Multi-peak Solitons Research Articles

Exploration of different multi-peak solitons and vibrant breather type waves’ solutions of nonlinear Schrödinger equations with advanced dispersion and cubic–quintic nonlinearity, unveiling their applications

Abstract The nonlinear Schrödinger equations (NLSEs) of higher order illustrate the transmission of extremely short light pulses in fiber optics. In this manuscript, we employ the two-variable (1/G, G′/G)-expansion technique to construct bright and multi-peak solitons, periodic multi-solitons, breather type solitary waves, periodic peakon solitons, and other wave solutions of higher-order NLSE in mono-mode optical fiber and generalized NLSE with cubic–quintic nonlinearity. The two-variable (1/G, G′/G)-expansion method is a generalization of the (G′/G)-expansion method, offering a more robust mathematical tool for solving various nonlinear partial differential equations (PDEs) in mathematical physics. We also analyze the characteristics of waves conducive to the formation of bright–dark and other soliton forms within this medium. Additionally, we provide graphical representations of the obtained results to visually depict the dynamical models under consideration. Our findings highlight the potency, reliability, and versatility of the proposed technique, which holds promise for solving a wide array of similar models encountered in applied sciences and engineering.

Pattern transformation and control of generalized multi-peak breathing solitons induced by transverse cross modulation

Based on the nonlocal nonlinear Schrödinger equation, the pattern transformation and control of transverse cross-modulated sine-Gaussian (TCMSG) breathing solitons during transmission are studied. Several expressions have been derived, including the transmission, soliton width, phase wavefront curvature, and so on. The study demonstrates that the coefficient of transverse cross modulation term controls the pattern transformation of the TCMSG breathing solitons. TCMSG breathing solitons can form generalized spatial solitons and breathers during transmission. The variation of the soliton width extrema and their change rates with the transverse cross modulation term coefficient is investigated. The influence of the initial incident power and the transverse cross modulation term coefficient on the soliton width change rate and phase wavefront curvature extrema is studied.

Multi-peak soliton dynamics and decoherence via the attenuation effects and trapping potential based on a fractional nonlinear Schrödinger cubic quintic equation in an optical fiber

Fiber optic research continues to develop due to the need for fast information technology. The input signal in an optical fiber is in the form of a soliton wave that propagates in the fiber with a balanced index of refraction, dispersion, and diffraction. Fiber optics may lose some of the signal over time if they solely use a nonlinear refractive index. This study aims to examine into the dynamics and decoherence of multi-peak solitons caused by electromagnetic signal attenuation and potential trap in the propagation via optical fibers. The methods used start with the stationary solution of the nonlinear Schrödinger Cubic Quintic equation, the Newton-Raphson method, and the fourth-order Runge-Kutta. In stationary solutions, the p (state energy) and ρ (Lèvy index) values affect the number, pulse width, and height of soliton peaks. Increased p and ρ values lead to increased instability, causing the data transmitted into the optical fiber to collapse and become unreadable by the receiver. In the excited state (p > 0), the decoherence phenomenon occurs and causes wave instability and destruction. In term of energy intensity, a stable form of transmission in the ground state has a pattern of continuously weakening, decreasing, and smoothing, while in the excited state there is a spike in the intensity of the electromagnetic wave. Based on this study, multi-peak solitons can only be applied for propagation distances, which tend to be short due to their dynamics and decoherence.

Nonlinear waves and transitions mechanisms for (2+1)-dimensional Korteweg–de Vries-Sawada-Kotera-Ramani equation

In this paper, state transition waves are investigated in a (2+1)-dimensional Korteweg–de Vries-Sawada-Kotera-Ramani equation by analyzing characteristic lines. Firstly, the N-soliton solutions are given by using the Hirota bilinear method. The breather and lump waves are constructed by applying complex conjugation limits and the long-wave limit method to the parameters. In addition, the transition condition of breather and lump wave are obtained by using characteristic line analysis. The state transition waves consist of quasi-anti-dark soliton, M-shaped soliton, oscillation M-shaped soliton, multi-peak soliton, W-shaped soliton, and quasi-periodic wave soliton. Through analysis, when solitary wave and periodic wave components undergo nonlinear superposition, it leads to the formation of breather waves and transformed wave structures. It can be used to explain the deformable collisions of transformation waves after collision. Furthermore, the time-varying property of transformed waves are studied using characteristic line analysis. Based on the high-order breather solutions, the interactions involving breathers, state transition waves, and solitons are exhibited. Finally, the dynamics of these hybrid solutions are analyzed through symbolic computations and graphical representations.

