Nonlinear Wave Research Articles

Exploring soliton and soliton-type solutions to the modified Camassa-Holm and Schrödinger-Hirota equations: An analytical approach

Abstract In this work, we investigate the solitary wave solutions of two nonlinear evolution equations (NLEEs): the modified Camassa-Holm (MCH) equation and the Schrödinger-Hirota (SH) equation. Both are widely used, especially in fluid dynamics, nonlinear optics, and quantum mechanics. We analytically extract solitary wave solutions to the considered equations by applying the (G'⁄((G'+G+A)))-expansion method which enhances the understanding of wave dynamics in various fields of mathematical physics and engineering. The graphical representations of the obtained results of these two equations for certain values are shown in 3D, 2D, contour, and density plot diagrams, all of which are of the soliton types with one compacton, one anti-flat kink-shaped, one anti-bell-shaped, and one parabolic-shaped soliton. Moreover, we analyze the role of non-algebraic functions, specifically exponential and trigonometric functions, in characterizing the intricate waveforms and their stability properties. We systematically explore the applicability of the chosen expansion method, demonstrating its effectiveness in generating explicit solutions and examining the conditions under which these solitary wave phenomena arise. The novel findings contribute new analytical insights into these classical equations and pave the way for future research into the complex behaviors exhibited by nonlinear wave interactions.

Evolution dynamics of fundamental mechanisms of wave propagation in gas bubble-liquid interactions and soliton solutions

Abstract This paper investigates the dynamics of solitary wave solutions based on the (3+1)-dimensional nonlinear wave equation with variable coefficients expressed to describe gas-bubble-liquid interactions. The generalised (1/G’)-expansion method, new solitary wave solutions for this equation have been successfully derived to better understand the underlying dynamics of wave phenomena, especially in gas-bubble-liquid systems. Thanks to this method, the results of the variations of physical parameters in the generated solutions are also emphasised. The physical dynamics of each dimension in the generated solutions allow both the dimensions to be compared with each other and the equation to be compared with the existing literature with a holistic understanding. With this application, we have also analyzed the direct effects of the viscosity of the fluid on the dispersion of the bubble. These solitary solutions have helped us to better explain the wave behavior in gas-bubble-liquid systems and provided a new perspective on the solution of nonlinear wave equations. The structure of the study contributes to a deeper understanding of wave phenomena by discussing the (3+1) dimensional variable coefficient nonlinear wave equation within the framework of both mathematical analysis and physical analysis.

Field–particle energy transfer during chorus emissions in space

Chorus waves are some of the strongest electromagnetic emissions naturally occurring in space and can cause radiation that is hazardous to humans and satellites1, 2–3. Although chorus waves have attracted extreme interest and been intensively studied for decades4, 5, 6–7, their generation and evolution remain highly debated7. Here, in contrast to the conventional expectation that chorus waves are governed by planetary magnetic dipolar fields5,7, we report observations of repetitive, rising-tone chorus waves in the terrestrial neutral sheet, where the effects of the magnetic dipole are absent. Using high-cadence data from NASA’s MMS mission, we present ultrafast measurements of the wave fields and three-dimensional electron distributions within the waves, which provides evidence for chorus–electron interactions and the development of electron holes in the wave phase space. We found that the waves are associated with resonant currents antiparallel to the wave magnetic field, as predicted by nonlinear wave theory. We estimated the nonlinear field–particle energy transfer inside the waves, finding that the waves extract energy from local thermal electrons, in line with the positive growth rate of the waves derived from an instability analysis. Our observations may help to resolve long-standing controversies regarding chorus emissions and in gaining an understanding of the energy transport observed in space and astrophysical environments.

