De Vries Equation Research Articles

New type of solutions for the modified Korteweg–de Vries equation

In this letter we report a new type of multi-soliton solutions for the modified Korteweg–de Vries (mKdV) equation. These solutions contain τ functions of the trigonometric solitons and classical solitons simultaneously. A new bilinear form of the mKdV equation is introduced to derive these solutions. The obtained solutions display as solitons living on a periodic background, which are analyzed and illustrated.

A new study on fractional Schamel Korteweg–De Vries equation and modified Liouville equation

Traveling wave solutions of fractional partial differential equations have great importance in the literature. The diversity of solutions plays an important role in understanding the physical structure of the model it represents. For this reason, two important differential equations with the fractional order, which have a significant role in applied sciences and can model real-life problems most accurately, have been solved by the generalized (G′G)-expansion method in this study. This method is a generalization of the classical (G′G)-expansion method. With this developed method, the non-linear fractional Schmael Korteweg–De Vries equation and fractional modified Liouville differential equations are discussed to find their exact solutions. In this way, new exact solutions of these equations that were not previously included in the literature have been found. The presented method has been applied to these two equations for the first time, and various new traveling wave solutions have been obtained. Thus, the study goes beyond other studies. To understand the physical behavior of these new exact solutions, three-dimensional graphs have been drawn according to different parameter values.

Cauchy matrix approach for generalized semi-discrete lattice potential Korteweg–de Vries equations

Starting from a new scattering relation, we generate a generalized semi-discrete lattice potential Korteweg–de Vries(gsd-lpKdV) equation from a determining equation set(DES) and its Lax presentation. Further, the gsd-lpKdV equation with self-consistent sources(gsd-lpKdVSCS) is worked out by introducing two discrete variables to the plane wave factors. Specially, explicit formulae for kink solitons, Jordan-block solutions and diagonal-Jordan-block solutions of gsd-lpKdVSCS are obtained based on the Cauchy matrix approach. Moreover, We illustrate the dynamics of one-kink soliton, two-kink soliton, high order Jordan-block solutions and the diagonal-Jordan-block solution.

Landau quantization effects on damping Kawahara solitons in electron–positron–ion plasma in rotating ionized medium

For the dynamics of three-dimensional electron–positron–ion plasmas, a fluid quantum hydrodynamic model is proposed by considering Landau quantization effects in dense plasma. Ion–neutral collisions in the presence of the Coriolis force are also considered. The application of the reductive perturbation technique produces a wave evolution equation represented by a damped Korteweg–de Vries equation. This equation, however, is insufficient for describing waves in our system at very low dispersion coefficients. As a result, we considered the highest-order perturbation, which resulted in the damped Kawahara equation. The effects of the magnetic field, Landau quantization, the ratio of positron density to electron density, the ratio of positron density to ion density, and the direction cosine on linear dispersion laws as well as soliton and conoidal solutions of the damped Kawahara equation are explored. The understanding from this research can contribute to the broader field of astrophysics and aid in the interpretation of observational data from white dwarfs.

Bi-Hamiltonian structure of a super KdV equation of Kupershmidt

In this paper, we study a super Korteweg–de Vries (sKdV) equation proposed by Kupershmidt which possesses a Lax operator with three fully nonlocal terms. The Lax operator is reformulated so that it is of the super constrained modified Kadomtsev–Petviashvili (scmKP) type. By calculating the bi-Hamiltonian structure of the scmKP hierarchy and employing Dirac reduction, we obtain the bi-Hamiltonian structure of the sKdV equation. We also present a spectral problem of its modified system.

Diverse variety of exact solutions for some nonlinear models via the [formula omitted]-expansion method

In this article, we explore several significant nonlinear physical models, including the Benjamin–Bona–Mahony–Peregrine–Burgers (BBMPB) equation, the Burgers–Korteweg–De Vries (BK) equation, the one-dimensional Oskolkov (OSK) equation, the Klein–Gordon (KG) equation with quadratic non-linearity, and the improved Boussinesq (IB) equation. Utilizing the (G′G)-expansion method ansatz, we derive new exact traveling wave solutions for these models. These solutions, expressed in the forms of rational, hyperbolic, and trigonometric functions, present a novel contribution distinct from existing literature. The physical dynamics of these solutions are elucidated through Mathematica simulations.