Dimensionless dynamics: Multipeak and envelope solitons in perturbed nonlinear Schrödinger equation with Kerr law nonlinearity

In this work, the dimensionless form of the improved perturbed nonlinear Schrödinger equation with Kerr law of fiber nonlinearity is solved for distinct exact soliton solutions. We examined the multi-wave solitons and rational solitons of the governing equation using the logarithmic transformation and symbolic computation using an ansatz functions approach. Multi-wave solitons in fluid dynamics describe the situation in which a fluid flow shows several different regions (or peaks) of high concentration or intensity of a particular variable (e.g., velocity, pressure, or vorticity). Multi-wave solitons in turbulent flows might indicate the existence of several coherent structures, like eddies or vortices. These formations are areas of concentrated energy or vorticity in the turbulent flow. Understanding how these peaks interact and change is essential to comprehending the energy cascade and dissipation in turbulent systems. Furthermore, a sub-ordinary differential equation approach is used to create solutions for the Weierstrass elliptic function, periodic function, hyperbolic function, Chirped free, dark-bright (envelope solitons), and rational solitons, as well as the Jacobian elliptic function, periodic function, and rational solitons. Also, as the Jacobian elliptic function's' modulus m approaches values of 1 and 0, we find trigonometric function solutions, solitons-like solutions, and computed chirp free-solitons. Envelope solitons can arise in stratified fluids and spread over the interface between layers, such as layers in the ocean with varying densities. Their research aids in the management and prediction of wave events in artificial and natural fluid settings. In fluids, periodic solitons are persistent, confined wave structures that repeat on a regular basis, retaining their form and velocity over extended distances. These structures occur in a variety of settings, including internal waves in stratified fluids, shallow water waves, and even plasma physics.

Lump Kink interactional and breather-type waves solutions of (3+1)-dimensional shallow water wave dynamical model and its stability with applications

The shallow water equations are used to describe the behavior of water waves in various shallow regions such as coastal areas, lakes, rivers, etc. These equations are derived by making simplifying assumptions about the water depth relative to the wavelength of the waves. In this paper, the generalized exponential rational function method (gERFM) is used to construct novel wave solutions of the (3+1)-dimensional shallow water wave ((3+1)-dSWW) dynamical model. These solutions encompass distinct kinds of waves, such as solitary waves, solitons, Kink and anti-kink solitons, lump Kink interactional waves, traveling breathers-type waves and multi-peak solitons. The dynamical behavior of these wave solutions is discussed, examining the influence of free parameters on the resulting wave shapes. Furthermore, to provide a scientific elucidation of the obtained results, the solutions are presented graphically, making it easy to distinguish the dynamical features, which have practical implications in different areas of applied sciences and engineering. The stability of this dynamical model is revealed via modulational instability analysis, signifying that all analytical results are stable. The obtained results show that the given technique is universal and efficient. Through comparing the projected technique with the existing techniques, the obtained results demonstrate that the given technique is universal, pithy and efficient.

Photovoltaic spatial gap solitons in biased photorefractive optical lattices

We investigate the existence and characteristics of spatially confined optical gap solitons within an optical lattice embedded in a biased bulk photovoltaic photorefractive crystal. The Floquet-Bloch theory is used to analyze the uniform lattice and derive the optical lattice band structure. In the photonic band gaps, which are typically opaque to light transmission, the photorefractive nonlinearity permits the formation of solitons. The paraxial Helmholtz equation is set up and solved thereby discovering single hump and double hump soliton states in both band gaps. Interestingly, we have not found any multi peak solitons to exist in this particular case. We examine the characteristics of these gap solitons, finding that the soliton width (an indicator of nonlinearity) and intensity depend on their location within the band gap. We find that the magnitude of the external electric field profoundly affects the gap soliton characteristics. Additionally, the Vakhitov-Kolokolov (VK) criterion, perturbation analysis and numerical techniques are used to analyze the stability of the various types of the spatial gap solitons in both the band gaps.

