Complex soliton wave patterns of Gross–Pitaevskii systems: application in quantum and optical engineering

In this article, four powerful techniques namely, the three wave method, double exponential, homoclinic breather, M-Shaped rational, M-Shaped with one kink, and M-Shaped with two kink are putting into practice to bolster the new optical solutions of the coupled form of the nonlinear (2+1)-dimensional Kundu-Mukherjee-Naskar equation in Fiber Bragg Gratings (FBGs). Implementation of the appropriate functions of the solutions results in a different form of multi-wave solutions with their interaction phenomena such as dark soliton solutions, singular soliton solutions, bright soliton solutions, kink soliton solutions, Multi-Peak soliton solutions, multi dark-bright soliton solutions, multi bright soliton solutions, multi dark soliton solutions with different structure, multi M-Shaped soliton solutions, multi M-Shaped soliton solutions, M-Shaped soliton solutions with different structure, kink soliton solutions with different structure, and periodical M-Shaped soliton solutions. Three-dimensional graphics demonstrating the dependability and productivity of the suggested methodologies are used to describe the dynamics of the produced solutions.

In recent years, discovering and examining novel soliton solutions to nonlinear models that emerge in the disciplines of science and engineering have received noticeable fascination from researchers, as these solutions shed light on the models' underlying phenomena. This study goals to establish and evaluate new plethora of soliton solutions in the context of (2+1)-dimensional Modified Zakharov-Kuznetsov Equation (MZKE), which arises in electrical engineering, using a novel analytical tool called the Riccati Modified Extended Simple Equation Method (RMESEM). A mathematical framework for MZKE is first modelled by applying Kirchhoff's law to the nonlinear electrical transmission line circuit. The suggested anstaz RMESEM then generates nonlinear ordinary differential equation (NODE) from the model using a sophisticated wave transformation. To find a new variety of soliton solutions, the resultant NODE is assumed to have a closed form solution that transforms it into a system of nonlinear algebraic equations by substitution. Solving the resultant system using maple yields novel families of soliton solutions for the aimed MZKE in the form of rational, hyperbolic, trigonometric, rational-hyperbolic and exponential functions. Graphical representations of the wave behavior of different soliton solutions are provided by three-dimensional, contour, and two-dimensional graphs. These graphs reveal that the found solitons significantly display the profiles of kink solitons, including bell-shaped, lump-like, twinning, and perturbed kinks. The obtained results contribute to a deeper understanding of the model and its potential applications in related domains by providing useful insights into the dynamics and behavior of the MZKE. Significantly, the results obtained also demonstrate that the proposed transformation-based RMESEM offers a straightforward and trustworthy strategy for exploring soliton phenomena in a wide range of nonlinear equations.

<abstract><p>The Phi-Four (also embodied as $ \phi^4 $) equation (PFE) is one of the most significant models in nonlinear physics, that emerges in particle physics, condensed matter physics and cosmic theory. In this study, propagating soliton solutions for the PFE were obtained by employing the extended direct algebraic method (EDAM). This transformational method reformulated the model into an assortment of nonlinear algebraic equations using a series-form solution. These equations were then solved with the aid of Maple software, producing a large number of soliton solutions. New families of soliton solutions, including exponential, rational, hyperbolic, and trigonometric functions, are included in these solutions. Using 3D, 2D, and contour graphs, the shape, amplitude, and propagation behaviour of some solitons were visualized which revealed the existence of kink, shock, bright-dark, hump, lump-type, dromion, and periodic solitons in the context of PFE. The study was groundbreaking as it extended the suggested strategy to the PFE that was being aimed at, yielding a significant amount of soliton wave solutions while providing new insights into the behavioral characteristics of soliton. This approach surpassed previous approaches by offering a systematic approach to solving nonlinear problems in analogous challenging situations. Furthermore, the results also showed that the suggested method worked well for building families of propagating soliton solutions for intricate models such as the PFE.</p></abstract>

This study is based on the Lie group method for the nonlinear elastic structural element equation (ESE Equation). We obtain a three-dimensional Lie algebra. By utilizing this Lie algebra a four-dimensional optimal system is constructed. The governing ESE Equation is converted to nonlinear ordinary differential equations (ODEs) via symmetry reduction. We use a modified auxiliary equation (MAE) procedure to deal with nonlinear ODEs. These ODEs reveal the dynamics of the periodic and soliton solutions. We obtain soliton solutions through rational, trigonometric, and hyperbolic functions. Wolfram Mathematica simulations vividly illustrate the wave characteristics of the derived solutions, affirming their properties as singular periodic solutions, a singular solution, an optical dark soliton solution, and a singular soliton solution. We also obtain the local conservation laws by a new conservation theorem introduced by Ibragimov.

