Search

The periodic and quasi-periodic solutions of the integrable system have been studied for four decades based on the Riemann theta functions. However, there is a fundamental difficulty in representing the solutions numerically because the Riemann theta function requires several transcendental parameters. This paper presents a novel method for the numerical representation of such solutions from the algebraic treatment of the periodic and quasi-periodic solutions of the Baker–Weierstrass hyperelliptic ℘ functions. We demonstrate the numerical representation of the hyperelliptic ℘ functions of genus two. • The hyperelliptic solutions of the KdV equation are displayed numerically. • It is proved algebraically that hyperelliptic p-functions obey the KdV equation. • The p-function is numerically expressed without theta functions. • The structure of the symmetric product of curves is expressed graphically. [ABSTRACT FROM AUTHOR]

We characterize the rational solutions to a KdV-like equation which are generated from polynomial solutions to the corresponding generalized bilinear equation. We use a particular class of polynomials satisfying a quadratic difference equation to obtain that there are no solutions of the bilinear equation with degree (in the spatial variable) greater than 5. As a byproduct, we answer positively a conjecture of Yi Zhang and Wen-Xiu Ma about these solutions. [ABSTRACT FROM AUTHOR]

The Korteweg–de Vries (KdV) equation can be derived from a plasma-fluid model via a reductive perturbation technique. The associated methodology is summarized, from first principles, focusing on the underlying physical assumptions involved in the plasma-theoretical framework. A beam permeated electron-ion plasma is assumed, although the main findings of this study may be extended to more complicated plasma configurations. Rather counter-intuitively, it is shown that either of the (two) real coefficients appearing in the KdV equation (actually, both depending parametrically on the plasma configuration and on the beam characteristics) may take either positive or negative values, a possibility overlooked in the past. Different possibilities are investigated, from first principles, regarding the sign of the nonlinearity coefficient A (that is determined by the electron background statistics, in combination with the beam velocity) and the sign of the dispersion coefficient B (that is solely determined by the beam velocity and is always positive in its absence). The possibility of polarity reversal is investigated from first principles, in relation with both the electrostatic potential (pulse) profile and its associated electric field (bipolar pulse) E = − ∇ ϕ in the electrostatic approximation. Different types of excitations are shown to exist and the role of the (sign of the) various coefficients in the pulse-shaped solution's propagation characteristics is discussed. • The dispersion coefficient B in the KdV equation may be either positive or negative. • The nonlinearity coefficient A may also be either positive or negative. • Soliton (pulse) polarity is determined by the nonlinearity coefficient. • In plasmas, A dictates the amplitude and polarity of an electrostatic potential pulse. • The electric field associated with an electrostatic soliton is derived analytically. [ABSTRACT FROM AUTHOR]

The currents in the ocean have a serious impact on ocean dynamics, since they affect the transport of mass and thus the distribution of salinity, nutrients and pollutants. In many physically important situations the current depends quadratic-ally on the depth. We consider a single layer of fluid and study the propagation of the surface waves in the presence of depth-dependent current with quadratic profile. We select the scale of parameters and quantities, which are typical for the Boussinesq propagation regime (long wave and small amplitude limit) and we also derive the well known KdV model for the surface waves interacting with current. • The problem of ocean waves interacting with currents is studied. • The depth-dependence of the current is given by a quadratic function. • A simple KdV type model is derived in the long-wave approximation. • The emerging solitary waves are influenced by the current. [ABSTRACT FROM AUTHOR]

Soliton gas or soliton turbulence is a subject of intense studies due to its great importance to optics, hydrodynamics, electricity, chemistry, biology and plasma physics. Usually, this term is used for integrable models where solitons interact elastically. However, soliton turbulence can also be a part of non-integrable dynamics, where long-lasting solutions in the form of almost solitons may exist. In the present paper, the complex dynamics of ensembles of solitary waves is studied within the Schamel equation using direct numerical simulations. Some important statistical characteristics (distribution functions, moments, etc.) are calculated numerically for unipolar and bipolar soliton gases. Comparison of results with integrable Korteweg–de Vries (KdV) and modified KdV (mKdV) models are given qualitatively. Our results agree well with the predictions of the KdV equation in the case of unipolar solitons. However, in the bipolar case, we observed a notable departure from the mKdV model, particularly in the behavior of kurtosis. The observed increase in kurtosis signifies the amplification of distribution function tails, which, in turn, corresponds to the presence of high-amplitude waves. • Schamel equation. • Soliton gas. • Soliton collisions. • Non-integrable equations. [ABSTRACT FROM AUTHOR]

