Fifth-order KdV Equation Research Articles

Variants of the generalized fifth-order KdV equation with compact and noncompact structures

In this paper we study compact and noncompact solutions for variants of the generalized fifth-order KdV equation. The analysis reveals compact and noncompact solutions for some of these variants, whereas other variants give rise to either compact or noncompact structures. The results emphasize the different physical structures of the compact and the noncompact solutions.

On asymptotically equivalent shallow water wave equations

The integrable third-order Korteweg–de Vries (KdV) equation emerges uniquely at linear order in the asymptotic expansion for unidirectional shallow water waves. However, at quadratic order, this asymptotic expansion produces an entire family of shallow water wave equations that are asymptotically equivalent to each other, under a group of nonlinear, non-local, normal-form transformations introduced by Kodama in combination with the application of the Helmholtz-operator. These Kodama–Helmholtz (KH) transformations are used to present connections between shallow water waves, the integrable fifth-order KdV equation, and a generalization of the Camassa–Holm (CH) equation that contains an additional integrable case. The dispersion relation of the full water wave problem and any equation in this family agree to fifth order. The travelling wave solutions of the CH equation are shown to agree to fifth order with the exact solution.

Local and global well-posedness for a class of nonlinear dispersive equations

We study the local and global well-posedness of the initial-value problem for the class of nonlinear dispersive PDEs of the form \[ u_t - M u_x + F(u)_x = 0, \;\;\; t\in \mathbb{R}, \] where $u=u(x,t)$, $x\in \mathbb{R} $ or $x\in \mathbb{T} $. Here $M$ is a linear operator, given in the Fourier space by the multiplication operator: $\hat{Mv} ( \xi ) = |\xi |^{2\beta } \hat{v} (\xi ),$ $ \beta \ge \frac12 $ and $F$ is a nonlinear, (sufficiently) smooth function. This equation is a generalization of the Korteweg--de Vries (KdV) equation ($\beta = 1 $), the Benjamin--Ono (BO) equation ($\beta = \frac12$) and the fifth-order KdV equation ($\beta = 2$). The nonlinearity can be very general, but a certain growth condition must be imposed in order to obtain global results. Roughly speaking, we impose that $(F'(r))_{+}$ grows at most like $\left| r\right| ^p$ as $r\to\infty $, for some $p < 4\beta $. Global existence of solutions is, therefore, intimately related to the balance between the strength of the nonlinearity and the dispersion relation. The semigroup methods developed by Goldstein--Oharu--Takahashi are being successfully applied here. Most of the results are presented in the periodic case (i.e., $x\in\mathbb{T} $), but they are also valid in the real-line case (when $x\in \mathbb{R} )$.

Analytical solutions for some nonlinear evolution equations

The following partial differential equations are studied: generalized fifth-order KdV equation, water wave equation, Kupershmidt equation, couples KdV equation. The analytical solutions to these problems via using various ansatzes by introducing a secondorder ordinary differential equation are found out.

An explicit and numerical solutions of some fifth-order KdV equation by decomposition method

In this paper, by considering the Adomian decomposition method, explicit and numerical solutions are calculated for a various fifth-order Korteweg-de Vries equations with initial condition. The method does not need linearization or weak nonlinearity assumptions, perturbation theory. The decomposition series explicit solution of the equation is quickly obtained by observing the existence of the self-canceling “noise” terms where sum of components vanishes in the limit.

When gap solitons become embedded solitons: a generic unfolding

A two-parameter unfolding is considered of single-pulsed homoclinic orbits to an equilibrium with two real and two zero eigenvalues in fourth-order reversible dynamical systems. One parameter controls the linearisation, with a transition occurring between a saddle-centre and a hyperbolic equilibrium. In the saddle-centre region, the homoclinic orbit is of codimension-one, which is controlled by the second generic parameter, whereas when the equilibrium is hyperbolic the homoclinic orbit is structurally stable. A geometric approach reveals the homoclinic orbits to the saddle to be generically destroyed either by developing an algebraically decaying tail or through a fold, depending on the sign of the perturbation of the second parameter. Special cases of different actions of Z 2 -symmetry are considered, as is the case of the system being Hamiltonian. Application of these results is considered to the transition between embedded solitons (corresponding to the codimension-one-homoclinic orbits) and gap solitons (the structurally stable ones) in nonlinear wave systems. The theory is shown to match numerical experiments on two models arising in nonlinear optics and on a form of fifth-order Korteweg de Vries equation.

Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework

The spectral problem associated with the linearization about solitary waves of the generalized fifth-order KdV equation is formulated in terms of the Evans function, a complex analytic function whose zeros correspond to eigenvalues. A numerical framework, based on a fast robust shooting algorithm on exterior algebra spaces is introduced. The complete algorithm has several new features, including a rigorous numerical algorithm for choosing starting values, a new method for numerical analytic continuation of starting vectors, the role of the Grassmannian G 2( C 5) in choosing the numerical integrator, and the role of the Hodge star operator for relating ⋀ 2( C 5) and ⋀ 3( C 5) and deducing a range of numerically computable forms for the Evans function. The algorithm is illustrated by computing the stability and instability of solitary waves of the fifth-order KdV equation with polynomial nonlinearity.

Nonlinear interaction of ice cover with shallow water waves in channels

A nonlinear analysis of the interaction between a water wave and a floating ice cover in river channels is presented. The one-dimensional weakly nonlinear equation for shallow water wave propagation in a uniform channel with a floating ice cover is derived. The ice cover is assumed to be a thin uniform elastic plate. The weakly nonlinear equation is a fifth-order KdV equation. Analytical solutions of the nonlinear periodic wave equation are obtained. These solutions show that the shape, wavelength and celerity of the nonlinear waves depend on the wave amplitude. The wave celerity is slightly smaller than the open water wave celerity. The wavelength decreases as the wave amplitude increases. Based on these solutions the fracture of the ice cover is analysed. The spacing between transverse cracks varies form 50 m to a few hundred metres with the corresponding wave amplitude varying from 0.2 to 0.8 m, depending on the thickness and strength of the cover. These results agree well with limited field observations.

Solitary patterns solutions having infinite slopes or cusps for fifth-order KdV like equations in higher dimensions

In this paper we consider a modified form of fifth-order KdV equation (fKdV) that exhibits solutions with solitary patterns having cusps or infinite slopes. We formally construct a framework for studying this problem in higher dimensions. Two distinct sets of general formulas, that are of substantial interest, are developed for this variant fK( n, n) for all positive integers n, n⩾1.

Compactons structures for fifth-order KdV like equations in higher dimensions

In this paper we study a variant of the fifth-order KdV equation (fKdV) that exhibits compactons: solitons with finite wave lengths. The work formally shows how to construct compact dispersive structures in higher dimensions. Two sets of general formulas for compactons solutions, that are of substantial interest, are developed for this variant fK( n, n) for all positive integers n, n⩾1.

Semi-stability of embedded solitons in the general fifth-order KdV equation

Evolution of perturbed embedded solitons in the general Hamiltonian fifth-order Korteweg–de Vries (KdV) equation is studied. When an embedded soliton is perturbed, it sheds a one-directional continuous-wave radiation. It is shown that the radiation amplitude is not minimal in general. A dynamical equation for velocity of the perturbed embedded soliton is derived. This equation shows that a neutrally stable embedded soliton is in fact semi-stable. When the perturbation increases the momentum of the embedded soliton, the perturbed state approaches asymptotically the embedded soliton, while when the perturbation reduces the momentum of the embedded soliton, the perturbed state decays into radiation. Classes of initial conditions to induce soliton decay or persistence are also determined. Our analytical results are confirmed by direct numerical simulations of the fifth-order KdV equation.

Do true elevation gravity–capillary solitary waves exist? A numerical investigation

