Poisson’s theorem and integrals of KdV equation

Water-wave experiments are presented showing the evolution of finite amplitude waves in relatively shallow water when no solitons are present. In each case, the initial wave is rectangular in shape and wholly below the still water level; the amplitude of the wave is varied. The asymptotic solution of the Korteweg-de Vries (KdV) equation in the absence of solitons (Ablowitz & Segur 1976) is compared with observed evolution. In addition, the asymptotic solution of the linearized KdV equation (a linear dispersive model) is compared with both the KdV solution and experiments. This comparison is a natural consequence of the fact that, in the absence of solitons, the asymptotic solutions of the KdV equation and its linearized version are qualitatively similar. Both the experiments and the model equations suggest that the asymptotic wave structure consists of a negative triangular wave, travelling with a speed (gh)½, followed by a train of modulated oscillatory waves which travel more slowly. Quantitative comparisons are made for the amplitude, shape and decay rate of the leading wave and the amplitude, dominant wavenumbers and velocities of the trailing wave groups. Over the parameter range of the experiments, asymptotic KdV theory predicts more closely the observed behaviour. The leading wave is observed to decay more rapidly than the trailing wave groups; hence the leading wave becomes less prominent with time. This is in agreement with the KdV solution, whereas just the opposite is predicted by linear theory. Linear predictions for the trailing wave groups are accurate only when they agree with the KdV predictions. Both models predict the evolution of short waves in the trailing wave region. When the short waves are unstable (k gt; 1·36), either group disintegration or focusing into envelope solitons is possible. Both of these phenomena are observed in the experiments; neither is predicted by long-wave models. The nonlinear Schrödinger equation is reviewed and tested as a model of these unstable wave groups. There is some evidence that the KdV equation and the nonlinear Schrödinger equation can be patched together to provide an asymptotic description of these unstable groups.

An application of the decomposition method for the generalized KdV and RLW equations

In this paper, an analytical and numerical computation of multi-solitons in Korteweg-de Vries (KdV) equation is presented. The KdV equation, which is classic of all model equations of nonlinear waves in the soliton phenomena, is described. In the analytical computation, the multi-solitons in KdV equation are computed symbolically using computer symbolic manipulator—Wolfram Mathematica via Hirota method because of the lengthy algebraic computation in the method. For the numerical computation, Crank-Nicolson implicit scheme is used to obtain numerical algorithm for the KdV equation. The simulations of solitons in MATLAB as well as results concerning collision or interactions between solitons are presented. Comparing the analytical and numerical solutions, it is observed that the results are identically equal with little ripples in solitons after a collision in the numerical simulations; however there is no significant effect to cause a change in their properties. This supports the existence of solitons solutions and the theoretical assertion that solitons indeed collide with one another and come out without change of properties or identities.

The excitation of nonlinear electron-acoustic solitary waves (SWs) is studied theoretically in an unmagnetized, uniform, and collisionless plasma. The plasma is composed of two types of electrons referred to as inertial cold electrons and inertialess Kappa (κ) distributed superthermal hot electrons along with stationary ions. Two parameters κ and β are introduced in the governing equations, where the density related parameter is defined by β = n co /n ho (n co and n ho are the cold and hot electron number densities, respectively). Using a reductive perturbation technique, nonlinear Korteweg-de Vries (KdV), modified KdV (mKdV), and standard Gardner (sG) equations have been derived, which admit SW solutions. The KdV equation allows to analyze both compressive (negative) and rarefactive (positive) solitons. But a significant solitary structure vanishes around a critical region corresponding to critical κ c and β c . To look a solitary structure in the critical region, it is proceeded to the next approach of mKdV solution with more higher order calculation of perturbation. But only rarefactive soliton is found in this approach. Furthermore, having higher order perturbation, the next approach of sG equation allows to rescue both compressive and rarefactive solitons even in the critical region. It is observed that the excitation of soliton pulses with various shapes depends on both parameters κ and β.

In this short communication, we consider a discrete example of how to perform multiple scale expansions and by starting from the discrete nonlinear Schroodinger equation (DNLS) as well as the Ablowitz-Ladik nonlinear Schrodinger equation (AL-NLS), we obtain the corresponding discrete versions of a Korteweg-de Vries (KdV) equation. We analyze in particular the equation obtained from the AL-NLS and discuss its integrability, as well as its connections with previously studied discrete versions of the KdV equation.

This study investigates the impact of multiplicative white noise on soliton solutions of the Korteweg–de Vries (KdV) equation, a classical model for nonlinear wave propagation. While the deterministic KdV framework effectively captures soliton dynamics in idealized settings, real-world systems often experience stochastic fluctuations that can significantly affect wave behavior. By introducing space–time-dependent multiplicative noise, we formulate a stochastic KdV (SKdV) equation and derive analytical expressions for both single and multi-soliton solutions using a novel wave transformation in conjunction with the Hirota bilinear method. Numerical simulations across varying noise intensities reveal that stochastic perturbations lead to soliton deformation, amplitude modulation, phase shifts, and disrupted coherence. These effects become more pronounced with stronger noise, altering soliton interactions and long-term stability. The findings emphasize the importance of incorporating stochastic effects into nonlinear wave models, with implications for applications in fluid dynamics, plasma physics, and optical communications.