Highlights • We generalize the second integral from ordinary differential equations to partial differential equations. • We put forward the concept of special class of second integrals for integrable systems. • For the KdV equation, we find a new 2-parameter special class, which generalize the special class of conserved quantities. • For the KdV equation, the special class for the multi-soliton solutions is presented. Abstract The special class of second integrals, distinguished for its infinite dimension, emerges naturally when generalizing the second integrals of ordinary differential equations to partial differential equations. The conserved quantities of the KdV equation are a special class of second integrals. We proved its uniqueness under the assumption on the cofactor operator. The special class is so peculiar that to find it is almost an algorithm. Thus we managed to generalize the special class of the conserved quantities of the KdV equation to a new 2-parameter special class. Among the special classes, the special class of nonlocal second integrals plays an extra role. As an example, the special class that generates the multi-soliton solutions of the KdV equation is presented. [ABSTRACT FROM AUTHOR]

We study the Hamiltonian formalism for second and fourth order nonlinear Schrödinger equations. In the case of the second order equation, we consider cubic and logarithmic nonlinearities. Since the Lagrangians generating these nonlinear equations are degenerate, we follow the Dirac–Bergmann formalism to construct their corresponding Hamiltonians. In order to obtain consistent equations of motion, the Dirac–Bergmann formalism imposes some set of constraints that contribute to the total Hamiltonian along with their Lagrange multipliers. The order of the Lagrangian degeneracy determines the number of primary constraints. If a constraint is not a constant of motion, a secondary constraint is introduced to force the consistency condition. We show that for second order and fourth order nonlinear Schrödinger equations we only have primary constraints, and the form of nonlinearity or the order of derivatives does not change the constraint dynamics of the system. However, we observe that introducing new fields to treat higher derivatives in the Lagrangians of these equations changes the constraint dynamics, and secondary constraints are needed to construct a consistent set of Hamilton equations. • The Dirac–Bergmann algorithm is used for the nonlinear Schrodinger equations. • The algorithm introduces a set of constraints in order to obtain consistent Hamilton equations. • Several integrable nonlinear equations are considered from the Lagrangian point of view. [ABSTRACT FROM AUTHOR]

Particle trajectories beneath depression solitary gravity-capillary waves in the weakly nonlinear weakly dispersive regime in a sheared channel with finite depth and constant vorticity are investigated. A fifth-order Korteweg–de Vries equation that incorporates the surface tension and the vorticity effects is derived asymptotically from the full Euler equations when the Bond number is nearly a critical value that depends on the vorticity. The velocity field in the bulk fluid is approximated which allow us to study the submarine structures beneath depression solitary waves. The dynamical system of particle trajectories is reformulated in the moving wave frame and its features are seen through the streamlines. We show that for large values of the vorticity parameter there are always stagnation points in the bulk fluid, which produce rich cat's-eyes structures. In this case, particle trajectories can undergo vertical excursions, describing closed orbits or undergo pure horizontal transport. Furthermore, bifurcation diagrams according to the number of stagnation points and complex flow structures beneath depression solitary waves with multiple local extrema are shown. • Fifth-order rotational KdV. • Depression solitary waves. • Kelvin cat's-eye structures. [ABSTRACT FROM AUTHOR]

Using the variational-asymptotic method we develop the theory of amplitude modulation of waves governed by Korteweg-de Vries (KdV) equation. Two asymptotic solutions describing the amplitude modulation of trains of solitons and of positons are obtained. The comparison with the exact solutions of KdV equation shows quite excellent agreement. [ABSTRACT FROM AUTHOR]

Abstract: Solutions of the KdV equation are derived by the algebraic operator method based on generalized operators of differentiation. The algebraic operator method based on the generalized operator of differentiation is exploited for the derivation of analytic solutions to the KdV equation. The structure of solitary solutions and explicit conditions of existence of these solutions in the subspace of initial conditions are derived. It is shown that special solitary solutions exist only on a line in the parameter plane of initial and boundary conditions. This new theoretical result may lead to important findings in a variety of practical applications. [Copyright &y& Elsevier]