Self-similar blow-up solutions in the generalised Korteweg-de Vries equation: spectral analysis, normal form and asymptotics

In the present work we revisit the problem of the generalised Korteweg–de Vries equation parametrically, as a function of the relevant nonlinearity exponent, to examine the emergence of blow-up solutions, as traveling waveforms lose their stability past a critical point of the relevant parameter p, here at p = 5. We provide a normal form of the associated collapse dynamics, and illustrate how this captures the collapsing branch bifurcating from the unstable traveling branch. We also systematically characterise the linearisation spectrum of not only the traveling states, but importantly of the emergent collapsing waveforms in the so-called co-exploding frame where these waveforms are identified as stationary states. This spectrum, in addition to two positive real eigenvalues which are shown to be associated with the symmetries of translation and scaling invariance of the original (non-exploding) frame features complex patterns of negative eigenvalues that we also fully characterise. We show that the phenomenology of the latter is significantly affected by the boundary conditions and is far more complicated than in the corresponding symmetric Laplacian case of the nonlinear Schrödinger problem that has recently been explored. In addition, we explore the dynamics of the unstable solitary waves for p > 5 in the co-exploding frame.

Cauchy Problem for the Korteweg–De Vries Equation in the Class of Periodic Infinite-Gap Functions

Cauchy Problem for the Korteweg–De Vries Equation in the Class of Periodic Infinite-Gap Functions

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.

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.

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.

Variable dust charge generates multi solitons, breather soliton structures in Saturn’s ring

In this study, we have investigated nonlinear wave structures that are localized along with the exact stationary wave solutions with oscillating, i.e. the breather structures in an unmagnetized dusty plasma with variable dust charge concentration with two temperatures of ions. We have derived GE from the normalized governing equations employing the reductive perturbative technique (RPT). The GE is the extended Korteweg–de Vries (KdV) equation that includes the united effect of quadratic and cubic nonlinearities. Employing the Hirota bilinear method (HBM), multi-soliton solutions and the breather soliton solutions have been obtained. Breathers (oscillating wave packets) play a significant role in hydrodynamics and optics, and their interaction can change the dynamics of the wave fields. It is also important to propagate finite-amplitude waves in the astrophysical atmosphere, plasma dynamics, ocean, optic fibers, signal processing, etc. In the circumstances of Saturn’s ring, it is noticed that the breather soliton structures are modified significantly with variations in dust charge concentration, ion temperature ratio, and other physical parameters.

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.

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.

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.

On the analytical soliton approximations to fractional forced Korteweg–de Vries equation arising in fluids and plasmas using two novel techniques

The current investigation examines the fractional forced Korteweg–de Vries (FF-KdV) equation, a critically significant evolution equation in various nonlinear branches of science. The equation in question and other associated equations are widely acknowledged for their broad applicability and potential for simulating a wide range of nonlinear phenomena in fluid physics, plasma physics, and various scientific domains. Consequently, the main goal of this study is to use the Yang homotopy perturbation method and the Yang transform decomposition method, along with the Caputo operator for analyzing the FF-KdV equation. The derived approximations are numerically examined and discussed. Our study will show that the two suggested methods are helpful, easy to use, and essential for looking at different nonlinear models that affect complex processes.

Interfacial conditioning in physics informed neural networks

Physics informed neural networks (PINNs) have effectively demonstrated the ability to approximate the solutions of a system of partial differential equations (PDEs) by embedding the governing equations and auxiliary conditions directly into the loss function using automatic differentiation. Despite demonstrating potential across diverse applications, PINNs have encountered challenges in accurately predicting solutions for time-dependent problems. In response, this study presents a novel methodology aimed at enhancing the predictive capability of PINNs for time-dependent scenarios. Our approach involves dividing the temporal domain into multiple subdomains and employing an adaptive weighting strategy at the initial condition and at the interfaces between these subdomains. By employing such interfacial conditioning in physics informed neural networks (IcPINN), we have solved several unsteady PDEs (e.g., Allen–Cahn equation, advection equation, Korteweg–De Vries equation, Cahn–Hilliard equation, and Navier–Stokes equations) and conducted a comparative analysis with numerical results. The results have demonstrated that IcPINN was successful in obtaining highly accurate results in each case without the need for using any labeled data.