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ABSTRACT This article concerns the use of the residual power series (RPS) approach to get the semi‐analytical solutions of multi‐dimensional Korteweg‐de Vries (KdV), that is, shallow water wave equations with any arbitrary time‐fractional order (γ$$ \gamma $$). The time‐fractional derivative is considered in the Caputo sense. The solutions of the fractional KdV equation obtained using the RPS method for γ=1$$ \gamma =1 $$ are compared with the exact solutions of the integer‐order KdV equation to show the RPS method's effectiveness, accuracy, and convergence. Some examples are also solved to study the effects of the fractional derivative (γ)$$ \left(\gamma \right) $$ on the wave's amplitude. [ABSTRACT FROM AUTHOR]

The characteristics of kinetic Alfvénic cnoidal waves (KACWs) in Saturn's magnetosphere plasma composed of two temperature superthermal electrons and inertial ions are presented. The Korteweg–de Vries (KdV) equation and its cnoidal wave solution are derived by adopting reductive perturbation technique. Only positive potential kinetic Alfvénic cnoidal and solitary waves are evolved in Saturnian magnetospheric plasma. The influence of superthermality of cold electrons, concentration of hot electrons, plasma beta and angle of propagation of the wave with respect to the magnetic field has been analyzed on the characteristics of KACWs. The findings of this investigation may shed the light on the possible existence of KACWs and acceleration as well as energy transportation in space plasmas especially in Saturn's magnetosphere. [ABSTRACT FROM AUTHOR]

Flow structures beneath a moving disturbance along a water free surface in the weakly nonlinear weakly dispersive regime in a sheared channel with finite depth and constant vorticity are investigated. We compute the exact two branches of steady solutions in the disturbance moving frame. The velocity field in the bulk fluid is approximated which allows us to compute the flow structures beneath the free surface including stagnation points and Kelvin cat-eyes structures. We show that stagnation points exist only in one branch of solutions. The bifurcation of the flow is analyzed according to the intensity of the vorticity and the speed of the moving disturbance. Differently from the unforced problem, stagnation points can arise for small values of the vorticity as long as the moving disturbance travels sufficiently fast. [ABSTRACT FROM AUTHOR]

Abstract We propose a general approach to quasi-deform the Korteweg–De Vries (KdV) equation by deforming its Hamiltonian. The standard abelianization process based on the inherent sl(2) loop algebra leads to an infinite number of anomalous conservation laws, that yield conserved charges for definite space-time parity of the solution. Judicious choice of the deformed Hamiltonian yields an integrable system with scaled parameters as well as a hierarchy of deformed systems, some of which possibly are quasi-integrable. One such system maps to the known quasi-deformed nonlinear Schrödinger (NLS) soliton in the already known weak-coupling limit, whereas a generic scaling of the KdV amplitude $$u\rightarrow u^{1+\varepsilon }$$ also suggests quasi-integrability under an order-by-order expansion. In general, these deformed KdV solutions need to be parity-even for quasi-conservation that agrees with our analytical results. Following the recent demonstration of quasi-integrability in regularized long wave (RLW) and modified regularized long wave (mRLW) systems by ter Braak et al. (Nucl Phys B 939:49–94, 2019), that are particular cases of the present approach, general soliton solutions should numerically be accessible.

Internal solitary waves in polar regions have attracted much interest recently. It is important to understand how sea ice affects them as this may have a profound influence on human activities and the environment. In this study, experiments on internal solitary waves with and without two types of sea ice (ice sheet and ice keel) are presented, as well as corresponding simulations using the Korteweg-de Vries (KdV) equation, the Benjamin-Ono (BO) equation, and the variable-coefficient Korteweg-de Vries (vKdV) equation, which is a derivation of the KdV equation. Comparison between experiments without sea ice and simulations using the KdV and BO equations proves the suitability of the former over the latter for this study. The experiments with sea ice and theoretical simulations using the vKdV equation provide evidence for wave deformation, oscillation occurring in the rear of the wave, and a decrease in amplitude. The latter suggests possibilities of energy dissipation or the emission of small amplitude linear waves. The sharp vertices of the ice result in occasional inconsistency with the vKdV predictions. Nonetheless, the vKdV equation is still suitable for modeling internal solitary waves under sea ice, giving generally accurate results that can assist further studies. This is the first time the vKdV equation has been applied to investigate the impacts of sea ice on internal solitary waves. [ABSTRACT FROM AUTHOR]

