Narrow Results

Tensor-scalar theories of gravitation are commonly employed as extensions of General Relativity that allow to describe a much wider phenomenology. They are also naturally generated as low energy limits of higher-dimensional or unified theories, and the gravitational scalar components can represent quantum corrections to the Einstein theory. The coupling of the scalars to an e.m. field does not introduce any relevant new physics if the e.m. action has the usual Maxwell form, implying a vanishing trace of the e.m. energy–momentum tensor. In the case of the extended Aharonov–Bohm electrodynamics, some interesting new situations are possible, which in this work are analyzed in the gravitational weak-field approximation and for a basic version of tensor-scalar gravity involving only a Brans–Dicke field plus another scalar. Since Aharonov–Bohm theory differs from Maxwell theory only in the presence of anomalous sources with local violation of charge conservation, which is thought to be possible only at a quantum level, the resulting formal framework can be useful for modeling interactions between gravitational and physical systems with macroscopic quantization. The theory contains some unknown parameters, the most important being the vacuum expectation value (VEV) ${\psi }_0$ of the second gravitational scalar and the level $\gamma$ of violation of local charge conservation in the e.m. sector. An attempt is made to relate these parameters to some experimental constraints. However, presently there is much space left for uncertainty.

This paper thoroughly investigates the integrable Akbota equation, a Heisenberg ferromagnet-type equation that plays a crucial role in exploring curve and surface geometry. The ${\varphi }^6$-model expansion method, known for its proficiency and reliability, is applied to generate generalized solitonic wave profiles, spanning a diverse range of soliton families. Prior to this study, there is no existing study in which this technique is utilized that ensured the existence of solution via development of conditions. On the other hand, this method is provided with the Jacobi elliptic function-based solutions with the limiting cases. The analytical strategy presented a notable advantage by specifying a constraint for each solution, ensuring its existence. Consequently, the solitonic wave structures exhibit various attributes, including the Jacobi elliptic function, periodicity, brightness, dark-brightness, singularity, exponential, trigonometry, and rational solitonic structures. These characteristics emerge under previously unexplored existence conditions. The results are visually depicted through 2D, 3D, and contour plots, providing a clear illustration of the behavioral responses to pulse propagation and allowing for the inference of fitting values for system parameters. This visualization offers valuable insights into the characteristics and dynamics of soliton solutions derived from the integrable Akbota equation.

The generalized $mK(m,n)$ model is used to portray the movement of nonlinear waves by applying four mathematical methods with assistance of Mathematica software. These methods are called enhanced simple equation (SE) method, Exp$(-\mathrm{\Psi }(\xi ))$-expansion method, $(G^{\prime }∕G)$-expansion method and modified F-expansion method. For physical behavior, few solutions are plotted graphically in the form of two-dimensional (2D) and three-dimensional (3D). The techniques and soliton solutions that are developed provide computational tools for additional research in this area. With this work, we have gained a better knowledge of soliton motion and how it is used in dynamical systems and other related technologies. Hence, this work will help physicists envision some new ideas and hypotheses in nonlinear science.

The objective of this research is to investigate the initial value problem of a higher-order matrix-type nonlinear Schrödinger (NLS) equation with discrete spectrum as simple and double poles (simple and second-order zeros $\mathrm{\text{rank}}(P(x,t,k_n))=3$ and second- and third-order zeros under $\mathrm{\text{rank}}(P(x,t,k_n))=2)$ under zero boundary conditions (ZBCs). Specifically, we not only consider the analyticity, symmetries and asymptotics of the Jost function, scattering and reflection coefficients in the process of direct scattering, but also residue conditions, norming constants, RH problem and the reconstruction formula in the process of inverse scattering. There exist some differences between it and the RH method in the study of vector and scalar equations, like the order of each pole of $a^{-1}(k)$ being less than or equal to the order of the zeros of $deta(k)$ (assuming $k=k_n\in {\text{ℂ}}^{+}$ is a second- or third-order zero of $deta(k)$ under $\mathrm{\text{rank}}(P(x,t,k_n))=2(\iff \mathrm{\text{rank}}(a(k_n))=0)$), etc.

We present a systematic and detailed review of the application of the method of Hirota and the rational dressing method to abelian Toda systems associated with the untwisted loop groups of complex general linear groups. Emphasizing the rational dressing method, we compare the soliton solutions constructed within these two approaches, and show that the solutions obtained by the Hirota's method are a subset of those obtained by the rational dressing method.

