Higher Order Korteweg-de Vries Equation Research Articles

The modulations of higher order solitonic pressure and energy of fluid filled elastic tubes

In this article, we investigate the weak higher order nonlinear solitonic pressure waves in elastic, incompressible, nonviscous fluid-filled tubes. The higher order Korteweg–de Vries equation has been developed from the perturbed nonlinear equation. The dependence of higher-order solitonic pressure, higher order fluid velocity, and higher order solitonic energy on the nonlinear coefficient and the perturbation parameter has been established. The presented solutions will be more beneficial in the several disciplines of blood flow and many other related fields of applied science.

Whitham shocks and resonant dispersive shock waves governed by the higher order Korteweg–de Vries equation

The addition of higher order asymptotic corrections to the Korteweg–de Vries equation results in the extended Korteweg–de Vries (eKdV) equation. These higher order terms destabilize the dispersive shock wave solution, also termed an undular bore in fluid dynamics, and result in the emission of resonant radiation. In broad terms, there are three possible dispersive shock wave regimes: radiating dispersive shock wave (RDSW), cross-over dispersive shock wave (CDSW) and travelling dispersive shock wave (TDSW). While there are existing solutions for the RDSW and TDSW regimes obtained using modulation theory, there is no existing solution for the CDSW regime. Modulation theory and the associated concept of a Whitham shock are used to obtain this CDSW solution. In addition, it is found that the resonant wavetrain emitted by the eKdV equation with water wave coefficients has a minimal amplitude. This minimal amplitude is explained based on the developed Whitham modulation theory.

Constructive Study of Modulational Instability in Higher Order Korteweg-de Vries Equations

Our present study is devoted to the constructive study of the modulational instability for the Korteweg-de Vries (KdV)-family of equations u t + s u p u x + u x x x (here s = ± 1 and p > 0 is an arbitrary integer). For deducing the conditions of the instability, we first computed the nonlinear corrections to the frequency of the Stokes wave and then explored the coefficients of the corresponding modified nonlinear Schrödinger equations, thus deducing explicit expressions for the instability growth rate, maximum of the increment and the boundaries of the instability interval. A brief discussion of the results, open questions and further research directions completes the paper.

Travelling Solitary Wave Solutions to Higher Order Korteweg-de Vries Equation

The travelling solitary wave solutions to the higher order Korteweg-de Vries equation are obtained by using tanh-polynomial method. The method is effective and concise, which is also applied to various partial differential equations to obtain traveling wave solutions. The numerical simulation of the solutions is given for completeness. Numerical results show that the tanh-polynomial method works quite well.

A Note on the Fractional Generalized Higher Order KdV Equation

We obtain exact solutions to the fractional generalized higher order Korteweg-de Vries (KdV) equation using the complex method. It has showed that the applied method is very useful and is practically well suited for the nonlinear differential equations, those arising in mathematical physics.

Conditions for modulation instability in higher order Korteweg–de Vries equations

The universal mechanism of modulation instability (MI) has been discovered first for the Nonlinear Schrödinger equation (NLS) and is well studied in the frame of the higher order NLS equations. Recent studies demonstrated by pure existence theorems that also the higher order Korteweg–de Vries (KdV) equations might possess the MI. In this Letter we present explicit form of the conditions for the MI to appear in the KdV-family of equations of the general form ut+supux+uxxx where s=±1 with p>0 being an arbitrary integer.

One- and two-soliton solutions to a new KdV evolution equation with nonlinear and nonlocal terms for the water wave problem

With the help of the Boussinesq perturbation expansion, a new basic equation describing the long, small-amplitude, unidirectional wave motion in shallow water with surface tension is derived to fourth order, namely a higher-order Korteweg–de Vries (KdV) equation. The procedure for deriving this equation assumes that the relation between the small parameter \(\alpha \), which measures the ratio of wave amplitude to undisturbed fluid depth, and the small parameter \(\beta \), which measures the square of the ratio of fluid depth to wave length, is taken in the form \(\beta = 0(\alpha ) = \varepsilon \), where \(\varepsilon \) is a small, dimensionless parameter which is the order of the amplitude of the motion. Hirota’s bilinear method is used to investigate one- and two-soliton solutions for this new higher-order KdV equation.

