Soliton Solutions Research Articles

New Soliton Solutions and Trajectory Equations of Soliton Collision to a (2+1)-Dimensional Shallow Water Wave Model

New Soliton Solutions and Trajectory Equations of Soliton Collision to a (2+1)-Dimensional Shallow Water Wave Model

On Sawada-Kotera and Kaup-Kuperschmidt integrable systems

To obtain new integrable nonlinear differential equations there are some well-known methods such as Lax equations with different Lax representations. There are also some other methods that are based on integrable scalar nonlinear partial differential equations. We show that some systems of integrable equations published recently are the M2 -extension of integrable such scalar equations. For illustration, we give Korteweg–de Vries, Kaup-Kupershmidt, and Sawada-Kotera equations as examples. By the use of such an extension of integrable scalar equations, we obtain some new integrable systems with recursion operators. We also give the soliton solutions of the systems and integrable standard nonlocal and shifted nonlocal reductions of these systems.

Dynamic Description of soliton solutions of nonlinear Schrödinger equation with higher order dispersion and nonlinear terms

Abstract The article presents an analytical solution for the higher-order nonlinear Schrödinger equation (NLSE), which describes the propagation of short light pulses in monomode optical fibers. Various traveling wave solutions are obtained using the generalized exponential rational function method, a technique with substantial applications in physics and mathematics. Additionally, the parameters leading to the occurrence of optical bright and multipeak solitons in this medium are provided along with their formation conditions. The derived solutions are graphically displayed to enhance the understanding of the model’s physical phenomena. This approach is credible, potent, and successful in solving a wide variety of different models of this kind that arise in the applied sciences. Its robustness, strength, and efficiency make it suitable for addressing various higher-order nonlinear problems in current research fields, extending beyond the models encountered in the applied sciences.

New dynamical behaviors and soliton solutions of the coupled nonlinear Schrodinger equation

New dynamical behaviors and soliton solutions of the coupled nonlinear Schrodinger equation

Generalized kudryashov and extended auxiliary equation methods for novel solitons solutions to (1+1)-dimensional doubly dispersive equation of murnaghan’s rod

Abstract In this study, the Generalized Kudryashov method and the Extended Auxiliary Equations method are employed to investigate the strongly nonlinear (1+1)-dimensional Doubly Dispersive equation model of inhomogeneous Murnaghan’s rod for developing novel soliton solutions. These symbolic methods are famous for solving various problems involving nonlinear partial differential equations. The study finds novel solitons like hyperbolic, exponential, rational, and trigonometric. Moreover, 2D, 3D, and contour plots under tunable parameters depict graphical representations of the solutions that furnish dynamic wave behavior and insights into the material’s elastic properties under strain and stress.

Computational analysis of the conformable nonlinear Schrödinger equation with Kudryashov’s refractive index model and generalized non-local nonlinearity

This paper investigates the nonlinear conformable Schrödinger equation with nonlocal nonlinearities and Kudryashov’s arbitrary refractive index. The study employs the Uniform Method (UM) and the Extended Hyperbolic Function Method (EHFM) to derive various optical soliton solutions for this present conformable equation. To demonstrate the importance of the novel solutions, the paper presents two-dimensional, three-dimensional, and contour plots, illustrating kink-type, wave, bright, bell-shaped, and mixed dark–bright soliton solutions. Further, the impact of the fractional order parameter, temporal parameter, and nonlinearity coefficient on these optical solutions is analyzed, providing valuable insights into the conformable nonlinear Schrödinger model. The behavior of these optical solutions is analyzed through illustrative graphs that account for variations in the conformable order derivative, temporal parameter, and nonlinearity coefficient. The methodologies applied in this research show potential for broader application to various nonlinear Schrödinger equations in fields such as applied mathematics and nonlinear optics. The research suggests that these methods could be applied in the future to investigate other differential equations with fractional and integer orders across different fields of applied sciences. This study expands our understanding of nonlinear optics and has potential practical implications in various fields like optical signal processing, laser technology, and telecommunications.