On the Extended Simple Equations Method (SEsM) and Its Application for Finding Exact Solutions of the Time-Fractional Diffusive Predator–Prey System Incorporating an Allee Effect

In this paper, I extend the Simple Equations Method (SEsM) and adapt it to obtain exact solutions of systems of fractional nonlinear partial differential equations (FNPDEs). The novelty in the extended SEsM algorithm is that, in addition to introducing more simple equations in the construction of the solutions of the studied FNPDEs, it is assumed that the selected simple equations have different independent variables (i.e., different coordinates moving with the wave). As a consequence, nonlinear waves propagating with different wave velocities will be observed. Several scenarios of the extended SEsM are applied to the time-fractional predator–prey model under the Allee effect. Based on this, new analytical solutions are derived. Numerical simulations of some of these solutions are presented, adequately capturing the expected diverse wave dynamics of predator–prey interactions.

Process, dynamics and bioeffects of acoustic droplet vaporization induced by dual-frequency focused ultrasound.

Acoustic droplet vaporization (ADV) plays a crucial role in ultrasound-related biomedical applications. While previous models have examined the stages of nucleation, growth, and oscillation in isolation, which may limit their ability to fully describe the entire ADV process. To address this, our study developed an integrated model that unifies these three stages of ADV, stimulated by a continuous nonlinear dual-frequency ultrasound wave. Using this integrated model, we investigated the influence of nonlinear dual-frequency ultrasound parameters on ADV dynamics and bioeffects by incorporating tissue viscoelasticity through parametric studies. Our results demonstrated that the proposed model accurately captured the entire ADV process, ensuring continuous vapor bubble formation and evolution throughout the phase transition process. Moreover, the applied unified theory for bubble dynamics can simulate intense bubble collapse with high Mach Number as a result of the nonlinear effects of dual-frequency ultrasound. In addition, cavitation-associated mechanical and thermal damage appeared to be more strongly correlated with rapid bubble collapse than with maximum bubble size. Our research also revealed that the mechanical and thermal effects could be regulated independently to some extent by adjusting dual-frequency ultrasound parameters, as they presented differing sensitivities to frequency and acoustic power. Importantly, dual-frequency combinations such as 1.5MHz+3 MHz (fundamental and second harmonic), which exhibit a higher Degree of Nonlinearity (DoN) can extend bubble lifespan, offering a potential pathway to the efficacy of ultrasound treatments.

Exploration of Soliton Solutions for the Kaup–Newell Model Using Two Integration Schemes in Mathematical Physics

ABSTRACTThis research deals with the Kaup–Newell model, a class of nonlinear Schrödinger equations with important applications in plasma physics and nonlinear optics. Soliton solutions are essential for analyzing nonlinear wave behaviors in different physical systems, and the Kaup–Newell model is also significant in this context. The model's ability to represent subpicosecond pulses makes it a significant tool for the research of nonlinear optics and plasma physics. Overall, the Kaup–Newell model is an important research domain in these areas, with ongoing efforts focused on understanding its various solutions and potential applications. A new version of the generalized exponential rational function method and ‐expansion function method are utilized to discover diverse soliton solutions. The generalized exponential rational function method facilitates the generation of multiple solution types, including singular, shock, singular periodic, exponential, combo trigonometric, and hyperbolic solutions in mixed forms. Thanks to ‐expansion function method, we obtain trigonometric, hyperbolic, and rational solutions. The modulation instability of the proposed model is examined, with numerical simulations complementing the analytical results to provide a better understanding of the solutions' dynamic behavior. These results offer a foundation for future research, making the solutions effective, manageable, and reliable for tackling complex nonlinear problems. The methodologies used in this study are robust, influential, and practicable for diverse nonlinear partial differential equations; to our knowledge, for this equation, these methods of investigation have not been explored before. The accuracy of each solution has been verified using the Maple software program.

