Order Korteweg-de Vries Equation Research Articles

Generalized solitary waves in a finite‐difference Korteweg‐de Vries equation

AbstractGeneralized solitary waves with exponentially small nondecaying far field oscillations have been studied in a range of singularly perturbed differential equations, including higher order Korteweg‐de Vries (KdV) equations. Many of these studies used exponential asymptotics to compute the behavior of the oscillations, revealing that they appear in the solution as special curves known as Stokes lines are crossed. Recent studies have identified similar behavior in solutions to difference equations. Motivated by these studies, the seventh‐order KdV and a hierarchy of higher order KdV equations are investigated, identifying conditions which produce generalized solitary wave solutions. These results form a foundation for the study of infinite‐order differential equations, which are used as a model for studying lattice equations. Finally, a lattice KdV equation is generated using finite‐difference discretization, in which a lattice generalized solitary wave solution is found.

Control and stabilization of the periodic fifth order Korteweg-de Vries equation

We establish local exact control and local exponential stability of periodic solutions of fifth order Korteweg-de Vries type equations in Hs(𝕋), s > 2. A dissipative term is incorporated into the control which, along with a propagation of regularity property, yields a smoothing effect permitting the application of the contraction principle.

Approximate Solution Method of the Seventh Order KdV Equations by Decomposition Method

In this paper, Adomian decomposition method (ADM) is implemented to approximate the solution of the Korteweg-de Vries (KdV) equations of seventh order, which are Kaup-Kuperschmidt equation and seventh order Kawahara equation. The results obtained by the ADM are compared with the exact solutions. It is found that the ADM is very efficient and convenient and can be applied to a large class of problems. The conservation properties of solution are examined by calculating the first three invariants.

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.

Modulation theory solution for nonlinearly resonant, fifth‐order Korteweg–de Vries, nonclassical, traveling dispersive shock waves

AbstractA new class of resonant dispersive shock waves was recently identified as solutions of the Kawahara equation— a Korteweg–de Vries (KdV) type nonlinear wave equation with third‐ and fifth‐order spatial derivatives— in the regime of nonconvex, linear dispersion. Linear resonance resulting from the third‐ and fifth‐order terms in the Kawahara equation was identified as the key ingredient for nonclassical dispersive shock wave solutions. Here, nonlinear wave (Whitham) modulation theory is used to construct approximate nonclassical traveling dispersive shock wave (TDSW) solutions of the fifth‐ order KdV equation without the third derivative term, hence without any linear resonance. A self‐similar, simple wave modulation solution of the fifth order, weakly nonlinear KdV–Whitham equations is obtained that matches a constant to a heteroclinic traveling wave via a partial dispersive shock wave so that the TDSW is interpreted as a nonlinear resonance. The modulation solution is compared with full numerical solutions, exhibiting excellent agreement. The TDSW is shown to be modulationally stable in the presence of sufficiently small third‐order dispersion. The Kawahara–Whitham modulation equations transition from hyperbolic to elliptic type for sufficiently large third‐order dispersion, which provides a possible route for the TDSW to exhibit modulational instability.

A MULTIPLE SCALES METHOD FOR SOLVING NONLINEAR KdV7 EQUATION

In this report, a method of multiple scales is presented for the analysis of the (1+1)-dimensional seven order Korteweg-de Vries (KdV7) equation and we derive nonlinear Schrödinger (NLS) type equation. Also we found the exact solutions for (1+1)-dimensional KdV7 equation by using the (G/G)-expansion method. These methods are very simple and effective for getting integrability and exact solutions of KdV type equations

New exact solutions for the conformable space-time fractional KdV, CDG, (2+1)-dimensional CBS and (2+1)-dimensional AKNS equations

ABSTRACTIn the present paper, expansion method is applied to the space-time fractional third order Korteweg-De Vries (KdV) equation, space-time fractional Caudrey-Dodd-Gibbon (CDG) equation, space-time fractional (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation and space-time fractional (2+1)-dimensional Ablowitz-Kaup-Newell-Segur (AKNS) equation. