $\omega$-Ricci–Bourguignon soliton, then it becomes a perfect fluid spacetime. Moreover, we have observed that a spacetime with $h$-almost conformal $\omega$-Ricci–Bourguignon soliton whose potential vector field is a conformal vector field is an $\omega$-Einstein manifold.

We prove the existence and stability of non-topological solitons in a class of weakly coupled non-linear Klein–Gordon–Maxwell equations. These equations arise from coupling non-linear Klein–Gordon equations to Maxwell's equations for electromagnetism.

The purpose of this article is to give a streamlined and self-contained treatment of the long-time asymptotics of the Toda lattice for decaying initial data in the soliton and in the similarity region via the method of nonlinear steepest descent.

We developed an efficient hybrid mode expansion method to study the maximum tunneling current as a function of the external magnetic field for a 2D large area lateral window junction. We consider the inhomogeneity in the critical current density, which is taken a piecewise constant. The natural modes of the expansion in y, are the linearized eigen-modes around a static solution which satisfies the 1D sine-Gordon equation with the critical current variation in y, and the boundary conditions determined by the overlap component of the bias current, which can be inline or overlap like. The magnetic field along with the inline component of the bias current enters as a boundary condition on the modal amplitudes. We obtain fast convergent results and for a ratio of idle to window widths of w₀/w = 4 (in units of λ_J), only two modes are needed. A simple scaling is obtained for the maximum tunneling current as we vary the idle region width. We also present the linear electromagnetic waveguide modes taking into account the variation normal to the waveguide of the critical current and the capacitance.

We study the nature of collective excitations in classical anharmonic lattices with aperiodic and pseudo-random harmonic spring constants. The aperiodicity was introduced in the harmonic potential by using a sinusoidal function whose phase varies as a power-law, ϕ ∝ n^ν, where n labels the positions along the chain. In the absence of anharmonicity, we numerically demonstrate the existence of extended states and energy propagation for a sufficiently large degree of aperiodicity. Calculations were done by using the transfer matrix formalism (TMF), exact diagonalization and numerical solution of the Hamilton's equations. When nonlinearity is switched on, we numerically obtain a rich framework involving stable and unstable solitons.

Two-point nonlinear boundary value problems (BVPs) in both unbounded and bounded domains are solved in this paper using fast numerical antiderivatives and derivatives of functions of L²(-∞, ∞). This differintegral scheme uses a new algorithm to compute the Fourier transform. As examples we solve a fourth-order two-point boundary value problem (BVP) and compute the shape of the soliton solutions of a one-dimensional generalized Korteweg–de Vries (KdV) equation.

Optical computing devices can be implemented based on controlled generation of soliton trains in single and multicomponent Bose–Einstein condensates (BEC). Our concepts utilize the phenomenon that the frequency of soliton trains in BEC can be governed by changing interactions within the atom cloud [F. Pinsker, N. G. Berloff and V. M. Pérez-García, Phys. Rev. A87, 053624 (2013), arXiv:1305.4097]. We use this property to store numbers in terms of those frequencies for a short time until observation. The properties of soliton trains can be changed in an intended way by other components of BEC occupying comparable states or via phase engineering. We elucidate, in which sense, such an additional degree of freedom can be regarded as a tool for controlled manipulation of data. Finally, the outcome of any manipulation made is read out by observing the signature within the density profile.

In this paper, a meshless spectral radial point interpolation (MSRPI) method using weighted $\theta$-scheme is formulated for the numerical solutions of a class of nonlinear Kawahara-type evolutionary equations. The formulated method is applied for simulation of single and double solitary waves motion, wave generation and oscillatory shock waves propagation. Quality of approximation is measured via discrete $L_{\infty }$, $L_2$ and $L_{\mathrm{\text{rms}}}$ error norms. Three invariant quantities corresponding to mass, momentum and energy are also computed for the method validation. Stability analysis of the proposed method is briefly discussed and verified computationally. Comparison of the obtained results are made with other existing results in the literature revealing the method superiority.

We consider the scattering of solitons and antisolitons in a class of models in (2+1) dimensions. We point out that although in general the interaction forces between solitons and antisolitons are attractive, in some σ models they depend on the relative orientation between the solitons and antisolitons in their internal space. We discuss the scattering properties of such solitons and antisolitons.

We tested the parallelization of explicit schemes for the solution of non-linear classical field theories of complex scalar fields which are capable of simulating hadronic collisions. Our attention focused on collisions in a fractional model with a particularly rich inelastic spectrum of final states. Relativistic collisions of all types were performed by computer on large lattices (64 to 256 sites per dimension). The stability and accuracy of the objects were tested by the use of two other methods of solutions: Pseudo-spectral and semi-implicit. Parallelization of the Fortran code on a 64-transputer MIMD Volvox machine revealed, for certain topologies, communication deadlock and less-than-optimum routing strategies when the number of transputers used was less than the maximum. The observed speedup, for N transputers in an appropriate topology, is shown to scale approximately as N, but the overall gain in execution speed, for physically interesting problems, is a modest 2–3 when compared to state-of-the-art workstations.

In this study we derive a semi-linear Elliptic Partial Differential Equation (PDE) problem that models the static (zero voltage) behavior of a Josephson window junction. Iterative methods for solving this problem are proposed and their computer implementation is discussed. The preliminary computational results that are given, show the modeling power of our approach and exhibit its computational efficiency.

We investigate the electromagnetic influence of the surrounding idle (no tunneling) region on static fluxons in window Josephson junctions. We calculated the fluxon width as a function of the size of the idle region for three different window (active tunneling area) geometries, namely elongated truncated rhombus, rectangular and bow-tie and derived approximate expressions for the case of small and large idle regions. The window geometry affects both the fluxon width and the fluxon stability. One can define an effective λ_J which depends on the junction width, the idle region width and the inductance ratio and has important consequences on the static and dynamic properties of window Josephson junctions. We also show the effect of the idle region on the maximum tunneling current as a function of the external magnetic field.

We introduce a new type of splitting method for semilinear partial differential equations. The method is analyzed in detail for the case of the two-dimensional static sine-Gordon equation describing a large area Josephson junction with overlap current feed and external magnetic field. The solution is separated into an explicit term that satisfies the one-dimensional sine-Gordon equation in the y-direction with boundary conditions determined by the bias current and a residual which is expanded using modes in the y-direction, the coefficients of which satisfy ordinary differential equations in x with boundary conditions given by the magnetic field. We show by direct comparison with a two-dimensional solution that this method converges and that it is an efficient way of solving the problem. The convergence of the y expansion for the residual is compared for Fourier cosine modes and the normal modes associated to the static one-dimensional sine-Gordon equation and we find a faster convergence for the latter. Even for such large widths as w=10 two such modes are enough to give accurate results.