Impact of relativistic positron beam on ion-acoustic solitary, periodic and breather waves in Earths’ ionospheric region through the framework of KdV and modified KdV equation

Abstract This study explores the propagation characteristics of ion-acoustic periodic, soliton, and breather waves in electron-positron-ion (EPI) plasma with a relativistic positron beam. The Korteweg–de Vries (KdV) equation is obtained by applying the traditional reductive perturbation method (RPM) to the fundamental set of fluid equations. When the KdV model is unable to accurately represent the nonlinear system’s evolution, a modified Korteweg–de Vries (mKdV) equation is constructed. In both models, Jacobi elliptic functions are used to derive periodic solutions, and a connection between periodic waves and soliton solutions is established. Hirota’s bilinear method is used to generate breathers directly from the KdV type framework without utilizing the modified Schrödinger framework inferred from the KdV type framework, which is a prevalent method in studies of nonlinear waves. Numerical knowledge of various physical factors in the ionospheric region is incorporated into the model to elucidate wave propagation in the Earth’s upper atmosphere.

Soliton, lump and hybrid solutions of a generalized (2+1)-dimensional Benjamin-Ono equation in fluids

Soliton, lump and hybrid solutions of a generalized (2+1)-dimensional Benjamin-Ono equation in fluids

Lump, lump-periodic, lump-soliton and multi soliton solutions for the potential Kadomtsev-Petviashvili type coupled system with variable coefficients

Lump, lump-periodic, lump-soliton and multi soliton solutions for the potential Kadomtsev-Petviashvili type coupled system with variable coefficients

Diffraction managed vortex soliton in competing cubic-quintic nonlinear media

We study analytically and numerically vortex solitons in competing cubic-quintic nonlinear media with normal and anomalous fourth order diffraction (FOD). Bifurcated solutions of vortex solitons are obtained with variational (Lagrangian) approach. Propagation dynamics are demonstrated with split-step Fourier transform, which show that normal (anomalous) FOD can weaken (enhance) the stable propagation distance of vortex solitons. Furthermore, stability of vortex solitons is determined by Vakhitov-Kolokolov (VK) criteria. The upper branch of vortex solitons is unstable, whereas, the lower branch of vortex solitons is robust against perturbations.

New analytical wave structures for generalized B-type Kadomtsev–Petviashvili equation by improved modified extended tanh function method

Abstract Recently, solving the complicated nonlinear partial differential equations has become very important demand in order to simulate their physical phenomena. This manuscript focuses on extracting the wave solutions of (3 + 1)-dimensional generalized B-type Kadomtsev-Petviashvili equation (GBKPE), which demonstrates the behavior of nonlinear waves in fluid mechanics. The improved modified extended Tanh function (IMETF) method is the suggested method to do this task as it gives different types of solutions. This method enables us to obtain many solutions, such as Jacobi elliptic, dark soliton, and singular soliton, exponential, and singular periodic wave solutions. Additionally, for more illustrations graphical visual representations of some solutions are provided.

Nonlinear waves for a variable-coefficient modified Kadomtsev–Petviashvili system in plasma physics and electrodynamics

In this paper, a variable-coefficient modified Kadomtsev–Petviashvili (vcmKP) system is investigated by modeling the propagation of electromagnetic waves in an isotropic charge-free infinite ferromagnetic thin film and nonlinear waves in plasma physics and electrodynamics. Painlevé analysis is given out, and an auto-Bäcklund transformation is constructed via the truncated Painlevé expansion. Based on the auto-Bäcklund transformation, analytic solutions are given, including the solitonic, periodic and rational solutions. Using the Lie symmetry approach, infinitesimal generators and symmetry groups are presented. With the Lagrangian, the vcmKP equation is shown to be nonlinearly self-adjoint. Moreover, conservation laws for the vcmKP equation are derived by means of a general conservation theorem. Besides, the physical characteristics of the influences of the coefficient parameters on the solutions are discussed graphically. Those solutions have comprehensive implications for the propagation of solitary waves in nonuniform backgrounds.

Retraction Note: Obtaining optical soliton solutions of the cubic–quartic Fokas–Lenells equation via three different analytical methods

Retraction Note: Obtaining optical soliton solutions of the cubic–quartic Fokas–Lenells equation via three different analytical methods

Conservation laws for a perturbed resonant nonlinear Schrödinger equation in quantum fluid dynamics and quantum optics

The current paper retrieves the conservation laws for the extended version of the resonant nonlinear Schrödinger's equation for description of physical processes in quantum fluid dynamics and quantum optics. The method of multipliers recovers three fundamental conservation laws. Analytical solutions of equation are found taking into account traveling wave reduction. The conserved quantities are subsequently computed from the soliton solution of the model equation that is derived in this work too.

