KdV Equation Research Articles

Dynamical behaviors of vector localized wave solutions for the coupled modified Korteweg-de Vries equations

Dynamical behaviors of vector localized wave solutions for the coupled modified Korteweg-de Vries equations

Exact solutions, wave dynamics and conservation laws of a generalized geophysical Korteweg de Vries equation in ocean physics using Lie symmetry analysis

An evolution equation is a partial diferential equation that describes the time evolution of a physical system starting from given initial data. Evolution equations arise from many areas of applied and engineering sciences. To this end, this article investigates the analytical studies of a generalized geophysical Korteweg-de Vries equation in ocean physics. The examination of this model is conducted via the Lie group theory of diferential equations. In the first place, point symmetries, which are constituent elements of a four-dimensional Lie algebra, are systematically computed. Thereafter, one-parameter transformation groups for the algebra are calculated. Besides, going forward, a one-dimensional optimal system of subalgebras is derived in a procedural manner. Sequel to this, the subalgebras and combination of the achieved symmetries are invoked in the reduction proces which enables the derivation of nonlinear ordinary diferential equations associated with the generalized geophysical Korteweg-de Vries equation under study. Most of the achieved nonlinear ordinary diferential equations are further solved either via direct integration or using a power series approach. Furthermore, travelling wave solutions are initially obtained. This is attained via direct integration and the use of Jacobi elliptic function approach. These techniques enable the attainment of various exact soliton solutions, including non-topological soliton solutions as well as general periodic function solutions of note, such as cosine amplitude, sine amplitude, and delta amplitude solutions of the model. Furthermore, numerical simulations of the solutions are invoked to gain a gross knowledge of the physical phenomena represented by the under-study generalized geophysical Korteweg-de Vries equation in ocean physics. In the end, the investigation further gives attention to the calculation of conserved vectors for the model using Ibragimov's theorem for conservation laws, as well as Noether's theorem.

Approximation of translation invariant Koopman operators for coupled non-linear systems.

Many physical systems exhibit translational invariance, meaning that the underlying physical laws are independent of the position in space. Data driven approximations of the infinite dimensional but linear Koopman operator of non-linear dynamical systems need to be physically informed in order to respect such physical symmetries. In the current work, we introduce a translation invariant extended dynamic mode decomposition (tieDMD) for coupled non-linear systems on periodic domains. This is achieved by exploiting a block-diagonal structure of the Koopman operator in Fourier space. Variants of tieDMD are applied to data obtained on one-dimensional periodic domains from the non-linear phase-diffusion equation, the Burgers equation, the Korteweg-de Vries equation, and a coupled FitzHugh-Nagumo system of partial differential equations. The reconstruction capability of tieDMD is compared to existing linear and non-linear variants of the dynamic mode decomposition applied to the same data. For the regarded data, tieDMD consistently shows superior capabilities in data reconstruction.

Transient modes for the coupled modified Korteweg-de Vries equations with negative cubic nonlinearity: Stability and applications of breathers.

Dynamics and properties of breathers for the modified Korteweg-de Vries equations with negative cubic nonlinearities are studied. While breathers and rogue waves are absent in a single component waveguide for the negative nonlinearity case, coupling can induce regimes of modulation instabilities. Such instabilities are correlated with the existence of rogue waves and breathers. Similar scenarios have been demonstrated previously for coupled systems of nonlinear Schrödinger and Hirota equations. Both real- and complex-valued modified Korteweg-de Vries equations will be treated, which are applicable to stratified fluids and optical waveguides, respectively. One special family of breathers for coupled, complex-valued equations is derived analytically. Robustness and stability of breathers are studied computationally. Knowledge of the growth rates of modulation instability of plane waves provides an instructive prelude on the robustness of breathers to deterministic perturbations. A theoretical formulation of the linear instability of breathers will involve differential equations with periodic coefficient, i.e., a Floquet analysis. Breathers associated with larger eigenvalues of the monodromy matrix tend to suffer greater instability and increased tendency of distortion. Predictions based on modulation instability and Floquet analysis show excellent agreements. The same trend is obtained for simulations conducted with random noise disturbances. Linear approaches like modulation instabilities and Floquet analysis, thus, generate a very illuminating picture of the nonlinear dynamics.

Extreme events in a damped Korteweg–de Vries (KdV) autonomous system: A comprehensive analysis

This manuscript explores extreme events in a three-dimensional, damped Korteweg–de Vries (KdV) autonomous system that is derived from the jerk system with a modified basin. Through rigorous stability analysis, we identify the extreme events, confirmed using phase portraits, Poincaré return maps, time series, and probability distribution functions. The Dragon-king phenomenon is obtained and self-organised criticality is discussed. The regime of extreme events in a wide parameter range was obtained through two-parameter scanning. The analog circuit was constructed to mimicking dynamics of damped KdV equation and demonstrated in real-time hardware experiment as well as PSpice simulation. The experimental results show excellent agreement with the numerically obtained results. This study is the first comprehensive examination of extreme events in the damped KdV system, highlighting their nature and potential applications in chaos-based dynamical systems.