Multiple localized nonlinear waves of a forced variable-coefficient Gardner equation in a fluid or plasma

Multiple localized nonlinear waves of a forced variable-coefficient Gardner equation in a fluid or plasma

Symplectic particle-in-cell methods for hybrid plasma models with Boltzmann electrons and space-charge effects

We study the geometric particle-in-cell methods for an electrostatic hybrid plasma model. In this model, ions are described by the fully kinetic equations, electron density is determined by the Boltzmann relation and space-charge effects are incorporated through the Poisson equation. By discretizing the action integral or the Poisson bracket of the hybrid model, we obtain a finite dimensional Hamiltonian system, for which the Hamiltonian splitting methods or the discrete gradient methods can be used to preserve the geometric structure or energy. The global neutrality condition is conserved under suitable boundary conditions. Moreover, the results are further developed for an electromagnetic hybrid model proposed by Vu (J. Comput. Phys., vol. 124, issue 2, 1996, pp. 417–430). Numerical experiments of finite grid instability, Landau damping and resonantly excited nonlinear ion waves illustrate the behaviour of the numerical methods constructed.

On Lie symmetries, soliton interaction nature and conservation laws of Broer-Kaup-Kupershmidt system in shallow water of uniform depth

Abstract The exact solutions of nonlinear partial differential equations are significant to analyze the soliton nature of physical phenomena. Therefore, the Lie symmetry method is introduced to obtain exact solutions of the Broer-Kaup-Kupershmidt system. It represents two-dimensional propagation of nonlinear and dispersive long gravity waves in shallow water with uniform depth. The present work is devoted to derive symmetry reductions, soliton solutions, and conserved vectors under one-parameter Lie group transformations. Meanwhile, the invariance property of Lie groups validates the reducibility of the system. Therefore, a continual process of reductions transmutes the test system into a system of ordinary differential equations. Thus, this integrability property leads to generalized exact solutions with parametric constraints. These solutions entail all three arbitrary functions, f 1(y), f 2(t), f 3(t) existing in infinitesimals and various free parameters. On account of existing functions and free parameters, the obtained solutions are more admirable than those of preceding works (Kassem and Rasheed 2019 Chinese J. Phys. 57 90–104, Kumar et al 2022 Math. Comput. Simulat. 196 319–35, Tanwar and Kumar 2024 J. Ocean Eng. Sci. 9 199–206) and helpful to describe significant physical nature under wide parametric ranges. The deductions of previous results on imposing the particular values to arbitrary functions and constants ensure the generalness and novelty of derived results. Moreover, adjoint equations and conserved quantities associated with Lagrangian formulation have been evolved via Lie symmetries. We numerically simulate these solutions to analyze their physical nature. Consequently, soliton fusion and fission, line solitons, doubly solitons, multisolitons, and bright and dark soliton nature are discussed, which validate the physical relevance of the solutions.

Nonlinear wave dynamics of (1+1)- dimensional conformable coupled nonlinear Higgs equation using modified ( G′G2)-expansion method

Abstract The conformable coupled nonlinear Higgs equation has been analyzed here using the modified \((\frac{G'}{G^2})\)-expansion method. This equation describes the dynamics of conserved scalar nucleons interacting with neutral scalar mesons. Accordingly, new exact solutions have been obtained. Functions derived from trigonometry, hyperbolic operations, and rational functions are used to define the traveling waveform responses. These solutions are illustrated with various interesting figures for different parameters. Additionally, physical interpretations of these results are provided. The solutions uncovered in this study are more thorough, wide-ranging, and, in some cases, novel, as they have not been discovered in previous studies. {\color{red}These solutions may be useful in understanding nonstandard interactions or forces and potentially addressing phenomena beyond the standard model.} The implemented strategies are efficient, align with computer algebraic expressions, and provide a sufficient number of diverse wave solutions.