Bell-Polynomial Approach and Soliton Solutions for Some Higher-Order Korteweg-de Vries Equations in Fluid Mechanics, Plasma Physics and Lattice Dynamics**Supported by the National Natural Science Foundation of China under Grant No. 11272023, the Open Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), and by the Fundamental Research Funds for the Central Universities

The Korteweg-de Vries (KdV)-type equations have been seen in fluid mechanics, plasma physics and lattice dynamics, etc. This paper will address the bilinearization problem for some higher-order KdV equations. Based on the relationship between the bilinear method and Bell-polynomial scheme, with introducing an auxiliary independent variable, we will present the general bilinear forms. By virtue of the symbolic computation, one- and two-soliton solutions are derived.

Riccati Equation Solutions to Higher Order Korteweg-de Vries Equation

The traveling wave solutions to the heigher order Korteweg-de Vries equation is obtained by using Riccati equation. The method is straightforward and concise, the applications are promising to obtain traveling wave solutions of various partial differential equations. It is shown that the Riccati equation method, with the symbolic computation, provide an effective and powerful mathematical tools for solving such systems. The numerical simulation of the solutions are given for completeness.

Solitary wave solution of higher-order Korteweg–de Vries equation

An attempt has been made to obtain exact analytical traveling wave solution or simple wave solution of higher-order Korteweg–de Vries (KdV) equation by using tanh-method or hyperbolic method. The higher-order equation can be derived for magnetized plasmas by using the reductive perturbation technique. It is found that the exact solitary wave solution of higher-order KdV equation is obtained by tanh-method. Using this method, different kinds of nonlinear wave equations can be evaluated. The higher-order nonlinearity and higher-order dispersive effect can be observed from the solutions of the equations. The method is applicable for other nonlinear wave equations.

An undular bore solution for the higher-order Korteweg–de Vries equation

Undular bores describe the evolution and smoothing out of an initial step in mean height and are frequently observed in both oceanographic and meteorological applications. The undular bore solution for the higher-order Korteweg–de Vries (KdV) equation is derived, using an asymptotic transformation which relates the KdV equation and its higher-order counterpart. The higher-order KdV equation considered includes all possible third-order correction terms (where the KdV equation retains second-order terms). The asymptotic transformation is then applied to the KdV undular bore solution to obtain the higher-order undular bore. Examples of higher-order undular bores, describing both surface and internal waves, are presented. Key properties, such as the amplitude and speed of the lead soliton and the width of the bore, are found. An excellent comparison is obtained between the analytical and numerical solutions. Also, it is illustrated how an asymptotic transformation and numerical solutions can be combined to generate hybrid asymptotic-numerical solutions, thus avoiding the severe instabilities associated with numerical schemes for the higher-order KdV equation.

On the asymptotic solutions of the KdV equation with higher-order corrections

A method for construction of new integrable PDEs, whose properties are related to an asymptotic perturbation expansion with the leading-order term given by an integrable equation, is developed. A new integrable equation is constructed by applying the properly defined Lie–Bäcklund group of transformations to the leading-order equation. The integrable equations related to the Korteweg–de Vries (KdV) equation with higher-order corrections are used to investigate the limits of applicability of the so-called asymptotic integrability concept. It is found that the solutions of the higher-order KdV equations obtained by a near identity transform from the normal form solitary waves cannot, in principle, describe some intrinsic features of the high-order KdV solitons.