Cauchy matrix approach for H1 a equation in the torqued Adler–Bobenko–Suris lattice list

As a torqued version of the lattice potential Korteweg–de Vries equation, the H1 a is an integrable nonsymmetric lattice equation with only one spacing parameter. In this paper, we present the Cauchy matrix scheme for this equation. Soliton solutions, Jordan-block solutions and soliton-Jordan-block mixed solutions are constructed by solving the determining equation set. All the obtained solutions have jumping property between constant values for fixed n and demonstrate periodic structure.

Ion-acoustic solitons in multicomponent plasma with two temperature non-Maxwellian electrons

The study of ion-acoustic solitons (IASs) in an unmagnetised collisionless plasma system having inertial fluid ions and two temperature hot as well as cold electrons obeying Cairns–Tsallis (CT) distribution is presented. Using reductive perturbation method, the KdV equation is derived. Further, introducing higher-order effects of nonlinearity and dispersion, a nonlinear KdV-type inhomogeneous equation is derived that supports the formation of dressed solitons. The combined effects of higher-order corrections and various plasma parameters are analysed on the characteristics of the different kinds of ion-acoustic solitons with implications to Saturn's magnetosphere.

Numerical solutions of KDV and mKDV equations: Using sequence and multi-core parallelization implementation

In this study, we use the explicit difference method and also the combination of the explicit difference method with the implicit method to numerically solve the third order of Korteweg–de Vries (KDV) and modified Korteweg–de Vries (mKDV) equations. This method is applied to obtain soliton solutions on several problems and the results obtained are compared with each other and the exact solution of the problem. Also, due to that these problems are very ill-posed, when the time increases, the waveform of solitons and the input parameters of these methods are carefully examined. The L∞, L2 and mean-square errors of the solutions show that these methods can be applied to different problems and have very good accuracy. These methods are widely used in combination with other algorithms, but their very high execution time, especially in this category of problems, is always a big limitation. In this paper, a parallel approach for implementing these methods is presented and very good results have been obtained compared to sequential implementation. This is a reference to further study solutions of other models of KDV and mKDV problems as well as solving their inverse problems.

Numerical Simulation of Kink Collisions, Analytical Solutions and Conservation Laws of the Potential Korteweg–de Vries Equation

Numerical Simulation of Kink Collisions, Analytical Solutions and Conservation Laws of the Potential Korteweg–de Vries Equation

On decay properties for solutions of the Zakharov–Kuznetsov equation

This work mainly focuses on spatial decay properties of solutions to the Zakharov–Kuznetsov equation. For the two- and three-dimensional cases, it was established that if the initial condition u0 verifies 〈σ⋅x〉ru0∈L2(σ⋅x≥κ), for some r∈N, κ∈R, being σ be a suitable non-null vector in the Euclidean space, then the corresponding solution u(t) generated from this initial condition verifies 〈σ⋅x〉ru(t)∈L2σ⋅x>κ−νt, for any ν>0. Additionally, depending on the magnitude of the weight r, it was also deduced some localized gain of regularity. In this regard, we first extend such results to arbitrary dimensions, decay power r>0 not necessarily an integer, and we give a detailed description of the gain of regularity propagated by solutions. The deduction of our results depends on a new class of pseudo-differential operators, which is useful for quantifying decay and smoothness properties on a fractional scale. Secondly, we show that if the initial data u0 has a decay of exponential type on a particular half space, that is, ebσ⋅xu0∈L2(σ⋅x≥κ), then the corresponding solution satisfies ebσ⋅xu(t)∈Hpσ⋅x>κ−t, for all p∈N, and time t≥δ, where δ>0. To our knowledge, this is the first study of such property. As a further consequence, we also obtain well-posedness results in anisotropic weighted Sobolev spaces in arbitrary dimensions.Finally, as a by-product of the techniques considered here, we show that our results are also valid for solutions of the Korteweg–de Vries equation.

Advanced Physics-informed neural networks for numerical approximation of the coupled Schrödinger–KdV equation

Physics-informed neural networks (PINNs) has been shown to be an effective tool for solving partial differential equations (PDEs). PINNs incorporate the PDEs residual into the loss function, seamlessly integrating it as part of the neural network architecture. This novel methodology has exhibited success in tackling a wide range of both forward and inverse PDE problems. However, a limitation of the first generation of PINNs is that they usually have limited accuracy even with many training points. Here, we propose two advanced methodologies: PINNs utilizing an H1 loss function and parallel physics-informed neural networks (P-PINNs). The former involves modifying the loss function with the H1 norm, while the latter entails solving the coupled Schrödinger–Korteweg–de Vries (Sch–KdV) equation separately with two parallel networks. These advancements are designed to enhance both accuracy and training efficiency. We have extensively tested these two advanced methods through a series of experiments and demonstrated the accuracy and effectiveness in approximation of the Sch–KdV system.