The forced Burgers equation works as a testing ground for a real turbulence, and as the qualitative model for a wide variety of problems including charge density waves, vortex lines in superconductors, disordered solids and epitaxial growth, etc. Its variable-coefficient generalizations call for better modeling of the physical situations. In this paper, we investigate a variable-coefficient generalization of the forced Burgers equation, and obtain several sets of exact soliton-like and other exact analytic solutions, via the extension of a generalized hyperbolic-function method with computerized symbolic computation. We also discuss the Wu method. We find some possibly observable effects, which might be discovered with the relevant experiments.

In this paper, a generalized sub-equation expansion method is presented for constructing some exact analytical solutions of nonlinear partial differential equations. Making use of the method and symbolic computation, we investigate the inhomogeneous nonlinear Schrödinger equation (INLSE) with the loss/gain and the frequency chirping and obtain rich exact analytical solutions. From our results, many known results of some nonlinear Schrödinger equations can be recovered by means of some suitable selections of the arbitrary functions and arbitrary constants. With computer simulation, the main soliton features of bright and dark solitons, Jacobi elliptic function solutions, and Weierstrass elliptic function solutions are shown by some figures. Nonlinear dynamics of the chirped soliton pulses is also investigated under the different regimes of soliton management. The method developed does provide a systematic way to generate various exact analytical solutions for INLSE with varying coefficients.

We study matrix Darboux transformation (MDT) for the generalized N-wave interactions (local/nonlocal) and express multi-soliton solutions in terms of ratios of determinants. In order to explain our construction, we obtain explicit expressions of single and double soliton solutions for local/nonlocal 3-wave and 4-wave interactions.

The extension of exponential rational function method is obtained to construct a series of exact solutions for higher-dimensional Broer–Kaup–Kupershmidt (BKK) dynamical system. New and general solutions are found. The solutions reported in this work are kink solutions, anti-kink solutions and bright solutions. They are expressed in terms of rational exponential functions. A confrontation of our results with the well-known results are done and it comes from this study that the solutions obtained here are new. The mathematical method applied to search for our solutions can be used for other nonlinear partial differential equations. The graphics of the obtained solutions in this paper are shown.

Generalized (3+1)-dimensional Boussinesq equation is investigated in this paper. Through the dependent variable transformation and symbolic computation, the one- and two-soliton solutions are obtained. With the one-soliton solution, the coefficient effects in the soliton propagation process are investigated. Through analyzing the two-soliton solution, two kinds of two-soliton interactions are presented: (i) Two solitons merge into a bigger one whose amplitude increases but does not exceed the sum of the two at the moment of the collision; (ii) Two solitons can pass through each other, and their shapes keep unchanged with a phase shift after the separation. In addition, two kinds of analytic solutions are discussed: (i) "Amplitudes" of the two analytic solutions immediately turn to negative (positive) infinity after the "collision"; (ii) Two analytic solutions are fused into a higher peak (valley) at the moment of "collision", whose "amplitudes" change to negative (positive) infinity after the separation.

In this paper, the time-dependent variable-coefficient KdV equation with a forcing term is considered. Based on the Hirota bilinear method, the bilinear form of this equation is obtained, and the multi-soliton solutions are studied. Then the periodic wave solutions are obtained by using Riemann theta function, and it is also shown that classical soliton solutions can be reduced from the periodic wave solutions.

In this paper, the Caudrey–Dodd–Gibbon–Kotera–Sawada (CDGKS) equation is analytically investigated using the Hirota bilinear method. Based on the bilinear form of the CDGKS equation, its N-soliton solution in explicit form is derived with the aid of symbolic computation. Besides the soliton solutions, several integrable properties such as the Bäcklund transformation, the Lax pair and the nonlinear superposition formula are also derived for the CDGKS equation.

In this paper, the auxiliary expansion equation method is applied to compute the analytical wave solutions for (3+1)-dimensional Boussinesq and Kadomtsev–Petviashvili (KP) equations. A simple transformation is carried out to reduce the set of nonlinear partial differential equations (NPDEs) into ODEs. These obtained results hold numerous traveling wave solutions that are of key importance in elucidating some physical circumstance.