Abstract: The KdV equation arises in the framework of the Boussinesq scaling as a model equation for waves at the surface of an inviscid fluid. Encoded in the KdV model are relations that may be used to reconstruct the velocity field in the fluid below a given surface wave. In this paper, velocity fields associated to exact solutions of the KdV equation are found, and particle trajectories are computed numerically. The solutions treated here comprise the solitary wave, periodic traveling waves, and the two-soliton solutions. For solitary waves and periodic traveling waves, approximate particle paths are found in closed form. [Copyright &y& Elsevier]

Abstract: A one-dimensional weakly-nonlinear model equation based on a Lagrangian-averaged Euler-α model of compressible flow in lossless fluids is presented. Traveling wave solutions (TWS)s, in the form of a topological soliton (or kink), admitted by this fourth-order partial differential equation are derived and analyzed. An implicit finite-difference scheme with internal iterations is constructed in order to study soliton collisions. It is shown that, for certain parameters, the TWSs interact as solitons, i.e., they retain their “identity” after a collision. Kink-like solutions with an oscillatory tail are found to emerge in a signaling-type initial-boundary-value problem for the linearized equation of motion. Additionally, connections are drawn to related weakly-nonlinear acoustic models and the Korteweg–de Vries equation from shallow-water wave theory. [Copyright &y& Elsevier]

Abstract: Using the solutions of an auxiliary differential equation, a direct algebraic method is described to construct several kinds of exact travelling wave solutions for some Wick-type nonlinear partial differential equations. By this method some physically important nonlinear equations are investigated and new exact travelling wave solutions are explicitly obtained. In addition, the links between Wick-type partial differential equations and variable coefficient partial differential equations are also clarified generally. [ABSTRACT FROM AUTHOR]

Abstract: In this work we study two completely integrable coupled KdV and coupled KP systems. The Hirota’s bilinear method is employed to formally derive multiple soliton solutions and multiple singular soliton solutions for each system. The resonance phenomenon will be examined. [Copyright &y& Elsevier]

Abstract: In this paper, the prolongation structures of a generalized coupled Korteweg-de Vries (KdV) equation are investigated and two integrable coupled KdV equations associated with their Lax pairs are derived. Furthermore, a Miura transformation related to a integrable coupled KdV equation is derived, from which a new coupled modified KdV equations is obtained. [Copyright &y& Elsevier]

Abstract: The variable-coefficient Korteweg-de Vries equation that governs the dynamics of weakly nonlinear long waves in a periodically variable dispersion management media is considered. For general bit patterns, an analytic expression describing the evolution of the timing shift produced by nonlinear interactions between neighboring solitons is derived. The general result with a Gaussian like solitary wave profile for a special case of negligible average dispersion is tested. By considering a piecewise-constant map, we found that the timing shift is reduced substantially in the dispersion-managed Korteweg-de Vries system. [Copyright &y& Elsevier]

Abstract: It is shown that if the dispersion of the KdV equation is replaced by a higher order dispersion , where is an odd integer, then the critical Sobolev exponent for local well-posedness on the circle does not change. That is, the resulting equation is locally well-posed in , . [Copyright &y& Elsevier]

Abstract: Solutions of a boundary value problem for the Korteweg–de Vries equation are approximated numerically using a finite-difference method, and a collocation method based on Chebyshev polynomials. The performance of the two methods is compared using exact solutions that are exponentially small at the boundaries. The Chebyshev method is found to be more efficient. [Copyright &y& Elsevier]

Abstract: The soliton perturbation theory is used to study the solitons that are governed by the generalized Korteweg–de Vries equation in the presence of perturbation terms. The adiabatic parameter dynamics of the solitons in the presence of the perturbation terms are obtained. [Copyright &y& Elsevier]

Abstract: In this work we use a modified tanh–coth method to solve the Korteweg-de Vries and Korteweg-de Vries–Burgers’ equations. The main idea is to take full advantage of the Riccati equation that the tanh-function satisfies. New multiple travelling wave solutions are obtained for the Korteweg-de Vries and Korteweg-de Vries–Burgers’ equations. [Copyright &y& Elsevier]

