Higher Order Korteweg-de Vries Research Articles

Higher‐order integrable models for oceanic internal wave–current interactions

AbstractIn this paper, we derive a higher‐order Korteweg–de Vries (HKdV) equation as a model to describe the unidirectional propagation of waves on an internal interface separating two fluid layers of varying densities. Our model incorporates underlying currents by permitting a sheared current in both fluid layers, and also accommodates the effect of the Earth's rotation by including Coriolis forces (restricted to the Equatorial region). The resulting governing equations describing the water wave problem in two fluid layers under a “flat‐surface” assumption are expressed in a general form as a system of two coupled equations through Dirichlet–Neumann (DN) operators. The DN operators also facilitate a convenient Hamiltonian formulation of the problem. We then derive the HKdV equation from this Hamiltonian formulation, in the long‐wave, and small‐amplitude, asymptotic regimes. Finally, it is demonstrated that there is an explicit transformation connecting the HKdV we derive with the following integrable equations of a similar type: KdV5, Kaup–Kuperschmidt equation, Sawada–Kotera equation, and Camassa–Holm and Degasperis–Procesi equations.

Proliferation of soliton, explosive, shocklike, and periodic ion-acoustic waves in Titan's ionosphere

We report on many electrostatic nonlinear structures, namely, soliton, explosive, shocklike, and periodic waves, which may exist due to the dynamics of different ion species in Titan's ionosphere. Using the latest available observations of Titan at an altitude of ≈ 1300 km, the main ion components are HCNH+, C2H5+, and C3H5+. The dynamics of these waves are described by the Korteweg–de Vries (KdV) equation. Using the G′/G–expansion method, we obtain a class of solutions that elucidate these nonlinear structures. As the wave amplitude increases, both the width and velocity of the wave deviate from the prediction of the KdV equation. To describe the waves of larger amplitude, the higher-order dispersion has to be taken into account. The G′/G–expansion method is used, for the first time, to solve the KdV equation with higher-order dispersion. We compare between both KdV and higher-order dispersive KdV solutions, i.e., soliton, explosive, shocklike, and periodic waves.

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.

Physical uniqueness of higher-order Korteweg-de Vries theory for continuously stratified fluids without background shear

The 2nd-order Korteweg-de Vries (KdV) equation and the Gardner (or extended KdV) equation are often used to investigate internal solitary waves, commonly observed in oceans and lakes. However, application of these KdV-type equations for continuously stratified fluids to geophysical problems is hindered by nonuniqueness of the higher-order coefficients and the associated correction functions to the wave fields. This study proposes to reduce arbitrariness of the higher-order KdV theory by considering its uniqueness in the following three physical senses: (i) consistency of the nonlinear higher-order coefficients and correction functions with the corresponding phase speeds, (ii) wavenumber-independence of the vertically integrated available potential energy, and (iii) its positive definiteness. The spectral (or generalized Fourier) approach based on vertical modes in the isopycnal coordinate is shown to enable an alternative derivation of the 2nd-order KdV equation, without encountering nonuniqueness. Comparison with previous theories shows that Parseval’s theorem naturally yields a unique set of special conditions for (ii) and (iii). Hydrostatic fully nonlinear solutions, derived by combining the spectral approach and simple-wave analysis, reveal that both proposed and previous 2nd-order theories satisfy (i), provided that consistent definitions are used for the wave amplitude and the nonlinear correction. This condition reduces the arbitrariness when higher-order KdV-type theories are compared with observations or numerical simulations. The coefficients and correction functions that satisfy (i)-(iii) are given by explicit formulae to 2nd order and by algebraic recurrence relationships to arbitrary order for hydrostatic fully nonlinear and linear fully nonhydrostatic effects.

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.

Improved Bell-polynomial procedure for the higher-order Korteweg–de Vries equations in fluid dynamics

Korteweg–de Vries (KdV)-typed equations are seen in fluid dynamics, plasma physics and other fields. By means of the Bell-polynomial procedure, we take two higher-order members of the KdV hierarchy, the seventh- and ninth-order Lax’s KdV equations in fluid dynamics, as the examples for studying the integrable properties of the higher-order equations. Different from lower-order equations, two new partial differential operators in the Bell-polynomial procedure are introduced to construct the “multi-dimensional” bilinear forms of such equations with several auxiliary independent variables. Through the procedure simplified via the algebraic operation of the polynomials, the Bäcklund transformations, Lax pairs and infinite conservation laws of such equations are deduced.

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.

Amplitudes Estimation of Large Internal Solitary Waves in the Mid-Atlantic Bight Using Synthetic Aperture Radar and Marine X-Band Radar Images

The accurate estimation of internal solitary waves' (ISWs') amplitudes from radar images is important for understanding the ISW evolution, energy dissipation, and mixing processes. The in situ data from the Non-Linear Internal Wave Initiative experiment in the Mid-Atlantic Bight show many ISWs with amplitudes of 10 m or more in a shallow water depth of 80 m or less. Therefore, the higher order Korteweg-de Vries (KdV) equation in a two-layer system is needed to describe these large-amplitude ISWs instead of the classic KdV equation. Based on a simple theoretical radar imaging model, we develop a method to estimate large ISW amplitudes from distances between the positive and negative peaks of ISW signatures in radar images and a selection rule from the two possible amplitude solutions. Two groups of ISWs with large amplitudes, determined from the temperature records from nearby moorings, are observed in a RADARSAT synthetic-aperture-radar image and in marine X-band radar data collected during the experiment. We validate the method using the ISW signatures taken from these two cases. We find the estimated amplitudes to agree well with those determined from the moorings. The proposed method provides a relatively simple and accurate way to estimate large ISW amplitudes from radar images.

Spontaneous soliton generation in the higher order Korteweg–de Vries equations on the half-line

Some new effects in the soliton dynamics governed by higher order Korteweg-de Vries (KdV) equations are discussed based on the exact explicit solutions of the equations on the positive half-line. The solutions describe the process of generation of a soliton that occurs without boundary forcing and on the steady state background: the boundary conditions remain constant and the initial distribution is a steady state solution of the problem. The time moment when the soliton generation starts is not determined by the parameters present in the problem formulation, the additional parameters imbedded into the solution are needed to determine that moment. The equations found capable of describing those effects are the integrable Sawada-Kotera equation and the KdV-Kaup-Kupershmidt (KdV-KK) equation which, albeit not proven to be integrable, possesses multi-soliton solutions.

Static algebraic solitons in Korteweg–de Vries type systems and the Hirota transformation

Some effects in the soliton dynamics governed by higher-order Korteweg-de Vries (KdV) type equations are discussed. This is done based on the exact explicit solutions of the equations derived in the paper. It is shown that some higher order KdV equations possessing multisoliton solutions also admit steady state solutions in terms of algebraic functions describing localized patterns. Solutions including both those static patterns and propagating KdV-like solitons are combinations of algebraic and hyperbolic functions. It is shown that the localized structures behave like static solitons upon collisions with regular moving solitons, with their shape remaining unchanged after the collision and only the position shifted. These phenomena are not revealed in common multisoliton solutions derived using inverse scattering or Hirota's method. The solutions of the higher-order KdV type equations were obtained using a method devised for obtaining soliton solutions of nonlinear evolution equations. This method can be combined with Hirota's method with a modified representation of the solution which allows the results to be extended to multisoliton solutions. The prospects for applying the methods to soliton equations not of KdV type are discussed.

Application of higher-order KdV-mKdV model with higher-degree nonlinear terms to gravity waves in atmosphere

Higher-order Korteweg-de Vries (KdV)-modified KdV (mKdV) equations with a higher-degree of nonlinear terms are derived from a simple incompressible non-hydrostatic Boussinesq equation set in atmosphere and are used to investigate gravity waves in atmosphere. By taking advantage of the auxiliary nonlinear ordinary differential equation, periodic wave and solitary wave solutions of the fifth-order KdV-mKdV models with higher-degree nonlinear terms are obtained under some constraint conditions. The analysis shows that the propagation and the periodic structures of gravity waves depend on the properties of the slope of line of constant phase and atmospheric stability. The Jacobi elliptic function wave and solitary wave solutions with slowly varying amplitude are transformed into triangular waves with the abruptly varying amplitude and breaking gravity waves under the effect of atmospheric instability.

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.

Higher-order Korteweg-de Vries models for internal solitary waves in a stratified shear flow with a free surface

Abstract. A higher-order extension of the familiar Korteweg-de Vries equation is derived for internal solitary waves in a density- and current-stratified shear flow with a free surface. All coefficients of this extended Korteweg-de Vries equation are expressed in terms of integrals of the modal function for the linear long-wave theory. An illustrative example of a two-layer shear flow is considered, for which we discuss the parameter dependence of the coefficients in the extended Korteweg-de Vries equation.

Asymptotic solitons of the extended Korteweg–de Vries equation

The interaction of two higher-order solitary waves, governed by the extended Korteweg--de Vries (KdV) equation, is examined. A nonlocal transformation is used on the extended KdV equation to asymptotically transform it to the KdV equation. The transformation is used to derive the higher-order two-soliton collision and it is found that the interaction is asymptotically elastic. Moreover, the higher-order corrections to the phase shifts suffered by the solitary waves during the collision are found. Comparison is made with a previous result, which indicated that, except for a special case, the interaction of higher-order KdV solitary waves is inelastic, with a coupling, or interaction, term occuring after collision. It is shown that the two theories are asymptotically equivalent, with the coupling term representing the higher-order phase shift corrections. Finally, it is concluded, with the support of existing numerical evidence, that the interpretation of the coupling term as a higher-order phase shift is physically appropriate; hence, the interaction of higher-order solitary waves is asymptotically elastic.

An improved theory of long waves on the water surface

The Hamiltonian formalism is used to derive the equations of long waves on the surface of an infinite layer of water over a horizontal, smooth bottom, taking into account second-order terms with respell to small parameters of non-linearity and dispersion; in other words, the Boussinesq equations of shallow-water theory are improved. A non-linear evolution equation is derived for the elevation of the free surface and a transformation is obtained to convert it into one of the higher-order Korteweg-de Vries (KdV) equations. Single- and double-soliton solutions are used to demonstrate the special features of the behaviour of the waves described by the equation, which are a more accurate version of KdV solitons.