Mid-life: a critical window for intervention during aging

Mid-life represents a pivotal period marked by profound physiological and metabolic transitions, increasing susceptibility to chronic diseases. This review explores the molecular and systemic underpinnings of mid-life transition by integrating insights from recent studies that elucidate aging-associated changes in the plasma proteome, immune system, adipose tissue remodeling, and cellular senescence. Nonlinear waves of proteomic alterations have been identified as critical mid-life transitions in inflammatory and hormonal pathways. In addition, sex-specific immune aging trajectories have linked adaptive immunity decline and innate immune activation to metabolic vulnerabilities in mid-life. Moreover, adipose tissue’s central role has been established in mid-life transitions as its early remodeling and inflammatory cytokine secretion drive the systemic aging and metabolic stress. Furthermore, Glb1-2A-mCherry reporter has been introduced to monitor systemic aging, identifying mid-life as a crucial phase for cardiac hypertrophy and senescence-induced inflammation. Collectively, these findings have established our understanding of mid-life transitions, underscoring the interplay between aging processes and metabolic health, with mid-life emerging as a critical window for intervention. This review also underscores biomarkers and therapeutic strategies to alleviate the metabolic challenges of mid-life, thereby promoting healthy aging.

Exploring novel solitary wave phenomena in Klein–Gordon equation using ϕ6 model expansion method

In this study, the ϕ6-model expansion method is showed to be useful for finding solitary wave solutions to the Klein–Gordon (KG) equation. We develop a variety of solutions, including Jacobi elliptic functions, hyperbolic forms, and trigonometric forms, so greatly enhancing the range of exact solutions attainable. The 2D, 3D, and contour plots clearly show different types of solitary waves, like bright, dark, singular, and periodic solitons. This gives us a lot of information about how the KG equation doesn’t work in a straight line. Our findings highlight the ϕ6 model as a powerful tool to study nonlinear wave equations, improve our understanding of their complex dynamics, and increase the scope for theoretical exploration. The ϕ6 model expansion technique is exceptionally adaptable and may be utilised for a wide array of nonlinear partial differential equations. Despite its versatility, the technique may not be applicable to all nonlinear PDEs, especially those that do not meet the specified requirements or structures manageable by this technique. In theoretical physics, particularly in field theory and quantum mechanics, the Klein–Gordon equation is a classical model. By studying this model, we can illustrate the waves and particles movements at relativistic speeds. Among other areas, its significance in cosmology, quantum field theory, and the study of nonlinear optics are widely considered. Additionally, it provides exact solutions and nonlinear dynamics have various applications in applied mathematics and physics. The study is novel because it provides a new understanding of the complex behaviours and various waveforms of the controlling model by means of detailed evaluation. Future research could focus on further exploring the stability and physical implications of these solutions under different conditions, thereby advancing our knowledge of nonlinear wave phenomena and their applications in physics and beyond.

Dynamics of radial threshold solutions for generalized energy-critical Hartree equation

Abstract In this paper, we study long time dynamics of radial threshold solutions for the focusing, generalized energy-critical Hartree equation and classify all radial threshold solutions. The main arguments are the spectral theory of the linearized operator, the modulational analysis and the concentration compactness rigidity argument developed by T. Duyckaerts and F. Merle to classify all threshold solutions for the energy critical NLS and NLW in [T. Duyckaerts and F. Merle, Dynamic of threshold solutions for energy-critical NLS, Geom. Funct. Anal. 18 2009, 6, 1787–1840, T. Duyckaerts and F. Merle, Dynamics of threshold solutions for energy-critical wave equation, Int. Math. Res. Pap. IMRP 2008 2008, Art ID rpn002], later by D. Li and X. Zhang in [D. Li and X. Zhang, Dynamics for the energy critical nonlinear Schrödinger equation in high dimensions, J. Funct. Anal. 256 2009, 6, 1928–1961, D. Li and X. Zhang, Dynamics for the energy critical nonlinear wave equation in high dimensions, Trans. Amer. Math. Soc. 363 2011, 3, 1137–1160] in higher dimensions. The new ingredient here is to solve the nondegeneracy of positive bubble solutions with nonlocal structure in H ˙ 1 ⁢ ( ℝ N ) {\dot{H}^{1}(\mathbb{R}^{N})} (i.e. the spectral assumption in [C. Miao, Y. Wu and G. Xu, Dynamics for the focusing, energy-critical nonlinear Hartree equation, Forum Math. 27 2015, 1, 373–447]) by the nondegeneracy result of positive bubble solution in L ∞ ⁢ ( ℝ N ) {L^{\infty}(\mathbb{R}^{N})} in [X. Li, C. Liu, X. Tang and G. Xu, Nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equations, preprint 2023, https://arxiv.org/abs/2304.04139] and the Moser iteration method in [S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of ℝ n \mathbb{R}^{n} , Appunti. Sc. Norm. Super. Pisa (N. S.) 15, Edizioni della Normale, Pisa, 2017], which is related to the spectral analysis of the linearized operator with nonlocal structure, and plays a key role in the construction of the special threshold solutions, and the classification of all threshold solutions.