On a Hirota equation in oceanic fluid mechanics: Double-pole breather-to-soliton transitions

In oceanic fluid mechanics, a Hirota equation describing the propagation of the deep ocean broader-banded waves is studied in this paper. With respect to the envelope of the wave field, we simultaneously take the multi-pole phenomena and breather-to-soliton transitions into account to investigate the double-pole breather-to-soliton transitions via the second-order generalized Darboux transformation. Under certain parameter conditions, double-pole anti-dark solitons, periodic waves, W-shaped solitons and multi-peak solitons are derived, analyzed and graphically illustrated. We perform the asymptotic analysis on the double-pole anti-dark solitons to investigate their interaction properties, e.g., amplitudes, characteristic lines, slopes, phase shifts and position differences. Different from the known double-pole solitons of some other models, the double-pole anti-dark solitons, hereby, indicate that the two soliton components own unequal amplitudes.

Orthogonal multi-peak solitons from the coupled fractional nonlinear Schrödinger equation

In this paper, we investigate the dynamics and stability of multi-peak solitons from the coupled nonlinear Schrödinger equation with the fractional dimension based on Lévy random flights. By implementing linear stability analysis and direct simulations, we demonstrate regions where the single and multi-peak modes are stable. Analysis of perturbed coupled solitons confirms the stability of higher-order modes compared to lower-order modes under the same configurations. The stability diagrams show that the force coupling, Lévy index, power, and the nonlinear intensity significantly influence the stability of high-order modes. Our findings indicate that stability is favored in self-defocusing systems with high Lévy indices under weak coupling conditions, with higher-order states exhibiting smaller stability regions.

Multi-peak solitons in parity-time symmetry composite Mathieu lattices

This paper numerically investigates the existence, stability, and transmission characteristics of multi-peak solitons in parity-time (PT) symmetric composite Mathieu lattices in self-focusing Kerr nonlinear media. These multi-peak solitons are highly sensitive to the incident light power, position, and phase. Different incident lights generate solitons with different transmission characteristics, especially the generation of breather soliton transmission characteristics. These findings have significant theoretical and practical implications for the fields of photonics and nonlinear optics.

Study on Abundant Dust-Ion-Acoustic Solitary Wave Solutions of a (3+1)-Dimensional Extended Zakharov–Kuznetsov Dynamical Model in a Magnetized Plasma and Its Linear Stability

This article examines how shocks and three-dimensional nonlinear dust-ion-acoustic waves propagate across uniform magnetized electron–positron–ion plasmas. The two-variable (G′/G,1/G)-expansion and generalized exp(−ϕ(ξ))-expansion techniques are presented to construct the ion-acoustic wave results of a (3+1)-dimensional extended Zakharov–Kuznetsov (eZK) model. As a result, the novel soliton and other wave solutions in a variety of forms, including kink- and anti-kink-type breather waves, dark and bright solitons, kink solitons, and multi-peak solitons, etc., are attained. With the help of software, the solitary wave results (that signify the electrostatic potential field), electric and magnetic fields, and quantum statistical pressures are also constructed. These solutions have numerous applications in various areas of physics and other areas of applied sciences. Graphical representations of some of the obtained results, and the electric and magnetic fields as well as the electrostatic field potential are also presented. These results demonstrate the effectiveness of the presented techniques, which will also be useful in solving many other nonlinear models that arise in mathematical physics and several other applied sciences fields.