The modified Kawahara equation also called Korteweg-de Vries (KdV) equation of fifth-order arises in shallow water wave and capillary gravity water waves. This study is based on the generalized Riccati equation mapping and modified the F-expansion methods. Several types of solitons such as Bright soliton, Dark-lump soliton, combined bright dark solitary waves, have been derived for the modified Kawahara equation. The obtained solutions have significant applications in applied physics and engineering. Moreover, stability of the problem is presented after being examined through linear stability analysis that justify that all solutions are stable. We also present some solution graphically in 3D and 2D that gives easy understanding about physical explanation of the modified Kawahara equation. The calculated work and achieved outcomes depict the power of the present methods. Furthermore, we can solve various other nonlinear problems with the help of simple and effective techniques.

In this paper we study the nonlocal reductions for the non-isospectral Ablowitz-Kaup-Newell-Segur equation. By imposing the real and complex nonlocal reductions on the non-isospectral Ablowitz-Kaup-Newell-Segur equation, we derive two types of nonlocal non-isospectral nonlinear Schrödinger equations, in which one is real nonlocal non-isospectral nonlinear Schrödinger equation and the other is complex nonlocal non-isospectral nonlinear Schrödinger equation. Of both of these two equations, there are the reverse time nonlocal type and the reverse space nonlocal type. Soliton solutions in terms of double Wronskian to the reduced equations are obtained by imposing constraint conditions on the double Wronskian solutions of the non-isospectral Ablowitz-Kaup-Newell-Segur equation. Dynamics of the one-soliton solutions are analyzed and illustrated by asymptotic analysis.

In this paper, we show that complete integrability is preserved in a multicomponent differential-difference Volterra system with branched dispersion relation. Using the Hirota bilinear formalism, we construct multisoliton solutions for a system of coupled $M$ equations. We also show that one can obtain the same solutions through a periodic reduction starting from a two-dimensional completely integrable generalized Volterra system. For some particular cases, graphical representations of solitons are displayed and stability is discussed using an asymptotic analysis.

The multi-wave solutions for nonlinear Kundu–Eckhaus (KE) equation are obtained using logarithmic transformation and symbolic computation using the function method. Three-wave method, double exponential and homoclinic breather approach are used to get these solutions. We study the conflict between our results and considerably-known results and state that the solutions reached here are new. By specifying the suitable values for the parameter, the drawings of the solutions obtained are shown in this paper.

This paper introduces the fractal form of the generalized nonlinear Schrödinger equation, newly named as the Biswas–Milovic model (BM). The BM equation theoretically explains the transmission of solitons for transatlantic and transcontinental distances utilizing optical fibers. The BM equation relating to Kerr law, parabolic law and nonlinearity quadratic law was studied using a variational approach for optical soliton solutions. Essential novel conditions are presented that guarantee the existence of the appropriate solitons. Besides, the physical action of the solution obtained was recorded in terms of 3D and contour plots for distinct parameters for the three different nonlinearities. This study shows the relevance and huge potential of the variational approach to the generalized nonlinear Schrödinger equation.

This paper is about the study of space–time fractional Fokas–Lenells equation that describes nonlinear wave propagation in optical fibers. Three prominent schemes are employed for extracting different types of exact soliton solutions. In particular, the ${exp}_a$ function method, the hyperbolic function method and the simplest Riccati equation scheme are investigated for the said model. As a sequel, a series of soliton solutions are obtained and verified through MATHEMATICA. The obtained solutions are significant additions in some specific fields of physics and engineering. Furthermore, the 3D graphical descriptions are left to analyze the pulse propagation for the reader.

In this paper, we present a work on dynamic equation of Deoxyribonucleic acid (DNA) derived from the Peyrard–Bishop (PB) model oscillator chain for various dynamical solitary wave solutions. In order to construct novel soliton solutions in the DNA dynamic PB model with beta-derivative, the efficiency of the newly developed algorithms is being investigated, which could include a new auxiliary equation strategy (NAES). Some precise soliton solutions comprising dark, light and other forms of multi-wave soliton solutions are achieved via the proposed methods. Furthermore, mathematical models demonstrate the singularity of our work in comparison to current literary materials and even describe some results using the classic Peyrard–Bishop model. All the established results contribute to the possibility of extending the approach to solve other nonlinear equations of fractional space–time derivatives in nonlinear sciences. The strategy that has been proposed recently is specific and is being employed to produce novel closed-form solutions for all many other FNLEEs.