Noether and partial Noether approach for the nonlinear (3+1)-dimensional elastic wave equations

The Lie group method is a powerful technique for obtaining analytical solutions for various nonlinear differential equations. This study aimed to explore the behavior of nonlinear elastic wave equations and their underlying physical properties using Lie group invariants. We derived eight-dimensional symmetry algebra for the (3+1)-dimensional nonlinear elastic wave equation, which was used to obtain the optimal system. Group-invariant solutions were obtained using this optimal system. The same analysis was conducted for the damped version of this equation. For the conservation laws, we applied Noether’s theorem to the nonlinear elastic wave equations owing to the availability of a classical Lagrangian. However, for the damped version, we cannot obtain a classical Lagrangian, which makes Noether’s theorem inapplicable. Instead, we used an extended approach based on the concept of a partial Lagrangian to uncover conservation laws. Both techniques account for the conservation laws of linear momentum and energy within the model. These novel approaches add an application of variational calculus to the existing literature. This offers valuable insights and potential avenues for further exploration of the elastic wave equations.

Computational and numerical solutions to the Benney–Luke equation: Insights into nonlinear long wave dynamics in dispersive media

The current investigation seeks to explore the nonlinear Benney–Luke model, a governing equation pertinent to the propagation of extended waves within dispersive media, such as those observed in water waves and plasma waves, among other nonlinear wave phenomena. This model is distinguished by its incorporation of both dispersive and nonlinear effects, rendering it a valuable instrument for delving into intricate wave dynamics. Notably, it shares similarities with other prominent nonlinear evolution equations, including the Korteweg–de Vries (KdV) equation and the Boussinesq equation, both of which have been extensively examined within the domain of water waves and associated phenomena.Employing analytical methodologies, specifically the Khater (Khat III) and modified Kudryashov (MKud) methods, the research endeavors to derive exact solutions for the Benney–Luke equation. These solutions serve to elucidate various aspects of long wave behavior in dispersive media, encompassing their propagation characteristics, stability, and potential for inter-wave interactions. Furthermore, a trigonometric-quantic-B-spline (TQBS) scheme is adopted as a numerical approach to corroborate the precision of the derived analytical solutions and demonstrate their relevance across diverse physical scenarios linked to the model under examination.The research outcomes underscore the efficacy of the analytical techniques in generating dependable solutions for the Benney–Luke model. Moreover, through the utilization of the TQBS numerical scheme, the accuracy of the derived analytical solutions is affirmed, thereby ensuring their pragmatic utility across a spectrum of physical scenarios pertinent to the investigated model. This endeavor holds significance in its contribution towards advancing the comprehension of nonlinear wave phenomena within dispersive media, with ramifications spanning multiple disciplines, including fluid dynamics, plasma physics, and nonlinear optics. The novelty of this study lies in its application of the Khater (Khat III) and modified Kudryashov (MKud) methods to the Benney–Luke equation, alongside the integration of the TQBS numerical scheme for solution validation. Positioned within the domain of nonlinear partial differential equations, this research aligns with its broader applications in physics, particularly concerning the investigation of dispersive wave phenomena.