In this paper, we derive a higher‐order Korteweg–de Vries (HKdV) equation as a model to describe the unidirectional propagation of waves on an internal interface separating two fluid layers of varying densities. Our model incorporates underlying currents by permitting a sheared current in both fluid layers, and also accommodates the effect of the Earth's rotation by including Coriolis forces (restricted to the Equatorial region). The resulting governing equations describing the water wave problem in two fluid layers under a "flat‐surface" assumption are expressed in a general form as a system of two coupled equations through Dirichlet–Neumann (DN) operators. The DN operators also facilitate a convenient Hamiltonian formulation of the problem. We then derive the HKdV equation from this Hamiltonian formulation, in the long‐wave, and small‐amplitude, asymptotic regimes. Finally, it is demonstrated that there is an explicit transformation connecting the HKdV we derive with the following integrable equations of a similar type: KdV5, Kaup–Kuperschmidt equation, Sawada–Kotera equation, and Camassa–Holm and Degasperis–Procesi equations. [ABSTRACT FROM AUTHOR]

A weak formulation is devised for the K(m,n)$K(m,n)$ equation, which is a nonlinearly dispersive generalization of the gKdV equation having compacton solutions. With this formulation, explicit weak compacton solutions are derived, including ones that do not exist as classical (strong) solutions. Similar results are obtained for a nonlinearly dispersive generalization of the gKP equation in two dimensions, which possesses line compacton solutions. [ABSTRACT FROM AUTHOR]

The emergence of higher-dimensional evolution equations in dissimilar scientific arenas has been on the rise recently with a vast concentration in optical fiber communications, shallow water waves, plasma physics, and fluid dynamics. Therefore, the present study deploys certain improved analytical methods to perform a solitonic analysis of the newly introduced three-dimensional nonlinear dynamical equations (all within the current year, 2024), which comprise the new (3 + 1) Kairat-II nonlinear equation, the latest (3 + 1) Kairat-X nonlinear equation, the new (3 + 1) Boussinesq type nonlinear equation, and the new (3 + 1) generalized nonlinear Korteweg–de Vries equation. Certainly, a solitonic analysis, or rather, the admittance of diverse solitonic solutions by these new models of interest, will greatly augment the findings at hand, which mainly deliberate on the satisfaction of the Painleve integrability property and the existence of solitonic structures using the classical Hirota method. Lastly, this study is relevant to contemporary research in many nonlinear scientific fields, like hyper-elasticity, material science, optical fibers, optics, and propagation of waves in nonlinear media, thereby unearthing several concealed features. [ABSTRACT FROM AUTHOR]

In this plasma model, consisting of ions, electrons, and positrons have been theoretically investigated when both the electrons and positrons are obeying q-nonextensive velocity distribution. The reductive perturbation method is used to obtain Korteweg de Vries (KdV) equation describing the basic set of normalized fluid equations. The existence of ion-acoustic solitary waves depending on nonextensive parameter, electron to positron temperature ratio, ion to electron temperature ratio and streaming velocity are investigated numerically. It has been found that solely fast ion-acoustic modes can produce the coexistence of small amplitude rarefactive solitons. [ABSTRACT FROM AUTHOR]

This paper presents a comprehensive study in a collisionless plasma composed of charged state of heavy positive ion and light positive as wel as negative ion. By deriving the Korteweg-de Vries (KdV) equation and by using its standard solution we analyze the characteristics of the solitary profile under varying parameters. We found that the solution gives both rarefactive and compressive soliton. The compressive structures are formed for the slow mode, while rarefactive solitary structures are formed for the fast mode. Furthermore, with the application of planar dynamical systems bifurcation theory, the phase portraits have been analyzed. This dynamical system analysis allowed us to extract important information on the stability of these structures as represented by the KdV equation. [ABSTRACT FROM AUTHOR]