Abstract: In this paper, we first introduced improved projective Riccati method by means of two simplified Riccati equations. Applying the improved method, we consider the general types of KdV and KdV–Burgers equations with nonlinear terms of any order. As a result, many explicit exact solutions, which contain new solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions are obtained. Some of them are found for the first time. [Copyright &y& Elsevier]

Abstract: In this work we introduce new schemes, each combines two hyperbolic functions, to study the KdV, mKdV, and the generalized KdV equations. It is shown that this class of equations gives conventional solitons and periodic solutions. We also show that the proposed schemes develop sets of entirely new solitary wave solutions in addition to the traditional solutions. The analysis can be used to a wide class of nonlinear evolutions equations. [Copyright &y& Elsevier]

Abstract: The Gardner equation, also called combined KdV–mKdV equation, is studied. New hyperbolic ansatze are proposed to derive solitons solutions. The tanh method is used as well to obtain kink solutions. [Copyright &y& Elsevier]

Abstract: In this paper, a fractional Korteweg-de Vries equation (KdV for short) with initial condition is introduced by replacing the first order time and space derivatives by fractional derivatives of order and with , respectively. The fractional derivatives are described in the Caputo sense. The application of Adomian decomposition method, developed for differential equations of integer order, is extended to derive explicit and numerical solutions of the fractional KdV equation. The solutions of our model equation are calculated in the form of convergent series with easily computable components. [Copyright &y& Elsevier]

In this paper we study the Zakharov–Kuznetsov (ZK) equation and two of its modified forms. We employ the sine–cosine algorithm to back up our analysis. Exact solutions with solitons and periodic structures are obtained. [Copyright &y& Elsevier]

In this paper we study the nonlinear dispersion of new variants of the KdV and the KP equations with positive and negative exponents. The approach stems mainly from the sine–cosine method. The study reveals compactons, solitons, solitary patterns and periodic solutions for these models. [Copyright &y& Elsevier]

A separation method is introduced within the context of dynamical system for solving the non-linear Korteweg–de Vries equation (KdV). Best efficiency is obtained for the number of iterations (n⩽8). Comparisons with the solutions of the quintic spline, finite difference, moving mesh and pseudo-spectral are presented. [Copyright &y& Elsevier]

In the present work, by employing the nonlinear equations of motion of an incompressible, isotropic and prestressed thin elastic tube and the approximate equations of an incompressible inviscid fluid, we studied the existence of some possible evolution equations in the longwave approximation and their progressive wave solutions. It is shown that, depending on the set of values of the initial deformation, it might be possible to obtain the conventional Korteweg-deVries (KdV) and the modified KdV equations of various forms. Finally, a set of progressive wave solutions is presented for such evolution equations. [Copyright &y& Elsevier]

Numerical simulations show that higher order KdV equation under certain conditions has a self-focusing singularity, which means that the solution of the equation blows up in finite time. In this paper, two numerical schemes: the split-step Fourier transform and the pseudospectral methods are used to investigate this self-focusing singularity problem. Parallel algorithms for the proposed schemes are designed and implemented. FFTW-MPI algorithm designed by Matteo Frigo and Steven Johnson is used for parallel implementation of the discrete Fourier transform (DFT). The parallel algorithms are implemented on an SGI Origin 2000 multiprocessor computer and experiments show that a considerable speedup is attained. [Copyright &y& Elsevier]

• The Kudryashov technique is developed to a new technique called "the extended Kudryashov technique (EKT)". • The Wick-type stochastic models are examined in the frame of newly conformable derivatives. • New stochastic exact solutions for the conformable mixed KdV-mKd equation are established by the EKT. • Two new kinds of traveling wave solutions, inclusive periodic and soliton solutions are obtained. • Numerical examples in the cases of deterministic and stochastic functions are applied with 3D profiles for the acquired results. In this work, we utilize a new generalized derivative of conformable type to examine the nonlinear evolution equations in a Wick-type stochastic environment. By a new auxiliary equation, the Kudryashov technique is developed to a new technique called "the extended Kudryashov technique". The the extended Kudryashov technique is utilized to establish exact solutions for the mixed KdV-mKdV equation in a Wick-type stochastic environment and with a new generalized conformable type derivative. Two new kinds of traveling wave functional solutions, inclusive periodic and soliton wave solutions are obtained. Furthermore, numerical examples in the cases of deterministic and stochastic functions are applied with 3D profiles for the acquired results. [ABSTRACT FROM AUTHOR]