In this article we propose a reduced model of the input-output behaviour of an arterial compartment, including the short systolic phase where wave phenomena are predominant. The objective is to provide basis for model-based signal processing methods for the estimation from non-invasive measurements and the interpretation of the characteristics of these waves. Standard space discretizations of distributed models of the flow lead to high order models for the pressure wave transfer function, and low order rational transfer functions approximations give poor results. The main idea developed here to circumvent these problems is to explicitly use a propagation delay in the reduced model. Due to phenomena such that peaking and steepening, the considered pressure pulse waves behave more like solitons generated by a Korteweg de Vries (KdV) equation than like linear waves. So we start with a quasi-1D Navier-Stokes equation that takes into account a radial acceleration of the wall, in order to be able to recover, during the reduction process, the dispersive term of KdV equation which, combined with the nonlinear transport term gives rise to solitons. The radial and axial acceleration terms being supposed small, a multiscale singular perturbation technique is used to separate the fast wave propagation phenomena taking place in a boundary layer in time and space described by a KdV equation from the slow phenomena represented by a parabolic equation leading to two-elements windkessel models.

We study here the water-waves problem for uneven bottoms in a highly nonlinear regime where the small amplitude assumption of the Korteweg-de Vries (KdV) equation is enforced. It is known, that for such regimes, a generalization of the KdV equation (somehow linked to the Camassa-Holm equation) can be derived and justified by A. Constantin, D. Lannes "The hydrodynamical relevance of the Camassa-Holm and Degasperis-Processi equations" when the bottom is flat. We generalize here this result with a new class of equations taking into account variable bottom topographies. Of course, the many variable depth KdV equations existing in the literature are recovered as particular cases. Various regimes for the topography regimes are investigated and we prove consistency of these models, as well as a full justification for some of them. We also study the problem of wave breaking for our new variable depth and highly nonlinear generalizations of the KDV equations.

The Korteweg-de Vries (KdV) equation is a mathematical model that describes the propagation of long waves in dispersive media. It takes into account both nonlinearity and dispersion, and is particularly useful for modeling phenomena like solitons. The nonlinear Schrödinger (NLS) equation models the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves. It is a useful model for describing many physical systems, including Bose-Einstein condensates, optical fibers, and water waves. A system that couples the KdV and NLS equations can model the interaction of long and short waves. This system combines the strengths of both models. The long waves described by the KdV equation can affect the behavior of the short waves described by the NLS equation, while the short waves can in turn affect the behavior of the long waves. Such a coupled system has been studied extensively over the last few decades, and has led to important insights into many physical systems. This paper considers the existence of local solutions to the Cauchy problem of KdV-Schrödinger nonlinear system on the basis of literature (Bernard D, Nghiem V N, Benjamin L S <ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="http://doi.org/10.1088/1751-8113/49/41/415501">2016 <i>J. Phys. A: Math. Theor.</i> <b>49</b> 415501</ext-link>), and also gives the existence space of the local solutions.

The effect of changing the direction of motion of a defect (a soliton of small amplitude) in soliton lattices described by the Korteweg–de Vries and modified Korteweg–de Vries integrable equations (KdV and mKdV) was studied. Manifestation of this effect is possible as a result of the negative phase shift of a small soliton at the moment of nonlinear interaction with large solitons, as noted in [1], within the KdV equation. In the recent paper [2], an expression for the mean soliton velocity in a “cold” KdV-soliton gas has been found using kinetic theory, from which this effect also follows, but this fact has not been mentioned. In the present paper, we will show that the criterion of negative velocity is the same for both the KdV and mKdV equations and it can be obtained using simple kinematic considerations without applying kinetic theory. The averaged dynamics of the “smallest” soliton (defect) in a soliton gas consisting of solitons with random amplitudes has been investigated and the average criterion of changing the sign of the velocity has been derived and confirmed by numerical solutions of the KdV and mKdV equations.

The dynamics of localized waves is analyzed in the framework of a model described by the Korteweg-de Vries (KdV) equation with account made for the cubic positive nonlinearity (the Gardner equation). In particular, the interaction process of two solitons is considered, and the dynamics of a "breathing" wave packet (a breather) is discussed. It is shown that solitons of the same polarity interact as in the case of the Korteweg-de Vries equation or modified Korteweg-de Vries equation, whereas the interaction of solitons of different polarity is qualitatively different from the classical case. An example of "unpredictable" behavior of the breather of the Gardner equation is discussed.

The Hirota bilinear method is extended to find one- and two-hump exact bright and dark soliton solutions to a coupled system between the linear Schrödinger and Korteweg-de Vries (KdV) equations arising in the energy transfer problem along a cubic anharmonic crystal medium. The bilinear form associated to this system is found by imitating the well known bilinearizing transformations used in the standard nonlinear Schrödinger (NLS) and KdV equations. Proper finite exponential expansions in the transformed variables allow one to exhibit multihump soliton solutions as single entities resulting from the adjustment of appropriate dispersion relations between the wave parameters describing the profiles. It is found that such one- and two-hump solutions correspond to the one- and two-KdV solitons trapped by both the bright and dark-gray NLS solitons. Our two-hump bright and dark solutions represent novel solutions for the type of interactions and nonlinearities considered.

The Korteweg-de Vries (KdV) equation is a universal mathematical model for the description of weakly nonlinear long wave propagation in dispersive media. It is a most remarkable nonlinear partial differential equation in 1 + 1 dimensions whose solutions can be exactly specified; it has a soliton like solution or solitary wave of sech2 form. Various physical systems of dispersive waves admit solutions in the form of generalized solitary waves. The study of this equation is the archetype of an integrable system and is one of the most fundamental equations of soliton phenomena and a topic of active mathematical research. Our purpose here is to give a motivated and a sketchy overview of this interesting subject. This article will cover in detail: the KdV equation and the inverse scattering method (based on Schrodinger and Gelfand-Levitan equations) used to solve it exactly.

Dark ring soliton in two-dimensional nonlinear self-defocusing medium

A reduced high-order compact finite difference scheme based on proper orthogonal decomposition technique for KdV equation