Complex wave solutions described by a (3+1)-dimensional coupled nonlinear Schrödinger equation with variable coefficients

On complex soliton solutions, complex elliptic solutions and complex rational function solutions for the Sasa-Satsuma model equation with variable coefficients

In this study, we utilize the new direct mapping method to derive optical soliton solutions for the nonlinear conformable Schrödinger equation, which accounts for group velocity dispersion coefficients and second-order spatiotemporal terms. Leveraging bifurcation and chaos theories, we conduct an in-depth analysis of the planar dynamical system associated with the equation, providing detailed graphical representations of chaotic solutions for the perturbed system. The time series of the planar system, along with a sensitivity analysis of the perturbed system, is visualized through various graphical illustrations to underscore the model’s significance and its dynamic behavior in practical applications. We derive a new family of soliton solutions including bell-shaped, dark-bright, and mixed dark-bright solitons which have not been previously reported in the literature. Further, the influence of the conformable derivative parameter and temporal dynamics on these soliton solutions is systematically investigated, highlighting the system’s importance in real-world contexts. The results highlight the potential of the proposed model in advancing optical fiber communication technologies, especially for the efficient transmission and control of ultra-fast pulses. This work contributes both theoretical novelty and practical insights into nonlinear wave propagation in advanced photonic systems.

Nonlinear tunneling of optical soliton in 3 coupled NLS equation with symbolic computation

Exact traveling wave solutions and soliton solutions of conformable M-fractional modified nonlinear Schrödinger model

In this work, a proof of the orbital stability of the black soliton solution of the quintic Gross–Pitaevskii equation in one spatial dimension is obtained. We first build and show explicitly black and dark soliton solutions and we prove that the corresponding Ginzburg–Landau energy is coercive around them by using some orthogonality conditions related to perturbations of the black and dark solitons. The existence of suitable perturbations around black and dark solitons satisfying the required orthogonality conditions is deduced from an implicit function theorem. In fact, these perturbations involve dark solitons with sufficiently small speeds and some proportionality factors arising from the explicit expression of their spatial derivative.

Derivation of optical solitons and other solutions for nonlinear Schrödinger equation using modified extended direct algebraic method

This work investigates the coupled Ivancevic option-pricing model, a nonlinear wave alternative to the Black–Scholes model. By utilizing the recently developed Kumar-Malik method, modified Sardar sub-equation method and the generalized Arnous method, the substantial results of this research are the successful derivation of novel exact soliton solutions, including bright, singular, dark, combined dark–bright, singular-periodic, complex solitons, exponential and Jacobi elliptic functions. A detailed analysis of option price wave functions and modulation instability analysis is conducted, with the conditions for valid solutions outlined. Additionally, a mathematical framework is established to capture market price fluctuations. Numerical simulations, illustrated through 2D, 3D and contour graphs, highlight the effects of parameter variations. Our findings demonstrate the effectiveness of the coupled Ivancevic model as a fractional nonlinear wave system, providing valuable insights into stock volatility and returns. This study contributes to creating new option-pricing models, which affect financial market analysis and risk management.

PurposeThe purpose of this paper is to introduce a variety of new completely integrable Calogero–Bogoyavlenskii–Schiff (CBS) equations with time-dependent coefficients. The author obtains multiple soliton solutions and multiple complex soliton solutions for each of the developed models.Design/methodology/approachThe newly developed models with time-dependent coefficients have been handled by using the simplified Hirota’s method. Moreover, multiple complex soliton solutions are derived by using complex Hirota’s criteria.FindingsThe developed models exhibit complete integrability, for specific determined functions, by investigating the compatibility conditions for each model.Research limitations/implicationsThe paper presents an efficient algorithm for handling integrable equations with analytic time-dependent coefficients.Practical implicationsThe work presents new integrable equations with a variety of time-dependent coefficients. The author showed that integrable equations with time-dependent coefficients give real and complex soliton solutions.Social implicationsThis study presents useful algorithms for finding and studying integrable equations with time-dependent coefficients.Originality/valueThe paper gives new integrable CBS equations which appear in propagation of waves and provide a variety of multiple real and complex soliton solutions.

Two new types of nonlocal Boussinesq equations in water waves: Bright and dark soliton solutions

On complex wave solutions governed by the 2D Ginzburg–Landau equation with variable coefficients