From the finite gap solutions of the KdV equation expressed in terms of abelian functions we construct solutions to the Schrödinger equation with a KdV potential in terms of fourfold Fredholm determinants. For this we establish a connection between Riemann theta functions and Fredholm determinants and we obtain multi-parametric solutions to this equation. As a consequence, a double Wronskian representation of the solutions to this equation is constructed. We also give quasi-rational solutions to this Schrödinger equation with rational KdV potentials. [ABSTRACT FROM AUTHOR]

The dynamics of a disturbance with finite energy in an infinite monatomic nonlinear one-dimensional lattice are analyzed. Based on the energy dynamics approach proposed earlier, we focus on such disturbance spatial characteristic as the position of its energy center. Restricting our analysis to long-wave low-amplitude disturbances, we investigate the dynamics of the α-FPU chain using its continuous version described by the KdV equation. We establish a connection of the Lagrangian and the energy of the original chain with the two conserving quantities of the KdV equation. Using these two quantities and the known properties of the KdV equation, we propose a method for determining the velocity of the energy center of the disturbance at large times based on the initial conditions. [ABSTRACT FROM AUTHOR]

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The current study is important from two perspectives. Firstly, in this article, we suggest a novel analytical technique for creating the exact solutions to nonlinear partial differential equations (NLPDEs). In order to study the dynamical behaviors of various wave phenomena, we can construct the several exact solutions in the form of Jacobi elliptic solutions, hyperbolic solutions, trigonometric solutions, and exponential solutions by using this method. Secondly, we consider a more generalized form of the (2+1)‐dimensional Korteweg–de Vries (KdV) and modified Korteweg‐de Vries (mKdV) equations that plays an important role in describing the shallow water. We successfully extract several soliton solutions for the examined equation. By choosing the appropriate parameters, some graphs of the discovered solutions have been represented in the figures. Every obtained solution has been demonstrated to satisfy the corresponding equation. The findings demonstrate that the method can be applied to solve a number of nonlinear evolution equations. The new solutions and the paper's findings could improve our understanding of how the waves move through shallow water and open up new research avenues. [ABSTRACT FROM AUTHOR]

In this paper, we have investigated the propagation of dust-acoustic dressed solitons (DADS) which are influenced by the presence of dust concentrations and hot isothermal electrons in an inhomogeneous plasma. The dynamics of ions, electrons and dust grains are described by the governing equations of fluid, and on the basis of reductive perturbations technique (RPT), we have derived the Korteweg–de Vries (KdV) equation. Accordingly, the solitary solutions are obtained. The DADS wave profile shows the variations in the amplitude, width and velocity for various dust multiplicity of dust grains and dust concentrations in the system. In this investigation, we have observed that the DADS allow only the compressive solitons up to a certain critical value of dust concentration parameter as a consequence of the presence of negatively charged dust particles. We have further observed that the DADS also support the rarefactive solitons beyond the critical value of dust concentration parameter in the system. [ABSTRACT FROM AUTHOR]

In this paper we investigate some number theoretic properties of the frequencies of the Korteweg--de Vries equation on the torus, relevant for the stability of finite gap solutions. [ABSTRACT FROM AUTHOR]

The existence and propagating characteristics of small amplitude dust-ion-acoustic (DIA) Korteweg-de Vries (KdV) and modified KdV solitons in a three component magnetized plasma composed of positive inertial ions with pressure variation, noninertial electrons and negative charged immobile dust grains are theoretically and numerically investigated when the electrons obey a q-nonextensive velocity distribution. Utilizing the reductive perturbation method, to derive KdV and modified KdV equations and obtain the DIA soliton solutions along with the corresponding small amplitude potentials. This study shows that there are compressive and/or rarefactive solitons and no soliton at all, due to the parametric dependency on the first-order nonlinear coefficient through the number density of positive ions and negative dust grains and the electron nonextensivity. The coexistence of compressive and rarefactive solitons appears by raising the measure of nonlinearity coefficient to the second-order using the modified KdV equation. The properties such as speed, amplitude, width etc. of the propagating soliton are numerically discussed.