Superposition of soliton, breather and lump waves in a non-painlevé integrabale extension of the Boiti-Leon-Manna-Pempinelli equation

Abstract In this paper, we derive general $N$th-order Pfaffian solutions for a $(3+1)$-dimensional non-Painlev{'e} integrable extension of the Boiti-Leon-Manna-Pempinelli (BLMP) equation. Specifcally, we obtain $N$-soliton, higher-order breather, higher-order lump and hybrid solutions, and explore the superpositions of Y-shaped and X-shaped soliton-breather waves.
Moreover, we construct bilinear B"{a}cklund transformations, Lax pairs, and conservation laws using Bell polynomials.
Finally, we identify a similar equation in the literature and demonstrate that it represents another non-Painlevé integrable extension of the BLMP equation.

Time-fractional of cubic-quartic Schrödinger and cubic-quartic resonant Schrödinger equations with parabolic law: various optical solutions

Abstract Schrödinger's nonlinear equation is a fundamental model in fiber optics and many other areas of science. Using the Jacobi elliptic expansion function method, the time-fractional cubic-quartic nonlinear Schrödinger equation and cubic-quartic resonant nonlinear Schrödinger equation are investigated. By applying the effective Jacobi elliptic expansion function method, optical soliton solutions such as bright, dark, singular, periodic singular, exponential, and Jacobi elliptic function solutions have been obtained. The effect of the time-fractional derivative on the solutions is also revealed. Graphical representations are illustrated to showcase the physical properties of raised solutions, providing a comprehensive understanding of the solutions’ functionality.

THE DARBOUX TRANSFORMATION, POSITON SOLUTION AND BREATHER SOLUTION OF THE THIRD-ORDER FLOW GERDJIKOV-IVANOV EQUATION

Abstract In this paper, the third-order flow Gerdjikov-Ivanov (TOFGI) equation is studied, and the Darboux transformation (DT) is used to obtain the determinant expression of the solution of this equation. On this basis, the soliton solution, rational solution, positon solution and breather solution of the TOFGI equation are obtained by taking zero "seed" solution and non-zero "seed" solution. The exact solutions and dynamic properties of the Gerdjikov-Ivanov (GI) equation and the TOFGI equation are compared in detail under the same conditions, and it is found that there are some differences in the velocities and trajectories of the solutions of the two equations.

Dark Soliton and Breather Solutions to the Coupled Sasa–Satsuma Equation

Dark Soliton and Breather Solutions to the Coupled Sasa–Satsuma Equation

Optical solitons, qualitative analysis, and chaotic behaviors to the highly dispersive nonlinear perturbation Schrödinger equation

In this paper, we study the highly dispersive nonlinear perturbation Schrödinger equation, which has arbitrary form of Kudryashov's with sextic‐power law refractive index and generalized nonlocal laws. For the equation has highly dispersive nonlinear terms and higher order derivatives, it cannot be integrated directly, so we build an integrable factor equation for the approximated equation and apply the trial equation method and the complete discrimination system for polynomial method to create new soliton solutions. On the other hand, we use the bifurcation theory to qualitatively analyze the equation and find the model has periodic solutions, bell‐shaped soliton solutions, and solitary wave solutions via phase diagrams. The topological stability of the solutions with respect to the parameters is explored in order to better understand the effect of parameters perturbations on the stability of the model's solutions. Furthermore, we analyze the modulation instability and give the corresponding linear criterion. After accounting for external perturbation terms, we analyze the chaotic behaviors of the equation through the largest Lyapunov exponents and phase diagrams.

Superposition and Interaction Dynamics of Complexitons, Breathers, and Rogue Waves in a Landau–Ginzburg–Higgs Model for Drift Cyclotron Waves in Superconductors

This article implements the Hirota bilinear (HB) transformation technique to the Landau–Ginzburg–Higgs (LGH) model to explore the nonlinear evolution behavior of the equation, which describes drift cyclotron waves in superconductivity. Utilizing the Cole–Hopf transform, the HB equation is derived, and symbolic manipulation combined with various auxiliary functions (AFs) are employed to uncover a diverse set of analytical solutions. The study reveals novel results, including multi-wave complexitons, breather waves, rogue waves, periodic lump solutions, and their interaction phenomena. Additionally, a range of traveling wave solutions, such as dark, bright, periodic waves, and kink soliton solutions, are developed using an efficient expansion technique. The nonlinear dynamics of these solutions are illustrated through 3D and contour maps, accompanied by detailed explanations of their physical characteristics.