This paper extends the numerical results of Hunter &amp; Vanden-Broeck (1983) and Vanden-Broeck (1991) which were concerned with studies of solitary waves on the surface of fluids of finite depth under the action of gravity and surface tension. The aim of this paper is to answer the question of whether small-amplitude elevation solitary waves exist. Several analytical results have proved that bifurcating from Froude number F = 1, for Bond number τ between 0 and 1/3, there are families of ‘generalized’ solitary waves with periodic tails whose minimum amplitude is an exponentially small function of F−1. An open problem (which, for τ sufficiently close to 1/3, was recently proved by Sun 1999 to be false) is whether this amplitude can ever be zero, which would give a truly localized solitary wave.The problem is first addressed in terms of model equations taking the form of generalized fifth-order KdV equations, where it is demonstrated that if such a zero-tail-amplitude solution occurs, it does so along codimension-one lines in the parameter plane. Moreover, along solution paths of generalized solitary waves a topological distinction is found between cases where the tail does vanish and those where it does not. This motivates a new set of numerical results for the full problem, formulated using a boundary integral method, namely to probe the size of the tail amplitude as τ varies for fixed F &gt; 1. The strong conclusion from the numerical results is that true solitary waves of elevation do not exist for the steady gravity–capillary water wave problem, at least for 9/50 &lt; τ &lt; 1=3. This finding confirms and explains previous asymptotic results by Yang &amp; Akylas.

Analytical and numerical studies of a singularly perturbed Boussinesq equation

We study the singularly perturbed (sixth-order) Boussinesq equation recently introduced by Daripa and Hua [Appl. Math. Comput. 101 (1999) 159]. Motivated by their work, we formally derive this equation from two-dimensional potential flow equations governing the small amplitude long capillary-gravity waves on the surface of shallow water for Bond number very close to but less than 1/3. On the basis of far-field analyses and heuristic arguments, we show that the traveling wave solutions of this equation are weakly non-local solitary waves characterized by small amplitude fast oscillations in the far-field. We review various analytical and numerical methods originally devised to obtain this type of weakly non-local solitary wave solutions of the singularly perturbed (fifth-order) KdV equation. Using these methods, we obtain weakly non-local solitary wave solutions of the singularly perturbed (sixth-order) Boussinesq equation and provide estimates of the amplitude of oscillations which persist in the far-field.

The homogeneous balance method, Lax pair, Hirota transformation and a general fifth-order KdV equation

In this paper, some notes of the homogeneous balance (HB) method are discussed by a kind of general fifth-order KdV (fKdV) equation. First, the auto-Bäcklund transformation and Lax represents of the higher-order KdV equation (a specific form of fKdV equation) are obtained by the HB method. Then, the connection of the Hirota transformation and the HB method is discussed by two specific fKdV equations: the Sawada–Kotera equation and the Lax's equation. At the same time, the solitary wave solution of the general fKdV equation is obtained.

Compacton solutions in a class of generalized fifth-order Korteweg-de Vries equations.

Solitons play a fundamental role in the evolution of general initial data for quasilinear dispersive partial differential equations, such as the Korteweg-de Vries (KdV), nonlinear Schrödinger, and the Kadomtsev-Petviashvili equations. These integrable equations have linear dispersion and the solitons have infinite support. We have derived and investigate a new KdV-like Hamiltonian partial differential equation from a four-parameter Lagrangian where the nonlinear dispersion gives rise to solitons with compact support (compactons). The new equation does not seem to be integrable and only mass, momentum, and energy seem to be conserved; yet, the solitons display almost the same modal decompositions and structural stability observed in integrable partial differential equations. The compactons formed from arbitrary initial data, are nonlinearly self-stabilizing, and maintain their coherence after multiple collisions. The robustness of these compactons and the inapplicability of the inverse scattering tools, that worked so well for the KdV equation, make it clear that there is a fundamental mechanism underlying the processes beyond integrability. We have found explicit formulas for multiple classes of compact traveling wave solutions. When there are more than one compacton solution for a particular set of parameters, the wider compacton is the minimum of a reduced Hamiltonian and is the only one that is stable.

Embedded solitons: solitary waves in resonance with the linear spectrum

It is commonly held that a necessary condition for the existence of solitons in nonlinear-wave systems is that the soliton’s frequency (spatial or temporal) must not fall into the continuous spectrum of radiation modes. However, this is not always true. We present a new class of codimension-one solitons (i.e., those existing at isolated frequency values) that are embedded into the continuous spectrum. This is possible if the spectrum of the linearized system has (at least) two branches, one corresponding to exponentially localized solutions, and the other to radiation modes. An embedded soliton (ES) is obtained when the latter component exactly vanishes in the solitary-wave’s tail. The paper contains both a survey of recent results obtained by the authors and some new results, the aim being to draw together several different mechanism underlying the existence of ESs. We also consider the distinctive properties of semi-stability of ESs, and moving ESs. Results are presented for four different physical models, including an extended fifth-order KdV equation describing surface waves in inviscid fluids, and three models from nonlinear optics. One of them pertains to a resonant Bragg grating in an optical fiber with a cubic nonlinearity, while two others describe second-harmonic generation (SHG) in the temporal or spatial domain (i.e., respectively, propagating pulses in nonlinear-optical fibers, or stationary patterns in nonlinear planar waveguides). Special attention is paid to the SHG model in the temporal domain for a case of competing quadratic and cubic nonlinearities. In particular, a new result is that when both harmonics have anomalous dispersion, an ES can exist which is, virtually, completely stable.