Motivated by the question how to describe long‐wave dynamics on periodic networks, we consider a Boussinesq equation posed on the infinite periodic necklace graph. For the description of long‐wave traveling waves, we derive the KdV equation and establish the validity of this formal approximation by providing estimates for the error. The proof is based on suitable energy estimates. As a consequence of the approximation result, the soliton dynamics present in the KdV equation can approximately be seen for the original system, too. There are no serious obstacles to transfer the analysis to other dispersive systems or to other periodic quantum graphs for which the KdV equation can be derived. However, a general KdV validity theory on quantum graphs would be extremely abstract and of minor use. [ABSTRACT FROM AUTHOR]

In the research paper, the propagation of nonlinear dust ion acoustic solitary waves in a collisionless dusty plasma consisting of negatively charged dust grains, ions, and isothermal electrons in the background of neutral particles is investigated. The reductive perturbation method has been employed to derive the Korteweg–de Vries (KdV) equation for small but finite-amplitude dust ion acoustic waves. The Agrawal technique is used to convert the regular KdV equation to the space-time fractional KdV equation. The Riemann–Liouville definition of the fractional derivative is used to describe the time fractional operator. The Hirota bilinear method is used to derive the multi-soliton solution of the time fractional KdV equation. The effects of the plasma parameters and time fractional parameters on the multisoliton of dust ion acoustic waves for the system are studied. We have also reported the conversion of two-soliton structures to one soliton, depending on the fractional parameter α . This observation in the plasma literature has never been reported before. The result of the present investigation may be applicable to some plasma environments, such as the Earth's mesosphere, cometary tails, Jupiter's magnetosphere, etc. [ABSTRACT FROM AUTHOR]

Seismic oceanography has been widely used in the study of internal solitary waves (ISWs) in recent years, and has achieved remarkable results. In this paper, we analyzed the multi-channel seismic reflection data in the Canterbury Basin offshore New Zealand from January 9 to January 29, 2000, collected by R/V Maurice Ewing. We observed 4 groups of ISWs (labeled ISW1s to ISW4s) on 4 seismic survey lines. We studied their waveforms and propagation speeds in detail. There are two theoretical structures used to describe the vertical waveform of ISWs: the first-order nonlinear vertical structure and the linear vertical structure. We found that ISW1s fit the nonlinear structure well, ISW3s and ISW4s fit the linear structure, and ISW2 does not fit either one. As the water depth increases, the waveforms of all ISWs gradually widen. Two satellite SAR images reveal that all ISWs generally travel shoreward across the isobaths. However, the propagation direction of ISW1s is about 354°-360° (clockwise from due north), different from the propagation directions of other ISWs (about 22°-26°), which explains why ISW1s have the largest characteristic half-height width. The estimated propagation speeds are close to the theoretical speeds, confirming our speed correction method. In the end, we also discuss the interaction of ISWs and eddies. [ABSTRACT FROM AUTHOR]

We study the tritronquée solution u(x, t) of the PI² equation, the second member of the Painlevé I hierarchy. This particular solution is also known as the Gurevich--Pitaevskii solution of the KdV equation. It is pole-free on the real line and has various applications in mathematical physics. We obtain a full asymptotic expansion of u(x, t) as x → ± ∞, uniformly for the parameter t in a large interval. Based on this result, we successfully derive the total integrals of u(x, t) and the associated Hamiltonian with respect to x ∈ R . Surprisingly, although u(x, t) exhibits significant differences between t > 0 and t < 0, both integrals equal zero for all t. [ABSTRACT FROM AUTHOR]

We consider methods for constructing finite-gap solutions of the real classical modified Korteweg–de Vries equation and elliptic finite-gap potentials of the Dirac operator. The Miura transformation is used in both methods to relate solutions of the Korteweg–de Vries and modified Korteweg–de Vries equations. We present examples. [ABSTRACT FROM AUTHOR]

The periodic, quasi-periodic and solitonic properties of ion-acoustic wave (IAW) with ion pressure anisotropy are examined in a magnetized nonthermal electron-ion plasma. Using CGL theory, double adiabatic equation of state for ions is assumed and inertialess electrons are considered to follow nonthermal κ -distribution. A damping in the system due to ions and neutral atoms collision is also included in the presence of ion pressure anisotropy. The reductive perturbation (RP) method is used to derive Kortewge-de Vries (KdV) equation in a collisionless plasma case and the damped KdV equation with collisional effects for IAW propagation in a magnetized plasma. The effects of anisotropic ion pressure and spectral index κ for nonthermal electrons on the formation of quasi-periodic nonlinear IAW structures in the presence of external periodic force and their changing from period doubling to multi-periods are investigated in detail. [ABSTRACT FROM AUTHOR]

