Sharp global well-posedness for KdV and modified KdV on ℝ and 𝕋

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.

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.

Variable coefficient nonlinear systems, the Korteweg de Vries (KdV), the modified KdV (mKdV) and the nonlinear Schrödinger (NLS) type equations, are derived from the nonlinear inviscid barotropic nondivergent vorticity equation in a beta-plane by means of the multi-scale expansion method in two different ways, with and without the so-called y-average trick. The non-auto-Bäcklund transformations are found to transform the derived variable coefficient equations to the corresponding standard KdV, mKdV and NLS equations. Thus, many possible exact solutions can be obtained by taking advantage of the known solutions of these standard equations. Further, many approximate solutions of the original model are ready to be yielded which might be applied to explain some real atmospheric phenomena, such as atmospheric blocking episodes.

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.

KdV and kink–antikink solitons in car-following models

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

The theoretical analysis of the lattice hydrodynamic models for traffic flow theory

A Hamiltonian formalism has been introduced in (~,2) for periodic and solitonlike solutions of KdV (Korteweg-de Vries) equation. This allows one to use general theorems of classical dynamics (as done, for example, in (a) to obtain explicit solutions of the linearized evolution equations). Poisson's theorem (see e.g. ref, (4)) states that the Yoisson bracket of two constants of motion is also a constant of motion. This fascinating statement leads very often to trivial or dependent constants of motion. In the case of KdV, any two constants chosen among the well-known infinite sequence (5) of polynomials have zero Poisson brackets, as proved by G ~ D ~ , R (~) (see also LAX (e)). Fortunately, this does not hold for the only known (5) constant of motion depending explicitly on space and time, as we shall see here. KdV equation is given by

Overtaking collisions of ion-acoustic (IA) solitons are examined in an unmagnetized pair-ion plasma containing degenerate electrons. A Korteweg-de Vries (KdV) equation governing nonlinear IA solitons in such a degenerate plasma model is obtained. At critical compositions, the nonlinear coefficient of the KdV equation vanishes. Therefore, a modified KdV (mKdV) equation describing nonlinear IA solitons in the degenerate plasma model has been derived by extending the analytic analysis to this critical case. Hirota’s bilinear method has been applied to obtain the multisolitons solution and their corresponding phase shifts in these two situations. For an in-depth analysis, we investigated the dynamics of two solitons of the same polarity and opposite polarities resulting from the overtaking collision for discrete times. In particular, the phase shifts are obtained for the first time at critical components in degenerate plasmas. In these two cases, it is found that the magnitude of the phase shifts due to the overtaking interaction of two solitons increases with increasing the negative ion-to-positive ion number density ratio. The present study may help explain the properties of nonlinear waves in white dwarfs that support the propagation of solitons.

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.

A general similarity projective solution to the generalized Korteweg–de Vries (KdV) system with varying coefficients is derived with the aid of direct similarity projective approach. All the allowed exact solutions of the standard KdV equation can be converted into the corresponding exact solutions of the generalized KdV system. In terms of the known exact solution of the KdV equation, some typical localized excitations with self-similar behaviors and similaritons are revealed by prescribing appropriate system parameters.

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

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.