Camassa–Holm, Korteweg–de Vries and related models for water waves

In this paper we first describe the current method for obtaining the Camassa–Holm equation in the context of water waves; this requires a detour via the Green–Naghdi model equations, although the important connection with classical (Korteweg–de Vries) results is included. The assumptions underlying this derivation are described and their roles analysed. (The critical assumptions are, (i) the simplified structure through the depth of the water leading to the Green–Naghdi equations, and, (ii) the choice of submanifold in the Hamiltonian representation of the Green–Naghdi equations. The first of these turns out to be unimportant because the Green–Naghdi equations can be obtained directly from the full equations, if quantities averaged over the depth are considered. However, starting from the Green–Naghdi equations precludes, from the outset, any role for the variation of the flow properties with depth; we shall show that this variation is significant. The second assumption is inconsistent with the governing equations.)Returning to the full equations for the water-wave problem, we retain both parameters (amplitude, ε, and shallowness, δ) and then seek a solution as an asymptotic expansion valid for, ε → 0, δ → 0, independently. Retaining terms O(ε), O(δ2) and O(εδ2), the resulting equation for the horizontal velocity component, evaluated at a specific depth, is a Camassa–Holm equation. Some properties of this equation, and how these relate to the surface wave, are described; the role of this special depth is discussed. The validity of the equation is also addressed; it is shown that the Camassa–Holm equation may not be uniformly valid: on suitably short length scales (measured by δ) other terms become important (resulting in a higher-order Korteweg–de Vries equation, for example). Finally, we indicate how our derivation can be extended to other scenarios; in particular, as an example, we produce a two-dimensional Camassa–Holm equation for water waves.

Soliton Chain Solutions to the Two-Dimensional Higher Order Korteweg-de Vries Equations of Sawada-Kotera Hierarchy

A series of two-dimensional equations with the two-dimensional Korteweg-de Vries equation being their first one are constructed as Sawada-Kotera did in one-dimensional case. Their soliton chain solutions are obtained and investigated.

Gauge transformation and the higher order Korteweg–de Vries equation

To the family of higher order Korteweg–de Vries (KdV) equations a new family of higher order mKdV equations depending on a parameter η is constructed. They are connected by an η2-dependent Miura transformation. A Bäcklund transformation is also established. Furthermore, a new gauge transformation is associated to this Bäcklund transformation. It enables one to derive auto-Bäcklund transformations for the families of higher order η2-mKdV and KdV equations, respectively. For illustration, four generations of explicit solutions of the second order KdV equation are presented.

Lattice equations, hierarchies and Hamiltonian structures

By applying a suitable continuum limit to a lattice version of the Korteweg-de Vries (KdV) equation we obtain an infinite hierarchy of integrable partial difference equations for time-dependent fields, defined on the sites of a one-dimensional chain, together with the associated Hamiltonian structure. A second continuum limit applied to any member of the hierarchy yields the well-known hierarchy of higher-order KdV equations with the recursion operator, as well as the Hamiltonian structure. A similar procedure is worked out for the hierarchies related to the modified Korteweg-de Vries (mKdV) equation, starting from a lattice version of the mKdV. For all the equations a direct linearization is given of an integral equation with arbitrary measure and contour.

Bilinearization of Nonlinear Evolution Equations. III. Bäcklund Transformations of Higher-Order Korteweg-de Vries Equations

The Backlund transformations of the higher-order Korteweg-de Vries (KdV) equations are derived on the basis of the bilinear transformation method. It is shown that these transformations are closely related to the quantities obtained from the functional derivative of conserved quantities of the modified KdV equation. The inverse scattering transforms of the higher-order KdV equations are also presented.

Bilinearization of Nonlinear Evolution Equations

A new method for transforming the Lax hierarchy of nonlinear evolution equations u t =-(∂/∂ x )(δ I n /δ u ) into bilinear forms is developed on the basis of the bilinear transformation method, and the structure of these equations is clarified, where I n is the n -th conserved quantity of some nonlinear evolution equation and δ/δ u denotes the functional derivative. The explicit calculations are carried out in the case of the higher-order Korteweg-de Vries equations.

N-Soliton Solution of the Higher Order Wave Equation for a Fluid of Finite Depth

Exact N -soliton solution of the higher order wave equation for a stratified fluid layer of finite depth is obtained by employing Hirota's bilinear transformation method. The solution reduces to the N -soliton solution of the higher order Korteweg-de Vries equation in the shallow water limit and to that of the higher order Benjamin-One equation in the deep water limit. The asymptotic form of the solution for large time is derived.

The prolongation structure of a higher order Korteweg-de Vries equation

We have used Wahlquist & Estabrook’s prolongation method to construct a Lie algebra for the higher order Korteweg de Vries (K.deV.) equation q 4 x + α qq 2 x + β q 2 x + γ q 3 ) x + q t = 0 with ( q nx =