Internal solitary waves in polar regions have attracted much interest recently. It is important to understand how sea ice affects them as this may have a profound influence on human activities and the environment. In this study, experiments on internal solitary waves with and without two types of sea ice (ice sheet and ice keel) are presented, as well as corresponding simulations using the Korteweg-de Vries (KdV) equation, the Benjamin-Ono (BO) equation, and the variable-coefficient Korteweg-de Vries (vKdV) equation, which is a derivation of the KdV equation. Comparison between experiments without sea ice and simulations using the KdV and BO equations proves the suitability of the former over the latter for this study. The experiments with sea ice and theoretical simulations using the vKdV equation provide evidence for wave deformation, oscillation occurring in the rear of the wave, and a decrease in amplitude. The latter suggests possibilities of energy dissipation or the emission of small amplitude linear waves. The sharp vertices of the ice result in occasional inconsistency with the vKdV predictions. Nonetheless, the vKdV equation is still suitable for modeling internal solitary waves under sea ice, giving generally accurate results that can assist further studies. This is the first time the vKdV equation has been applied to investigate the impacts of sea ice on internal solitary waves.

It is necessary to utilize certain stochastic methods while finding the soliton solutions since several physical systems are by their very nature stochastic. By adding randomness into the modeling process, researchers gain deeper insights into the impact of uncertainties on soliton evolution, stability, and interaction. In the realm of dynamics, deterministic models often encounter limitations, prompting the incorporation of stochastic techniques to provide a more comprehensive framework. Our attention was directed towards the short-wave intermediate dispersive variable (SIdV) equation with the Wiener process. By integrating advanced methodologies such as the modified Kudrayshov technique (KT), the generalized KT, and the sine-cosine method, we delved into the exploration of diverse solitary wave solutions. Through those sophisticated techniques, a spectrum of the traveling wave solutions was unveiled, encompassing both the bounded and singular manifestations. This approach not only expanded our understanding of wave dynamics but also shed light on the intricate interplay between deterministic and stochastic processes in physical systems. Solitons maintained stable periodicity but became vulnerable to increased noise, disrupting predictability. Dark solitons obtained in the results showed sensitivity to noise, amplifying variations in behavior. Furthermore, the localized wave patterns displayed sharp peaks and periodicity, with noise introducing heightened fluctuations, emphasizing stochastic influence on wave solutions.

The Korteweg-de Vries (KdV) equation is a well-known model equation for unidirectional shallow-water (long) surface waves. The equation includes dispersion and weak non-linearity. The derivation of the equation originates in assuming that the velocity potential takes the form of an asymptotic expansion, and applying this in the classical surface wave problem. While a typical assumption on the relative size of non-dimensional key parameters introduced in the derivation will give the KdV equation as a final result, one can change the assumption on the relative size of parameters, and end up with an equation including terms in higher orders in desired parameters. The present article presents the derivation of an extended form referred to as the eeKdV equation. Information regarding various properties of the flow can be found by studying the derivation of the eeKdV equation itself, and some of the relations found can be used for studying the energy balance of a system modeled by the equation. In line with previous work for the KdV equation, we present here corresponding formulations of energy balance laws for an inertial reference frame, in context of the eeKdV equation. We also present a partial verification of the KdV and eeKdV energy flux expressions by looking at a far-field, uniform flow situation, as well as performing a numerical study to confirm the assumed behaviour of the error in the eeKdV energy equation in the context of a undular bore flow setup. Further, conserved integrals for the eeKdV equation are presented and numerically checked. [ABSTRACT FROM AUTHOR]