Nonlinear waves and modulation instability in the generalized Burger-BBM equation

Nonlinear waves and modulation instability in the generalized Burger-BBM equation

Extended spectral element formulation for modeling the propagation of nonlinear ultrasonic waves produced by multiple cracks in solid media

Extended spectral element formulation for modeling the propagation of nonlinear ultrasonic waves produced by multiple cracks in solid media

Wave-breaking phenomena around a wedge-shaped bow

Bow wave breaking has long been a research challenge in the research field of ship hydrodynamic. This study simplifies the ship's bow as a three-dimensional wedge-shaped structure and conducts experimental measurements to investigate the dynamic characteristics of bow wave breaking. The experiments are carried out in a recirculating water channel, with the force transducer and wave height probes used for measurement. The complex mechanisms influencing bow wave breaking in real-world scenarios can be influenced by factors such as flow velocity, draft, flooding angle, flare angle, and yaw angle. Experiments are carried out under various conditions by controlling one factor at a time. The results show that more intense wave breaking not only increases the resistance force on the wedge but also raises the wave height of the measuring point in the non-breaking region and the nonlinearity in the breaking region. The high-frequency fluctuations of wave height in the breaking region become more obvious. Through a quantitative comparison of different factors on the wave-breaking phenomena, it can be found that the flooding angle has a significant impact on resistance, which helps explain why slender hull designs are commonly used for fast ships. The yaw angle plays a crucial role in affecting the bow waves, influencing both stable wave height and nonlinear breaking waves. The measurement results presented in this paper can provide valuable validation data and flow insights for studies on ship bow wave breaking.

Nonlinear dynamic wave properties of travelling wave solutions in in (3+1)-dimensional mKdV-ZK model.

The (3+1)-dimensional mKdV-ZK model is an important framework for studying the dynamic behavior of waves in mathematical physics. The goal of this study is to look into more generic travelling wave solutions (TWSs) for the generalized ion-acoustic scenario in three dimensions. These solutions exhibit a combination of rational, trigonometric, hyperbolic, and exponential solutions that are concurrently generated by the new auxiliary equation and the unified techniques. We created numerous soliton solutions, including kink-shaped soliton solutions, anti-kink-shaped solutions, bell-shaped soliton solutions, periodic solutions, and solitary soliton solutions, for various values of the free parameters in the produced solutions. The attained solutions are displayed geometrically in the surface plot (3-D), contour, and combined two-dimensional (2-D) figures. The combined 2-D figure would make it easier to understand the impact of the speed of the wave. Based on time, the influence of the nonlinear parameter β on wave type is comprehensively investigated using various figures, demonstrating the significant impact of nonlinearity. These graphical representations are based on specific parameter settings, which help to grasp the model's intricate general behavior. However, the results of this research are compared with the outcomes obtained in published literature executed by other scholars. The results indicate the approach's effectiveness and reliability, making it suitable for widespread use in a range of sophisticated nonlinear models. These techniques successfully generate inventive soliton solutions for various nonlinear models, which are crucial in mathematical physics.

Numerical investigation on the hydrodynamic wave forces on the three barges in proximity

Two-dimensional nonlinear hydrodynamic wave forces acting on three side-by-side barges are numerically investigated in this work. A finite volume viscous flow model, incorporating the volume of fluid method for free-surface capturing, is implemented based on OpenFOAM® platform to simulate interactions between nonlinear free-surface waves and multiple barges. The study explores the impact of fluid resonances in the gaps between three barges on the hydrodynamic wave forces. The initial transient and final quasi-steady wave forces are first examined to highlight the significance of transient evolution stage. The steady-state wave forces are subsequently concerned with the amplitude-frequency response of wave forces, along with the pressure and viscous force components. Relationship between wave forces and fluid oscillations within the gaps is then clarified to explain the mechanism of wave forces on the barges associated with gap resonance. Significant nonlinear higher-order harmonics are further demonstrated, with their origins closely linked to fluid oscillations within gaps between the barges. Contributions of higher-order harmonics to the overall wave force amplitudes are explored through harmonic analysis utilizing Fourier transformations. Finally, effects of the normalized incident wave amplitudes on the amplitudes of wave forces and higher-order harmonics are found to be highly dependent on frequency bands.