Formation of multi-peak gap solitons and stable excitations for double-Lévy-index and mixed fractional NLS equations with optical lattice potentials

In this paper, we investigate the new two-Lévy-index fractional nonlinear Schrödinger (FNLS) equations with optical lattices, where two fractional derivative terms are introduced to control the diffractions. Especially, we explore the linear Bloch bandgaps and existence of gap solitons for the two-Lévy-index mixed FNLS equations with optical lattices. We analyse the effects of different coefficients of normal and abnormal diffraction terms on bandgap structures. Families of single-, double-, triple-, as well as quadruple-peak solitons are found in the first gap with the defocusing nonlinearity while such solitons can be found in the semi-infinite gap when it turns to the focusing regime. Meanwhile, we find that the above-obtained gap solitons can be roughly divided into two types, one of which branches out from the band edge, and another one cannot branch out from the band edge. Moreover, the stability of gap solitons is discussed via the linear stability analysis, and then their dynamical behaviours are verified by means of direct simulations. And stable multi-peak solitons can be obtained via analysing the phase structure. Meanwhile, we find the stable excitations of gap solitons via excitations of system parameters. The stable gap solitons are also verified to appear in the two-Lévy-index FNLS equation. These results may pave the way for the study of linear and nonlinear phenomena of two-Lévy-index and mixed FNLS equations or other mixed fractional physical models in optical lattices and the related physical experimental designs.

Envelope solitons, multi-peak solitons and breathers in optical fibers via Chupin Liu’s theorem and polynomial law of nonlinearity

Envelope solitons, multi-peak solitons and breathers in optical fibers via Chupin Liu’s theorem and polynomial law of nonlinearity

Multi-peak and rational soliton propagations for (3 + 1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup Kupershmidt model in fluid mechanics, ocean dynamics and plasma physics

This paper retrieves the investigation of rational solitons via symbolic computation with logarithmic transformation and ansatz functions approach for the [Formula: see text]-dimensional generalized Konopelchenko–Dubrovsky–Kaup-Kupershmidt (GKDKK) equation in fluid mechanics, ocean dynamics and plasma physics. We find two categories of M-shaped rational solitons and their dynamics will be revealed through graphs by choosing the suitable values of involved parameters. In addition, two categories of M-shaped rational solitons and their interactions with kink waves will be analyzed. Furthermore, homoclinic breathers, multi-wave and kink cross rational solitons will be investigated. The periodic, rational, dark, bright, Weierstrass elliptic function and positive soliton solutions will also be retrieved with the aid of Sub-ODE approach. Moreover, stability characteristics of solutions will be evaluated.

Discrete solitons in competitive zigzag waveguide arrays with cubic-quintic nonlinearity

In this paper, we study one-dimensional discrete solitons in zigzag waveguide arrays with competitive cubic-quintic nonlinearity and competitive linear mixing between the nearest-neighbor (NN) and next-nearest-neighbor (NNN) couplings. The competitive nonlinearity features a cubic self-focusing associated with a quintic self-defocusing nonlinearities. The competitive linear mixing between the NN and NNN couplings is induced by making the two coefficients opposite of each other, which is assumed to be induced by the embedding synthetic gauge phase within the coupling constants. The combination of these two types of competition, linear mixing and nonlinearity can create four types of soliton: multipeak bell-shaped solitons, multipeak flat-top solitons, staggered bell-shaped solitons, and staggered flat-top solitons. The stability and dynamics of these types of solitons are verified systematically through the paper. The total power and the types of competition between the linear mixing play important roles in tuning these solitons.

Breathers, Transformation Mechanisms and Their Molecular State of a (3+1)-Dimensional Generalized Yu–Toda–Sasa–Fukuyama Equation

A (3+1)-dimensional generalized Yu–Toda–Sasa–Fukuyama equation is considered systematically. N-soliton solutions are obtained using Hirota’s bilinear method. The employment of the complex conjugate condition of parameters of N-soliton solutions leads to the construction of breather solutions. Then, the lump solution is obtained with the aid of the long-wave limit method. Based on the transformation mechanism of nonlinear waves, a series of nonlinear localized waves can be transformed from breathers, which include the quasi-kink soliton, M-shaped kink soliton, oscillation M-shaped kink soliton, multi-peak kink soliton, and quasi-periodic wave by analyzing the characteristic lines. Furthermore, the molecular state of the transformed two-breather is studied using velocity resonance, which is divided into three aspects, namely the modes of non-, semi-, and full transformation. The analytical method discussed in this paper can be further applied to the investigation of other complex high-dimensional nonlinear integrable systems.