The KdV equation serves as a mathematical model for ion acoustic waves in a plasma, long internal waves in a density-stratified ocean, acoustic waves on a crystal lattice, and shallow-water waves with weakly non-linear restoring forces. Owing to its significance and numerous uses, a variety of solutions have been developed and examined in literature. The unexplained superposition of elliptic functions as solutions is one of them. In this article, certain theorems and a corollary on the existence of superposed and superposed-type solutions for KdV equations are discussed. The KdV equation’s six sets of superposed solutions are derived by applying the theorem and corollary on the existence of superposed solutions. It was demonstrated that superposed solutions of the KdV equation could be created by combining two elementary solutions that contained reciprocal Jacobi elliptic functions. Moreover, to explain the existing claimed superposed solutions of this equation in the literature, we proposed a few theorems and corollaries on the existence of these superposed solutions. Among them, the splitting technique theorem is the most significant and captivating. We obtained several superposed-type solutions of KdV equation in terms of the Jacobi elliptic function by using the splitting technique. It is further established that the splitting technique consists of the generalised Miura transformation. This represents an additional modification to the generalised Miura transformation. These theorems provide an explanation for the emergence of various seemingly strange superposition-type solutions to several recently published nonlinear equations. Additionally, the KdV equation was solved using the splitting technique and earlier theorems, which produced a large number of additional superposed-type solutions. It is additionally shown that solutions generated by the generalised Miura transformation are particular instances of solutions that are acquired by using the splitting technique. All of the derived solution characteristics have been demonstrated utilising 3D, and 2D plots. [ABSTRACT FROM AUTHOR]

In this research article, we study the (2+1)-dimensional Korteweg-de Vries equation (KdV), which is derived from Euler's equations related to ideal fluid systems. Our primary goal is to obtain explicit solutions utilizing novel methodologies, particularly the new auxiliary equation and the simplest equation methods. We concentrate on the search for solitary wave solutions inside the (2+1)-dimensional KdV, which is relevant in domains such as fluid dynamics, plasma physics and nonlinear waves. Our objective is to increase knowledge of this equation and give insight into the behavior of solitary waves by employing a novel mathematical technique. The bifurcation analysis is also presented for the model problem. This will be accomplished by displaying our findings in 2D and 3D graphics. [ABSTRACT FROM AUTHOR]

In this paper, the technique involving the e x p a function is employed to calculate analytical soliton solutions for three distinct equations: the (3+1)-dimensional mKdV–Zakharov–Kuznetsov equation, the KdV equation, and the (1+1)-dimensional Mikhailov Novikov–Wang integrable equation, which fractional-order in the sense of conformable derivatives. By selecting some parameter values, a diverse spectrum of soliton solutions is obtained, encompassing kink solitons, singular solitons, and periodic-singular solitons. The representation in physical terms enables the examination of authentic multispecies plasmas, plasma models, and frequency ranges. [ABSTRACT FROM AUTHOR]

In this study, we demonstrate that fixed points can occur in extended b-metric spaces that are not necessarily complete. We must employ a novel form of contraction conditions in order to relax the completeness criterion. It has been investigated how fixed points behave in both partially complete and orbitally complete spaces. We provide some instances to highlight our findings. [ABSTRACT FROM AUTHOR]

It is necessary to utilize certain stochastic methods while finding the soliton solutions since several physical systems are by their very nature stochastic. By adding randomness into the modeling process, researchers gain deeper insights into the impact of uncertainties on soliton evolution, stability, and interaction. In the realm of dynamics, deterministic models often encounter limitations, prompting the incorporation of stochastic techniques to provide a more comprehensive framework. Our attention was directed towards the short-wave intermediate dispersive variable (SIdV) equation with the Wiener process. By integrating advanced methodologies such as the modified Kudrayshov technique (KT), the generalized KT, and the sine-cosine method, we delved into the exploration of diverse solitary wave solutions. Through those sophisticated techniques, a spectrum of the traveling wave solutions was unveiled, encompassing both the bounded and singular manifestations. This approach not only expanded our understanding of wave dynamics but also shed light on the intricate interplay between deterministic and stochastic processes in physical systems. Solitons maintained stable periodicity but became vulnerable to increased noise, disrupting predictability. Dark solitons obtained in the results showed sensitivity to noise, amplifying variations in behavior. Furthermore, the localized wave patterns displayed sharp peaks and periodicity, with noise introducing heightened fluctuations, emphasizing stochastic influence on wave solutions. [ABSTRACT FROM AUTHOR]