Weakly non-local solitary wave solutions of a singularly perturbed Boussinesq equation

We study the singularly perturbed (sixth-order) Boussinesq equation recently introduced by Daripa and Hua [Appl. Math. Comput. 101 (1999) 159]. This equation describes the bi-directional propagation of small amplitude and long capillary-gravity waves on the surface of shallow water for bond number less than but very close to 1/3. On the basis of far-field analyses and heuristic arguments, we show that the traveling wave solutions of this equation are weakly non-local solitary waves characterized by small amplitude fast oscillations in the far-field. Using various analytical and numerical methods originally devised to obtain this type of weakly non-local solitary wave solutions of the singularly perturbed (fifth-order) KdV equation, we obtain weakly non-local solitary wave solutions of the singularly perturbed (sixth-order) Boussinesq equation and provide estimates of the amplitude of oscillations which persist in the far-field.

Solitary wave interaction for the extended BBM equation

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Marchant T. R. 2000Solitary wave interaction for the extended BBM equationProc. R. Soc. Lond. A.456433–453http://doi.org/10.1098/rspa.2000.0524SectionRestricted accessSolitary wave interaction for the extended BBM equation T. R. Marchant T. R. Marchant School of Mathematics and Applied Statistics, University of Wollongong, Northfields Avenue, Wollongong, NSW 2522, Australia Google Scholar Find this author on PubMed Search for more papers by this author T. R. Marchant T. R. Marchant School of Mathematics and Applied Statistics, University of Wollongong, Northfields Avenue, Wollongong, NSW 2522, Australia Google Scholar Find this author on PubMed Search for more papers by this author Published:08 February 2000https://doi.org/10.1098/rspa.2000.0524AbstractSolitary wave interaction is examined, for the case of surface waves on shallow water, by using an extended Benjamin–Bona–Mahony (eBBM) equation. This equation includes higher–order nonlinear and dispersive effects, and hence is asymptotically equivalent to the extended Korteweg–de Vries (eKdV) equation. However, it has certain numerical advantages as it allows the modelling of steeper waves, which, due to numerical instability, is not possible using the eKdV equation.Numerical simulations of a number of collisions of varying nonlinearity are performed. The numerical results show evidence of inelastic behaviour at high order. For waves of small amplitude the evidence of inelastic behaviour is indirect; after collision a dispersive wavetrain of extremely small amplitude is found behind the smaller solitary wave. For steeper waves, however, direct evidence of inelastic behaviour is found; the larger wave is increased and the smaller wave is decreased in amplitude after the collision.Conservation laws for the mass and energy, exact and asymptotic, respectively, are derived for the eBBM equation, and numerically verified. Data from the collisions, such as the change in solitary wave amplitudes, the higher–order phase shifts and the mass and energy of the dispersive wavetrains are all tabulated. These are used to show that the change in solitary wave amplitude is of O(α4), verifying previously obtained theoretical predictions. A good comparison is also obtained between the numerically obtained phase shifts and existing asymptotic results.Lastly, three different eBBM solitary wave interactions are examined in detail and compared with existing numerical data obtained from alternative weakly nonlinear models and the Euler water–wave equations. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Li F and Yao Y (2022) Multisoliton and rational solutions for the extended fifth-order KdV equation in fluids with self-consistent sources, Theoretical and Mathematical Physics, 10.1134/S0040577922020039, 210:2, (184-197), Online publication date: 1-Feb-2022. Bona J, Chen H, Hong Y and Karakashian O (2019) Numerical Study of the Second-Order Correct Hamiltonian Model for Unidirectional Water Waves, Water Waves, 10.1007/s42286-019-00003-y, 1:1, (3-40), Online publication date: 1-May-2019. Yıldırım Y and Yaşar E (2017) An extended Korteweg–de Vries equation: multi-soliton solutions and conservation laws, Nonlinear Dynamics, 10.1007/s11071-017-3749-x, 90:3, (1571-1579), Online publication date: 1-Nov-2017. Wang Y and Chen Y (2016) Bell polynomials approach for two higher-order KdV-type equations in fluids, Nonlinear Analysis: Real World Applications, 10.1016/j.nonrwa.2016.03.005, 31, (533-551), Online publication date: 1-Oct-2016. Meng G, Gao Y, Zuo D, Shen Y, Sun Y and Yu X (2015) Multi-Soliton and Rational Solutions for the Extended Fifth-Order KdV Equation in Fluids, Zeitschrift für Naturforschung A, 10.1515/zna-2015-0131, 70:7, (559-566), Online publication date: 1-Jul-2015. Miao Q, Wang Y, Chen Y and Yang Y (2014) PDEBellII: A Maple package for finding bilinear forms, bilinear Bäcklund transformations, Lax pairs and conservation laws of the KdV-type equations, Computer Physics Communications, 10.1016/j.cpc.2013.09.005, 185:1, (357-367), Online publication date: 1-Jan-2014. Wang P, Tian B, Liu W, Qu Q, Li M and Sun K (2011) Lax pair, conservation laws and N-soliton solutions for the extended Korteweg-de Vries equations in fluids, The European Physical Journal D, 10.1140/epjd/e2010-10357-x, 61:3, (701-708), Online publication date: 1-Feb-2011. Hiraoka Y and Kodama Y Normal Form and Solitons Integrability, 10.1007/978-3-540-88111-7_6, (175-214) Soares-Frazão S and Guinot V (2008) A second-order semi-implicit hybrid scheme for one-dimensional Boussinesq-type waves in rectangular channels, International Journal for Numerical Methods in Fluids, 10.1002/fld.1679, 58:3, (237-261), Online publication date: 30-Sep-2008. Porubov A, Maugin G, Gursky V and Krzhizhanovskaya V (2005) On some localized waves described by the extended KdV equation, Comptes Rendus Mécanique, 10.1016/j.crme.2005.06.003, 333:7, (528-533), Online publication date: 1-Jul-2005. Marchant T (2004) Asymptotic solitons for a third-order Korteweg–de Vries equation, Chaos, Solitons & Fractals, 10.1016/j.chaos.2004.02.033, 22:2, (261-270), Online publication date: 1-Oct-2004. Marchant T (2002) High‐Order Interaction of Solitary Waves on Shallow Water, Studies in Applied Mathematics, 10.1111/1467-9590.00001, 109:1, (1-17), Online publication date: 1-Jul-2002. Marchant T (2009) Numerical solitary wave interaction: the order of the inelastic effect, The ANZIAM Journal, 10.1017/S144618110000794X, 44:1, (95-102), Online publication date: 1-Jul-2002. This Issue08 February 2000Volume 456Issue 1994 Article InformationDOI:https://doi.org/10.1098/rspa.2000.0524Published by:Royal SocietyPrint ISSN:1364-5021Online ISSN:1471-2946History: Published online08/02/2000Published in print08/02/2000 License: Citations and impact Keywordsinelastic effectsnumerical solutionssolitary wave collisionshigher–order KdV and BBM equations

On the nonlinear stability of solitary wave solutions of the fifth-order Korteweg–de Vries equation

For the fifth-order Korteweg–de Vries equation it is demonstrated that the Hamiltonian is bounded from below for fixed momentum. If there exists a solitary wave solution which realizes this minimum, then it is stable with respect to not only small perturbations but also finite ones. The proof is based on both the Lyapunov theorem and an integral estimation of the Sobolev–Gagliardo–Neirenberg inequalities.

Time dependent gravity-capillary flows past an obstacle

The evolution of free surface gravity-capillary flows past a submerged obstacle in a shallow water channel is studied. For weakly nonlinear waves, the problem is formulated as a forced fifth-order Korteweg-de Vries (KdV) equation for the free-surface profile. Solutions to the initial value problem for this equation are computed numerically. It is found that there is a wealth of steady and unsteady behaviors, and several regimes are identified by varying the Froude and Bond numbers. This generalizes previous calculations in the absence of surface tension.