Abundant solitary wave solutions for space-time fractional unstable nonlinear Schrödinger equations and their applications

The unstable nonlinear Schrödinger models illustrate the time evolution of disturbances in marginally stable or unstable media. In this article, an improved auxiliary equation method is proposed for conformable fractional-order nonlinear evolution equations to obtain the novel wave solutions of space–time fractional unstable and modified unstable nonlinear Schrödinger equations. Several types of novel wave solutions are constructed in forms of some special Jacobi elliptic functions, rational, exponential, trigonometric and hyperbolic functions in which various are unique and having key utilization in quantum physics. With the aid of appropriate values to parameters, different shapes of bright, dark, kink type soliton, multi-peak solitons, periodic solitary waves etc. are depicted. These distinct physical structures are helpful in the understanding the complex physical interpretation of unstable dynamical models. Furthermore, this article gives an idea, how can reduce the conformable fractional order unstable nonlinear Schrödinger equations into an ODE of one variable to obtain the exact solutions. The numerous obtained waves and other solutions reveal the usefulness of the proposed technique which can be utilized to other nonlinear fractional models arising in quantum physics and other fields of applied sciences.

Localized wave solutions and their superposition and conversion mechanism for the (2+1)-dimensional Hirota’s system

Via Darboux transformation method, breather and rogue wave solutions for the (2+1)-dimensional Hirota’s system will be derived. Meanwhile, the dynamic features of those solutions, as well as their various special superpositions creating different sequences of rogue wave due to the different simplest ratio of two frequencies, will be analytically and graphically analyzed. In addition, we will study the conversion mechanism, which can convert breathers and rogue waves into solitons and periodic waves. Through this conversion mechanism, we will obtain a range of stationary nonlinear waves such as M-shaped and W-shaped multi-peak solitons, (quasi) periodic waves and (quasi) anti-dark solitons that exhibit the stationary feature. By virtue of the second-order solutions, we will consider all possible semi-transformed waves, interactions between two non-parallel transformed waves and superpositions of two parallel transformed waves.

Dynamic study of multi-peak solitons and other wave solutions of new coupled KdV and new coupled Zakharov–Kuznetsov systems with their stability

In this paper, our aim is to further expand the use of the two-variable ( G ′ / G , 1 / G ) -expansion approach to a new coupled KdV and Z-K system, which has various significant applications in different fields of applied sciences. The KdV equation, along with shallow-water waves and long internal waves in oceans, basically explains how long, one-dimensional waves propagate in a variety of physical conditions. The study of coastal waves on the basis of the ocean is done using the Zakharov–Kuznetsov (Z-K) equation and this model is utilized to illustrate ion-acoustic wave propagation. By using this method, different forms of analytical solutions of the new coupled KdV (NCKdV) system and the new coupled Z-K (NCZ-K) system, such as solitons, multi-peak solitons, solitary waves, trigonometric, hyperbolic and rational functions and other wave solutions are constructed. The significant features of multi-peak solitons induced by the higher-order effects, including velocity variations, localization or periodicity attenuation and state transitions, are revealed. When the localization disappears then the multi-peak soliton becomes a periodic wave. The constructed solutions are also presented graphically having their applications in engineering, etc. The stability of the solution is examined by utilizing modulation instability. The results obtained show that the proposed technique is universal and efficient. In addition, this technique can also be applied to lots of other new coupled systems arising in other areas of applied sciences.

Optical solitons in Fiber Bragg Gratings for the coupled form of the nonlinear [formula omitted]-dimensional Kundu-Mukherjee-Naskar equation via four powerful techniques

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.