In this paper, we consider a non-homogeneous generalized Kortewegde Vries problem with some hypotheses on the right-hand side, and we give a new regularity result of the solution in an anisotropic Sobolev space. Then we apply the obtained result to a non-homogeneous KdV problem. This work is an extension of solvability results for a right-hand side f in Lebesgue space. [ABSTRACT FROM AUTHOR]

We explore the dynamic characteristics of positron acoustic multiple-solitons in an unmagnetized plasma containing mobile cold positrons, Kappa-distributed superthermal hot electrons and positrons, and stationary positive ions. This study investigates the overtaking collisional effects, various parametric impacts, and phase shifts in the electron–positron-ion (e-p-i) plasma. Through the reductive perturbation technique, we derived the Korteweg-de Vries (KdV) equation and the modified Korteweg-de Vries (mKdV) equation. The multiple-soliton solutions (MSS) are then obtained using the simplified Hirota method and Cole-Holf transformation. The investigation shows that the amplitudes and widths of multi-solitons decrease with the increasing hot positron concentration but increase with the increasing hot electron index parameter. The results are expected to help us understand the dynamics of waves propagation in the pulsar magnetosphere, the active galactic nuclei, the quasar's relativistic jet, the inner region of the accretion disk surrounding a black hole, etc. [ABSTRACT FROM AUTHOR]

The existence and propagating characteristics of small amplitude dust-ion-acoustic (DIA) Korteweg-de Vries (KdV) and modified KdV solitons in a three component magnetized plasma composed of positive inertial ions with pressure variation, noninertial electrons and negative charged immobile dust grains are theoretically and numerically investigated when the electrons obey a q-nonextensive velocity distribution. Utilizing the reductive perturbation method, to derive KdV and modified KdV equations and obtain the DIA soliton solutions along with the corresponding small amplitude potentials. This study shows that there are compressive and/or rarefactive solitons and no soliton at all, due to the parametric dependency on the first-order nonlinear coefficient through the number density of positive ions and negative dust grains and the electron nonextensivity. The coexistence of compressive and rarefactive solitons appears by raising the measure of nonlinearity coefficient to the second-order using the modified KdV equation. The properties such as speed, amplitude, width etc. of the propagating soliton are numerically discussed. [ABSTRACT FROM AUTHOR]

The soliton resolution conjecture proposes that the initial value problem can evolve into a dispersion part and a soliton part. However, the problem of determining the number of solitons that form in a given initial profile remains unsolved, except for a few specific cases. In this paper, the authors use the deep learning method to predict the number of solitons in a given initial value of the Korteweg-de Vries (KdV) equation. By leveraging the analytical relationship between Asech2(x) initial values and the number of solitons, the authors train a Convolutional Neural Network (CNN) that can accurately identify the soliton count from spatio-temporal data. The trained neural network is capable of predicting the number of solitons with other given initial values without any additional assistance. Through extensive calculations, the authors demonstrate the effectiveness and high performance of the proposed method. [ABSTRACT FROM AUTHOR]

With the help of the variational approach, a detailed derivation of a generalized dynamical equation is given in a particular case describing a soliton and a shock wave. [ABSTRACT FROM AUTHOR]

We study the invariance properties of the fractional time version of the nonlinear class of equations $ u_{t}^{\alpha}-g(u)\; u_{x}-f(u)\; u_{xxx} = 0 $, where $ 0 < \alpha < 1 $ using some recently developed symmetry-based techniques. The equations reduce to ordinary fractional Airy type, Korteweg-de Vries (KdV) and modified KdV equations through the change of variables provided by the symmetries. Furthermore, we utilize the symmetries to construct conservation laws for the fractional partial differential equations.

Taylor series method is a simple analytical method, which is accessible to all non-mathematician, has slow convergence. This paper develops a new Taylor series based numerical method to overcome the shortcoming of the Taylor series while maintaining its simplicity. Some examples are given, showing its reliability and efficiency. The proposed method is also proved to be extremely effective for initial value problems and boundary value problems. The method provides a universal approach to various highly non-linear problems, and it sheds a bright light on numerical theories for practical engineering applications.