Rogue wave, lump, kink, periodic and breather-like solutions of the (2+1)-dimensional KdV equation

In this paper, the ([Formula: see text])-dimensional KdV equation is investigated by using the bilinear neural network method (BNNM). We construct six neural network models, extending beyond single hidden layer models to create deeper and broader network structures (e.g. [3-3-1], [3-4-1], [3-1-3-1], [3-4-1-1], [3-2-2-1] and [3-2-3-1-1] models). Introducing specific activation functions into the neural network model enables the generation of various test functions, resulting in novel solutions for equations that include rogue wave solutions, lump-kink solutions, periodic soliton solution, breather-like solutions and lump solutions. The physical properties of these novel solutions are vividly depicted through three-dimensional plots, density plots, and curve plots. The findings contribute to a better understanding of the propagation behavior of shallow water waves.

Travelling wave solutions and conservation laws of the (2+1)-dimensional new generalized Korteweg–de Vries equation

In this study, we investigate the travelling wave solutions of the (2+1)-dimensional new generalized Korteweg–de Vries equation by employing Lie group analysis along with various techniques which include direct integration, simplest equation method and Kudryashov’s method. The results obtained consists of periodic, kink, soliton and hyperbolic solutions. The symbolic computation software Maple is used to check the accuracy of all solutions obtained. Finally, 3D, density, and 2D plots of the derived solution are displayed to show the physical appearance of the model. Furthermore, we utilize the general multiplier technique and Ibragimov’s method to derive its conserved vectors. Conservation of energy and momentum, amongst others were found. Conservation laws have many significant uses with regards to integrability, linearization and analysis of solutions.

Integrable Couplings and Two-Dimensional Unital Algebras

The paper aims to demonstrate that a linear expansion in a unital two-dimensional algebra can generate integrable couplings, proposing a novel approach for their construction. The integrable couplings presented encompass a range of perturbation equations and nonlinear integrable couplings. Their corresponding Lax pairs and hereditary recursion operators are explicitly detailed. Concrete applications to the KdV equation and the AKNS system of nonlinear Schrödinger equations are extensively explored.

Long time bounds for coupled KdV equations

In this paper, we consider the coupled KdV equationηt+wx+(wη)x+16wxxx=0,wt+ηx+wwx+16ηxxx=0on T=R/2πZ with initial data of small amplitudes ɛ in Sobolev spaces. If the first three Fourier modes of initial data are of size ɛ1+μ for any 0≤μ≤12, we prove that the solutions remain smaller than 2ɛ for a time scale of order ɛ−(1+μ) via a normal form transformation. Further, we show this order of time scale is sharp.

Bilinear Bäcklund transformation, Lax pair, Painlevé integrability, and different wave structures of a 3D generalized KdV equation

Bilinear Bäcklund transformation, Lax pair, Painlevé integrability, and different wave structures of a 3D generalized KdV equation

Superposed and Superposed-type Double Periodic Jacobi Elliptic Function Solutions of Variable Coefficients KdV Equation

Superposed and Superposed-type Double Periodic Jacobi Elliptic Function Solutions of Variable Coefficients KdV Equation

Long‐time asymptotics of solution for the fifth‐order modified KdV equation in the presence of discrete spectrum

AbstractWe investigate the Cauchy problem of an integrable focusing fifth‐order modified Korteweg–de Vries (KdV) equation, which contains the fifth‐order dispersion and relevant higher order nonlinear terms. The long‐time asymptotics of solution is established in the case of initial conditions that lie in some low regularity weighted Sobolev spaces and allow for the presence of discrete spectrum. Our method is based on a generalization of the nonlinear steepest descent method of Deift and Zhou. We show that the solution decomposes in the long time into three main regions: (i) an expanding oscillatory region where solitons and breathers travel with positive velocities, the leading order term has the form of a multisoliton/breather and soliton/breather–radiation interactions; (ii) a Painlevé region, which does not have traveling solitons and breathers, the asymptotics can be characterized with the solution of a fourth‐order Painlevé II equation; (iii) a region of breathers traveling with negative velocities. Employing a global approximation via PDE techniques, the asymptotic behavior of solution is extended to lower regularity spaces with weights.

Nonlinear analysis of ion‐acoustic solitary waves in an unmagnetized highly relativistic quantum plasma

AbstractThe present paper explores the dynamics of ion‐acoustic solitary waves (IASWs) within an unmagnetized, highly relativistic quantum plasma containing positive and negative ions alongside electrons, employing the quantum hydrodynamic model. The treatment accounts for the inertial characteristics of negative and positive ions, considering electrons as inertialess. The nonlinear nature of quantum IASWs is investigated through the derivation of the Korteweg–de Vries equation using the reductive perturbation method. The investigation highlights how wave propagation characteristics, whether compressive or rarefactive, are substantially modulated by several parameters, such as the quantum diffraction parameter , relativistic effects , equilibrium density , and ion mass ratio . The analysis identifies the existence of both slow and fast wave modes, heavily influenced by the relativistic parameters involved. Our findings have significant implications for the understanding of both laboratory and space plasmas, especially in contexts where negative ions are present.

Multi-parametric solutions to the functional difference KdV equation

Using a specific Darboux transformation, we construct solutions to the functional difference KdV equation in terms of Casorati determinants. We give a complete description of the method and the corresponding proofs. We construct explicitly some solutions for the first orders.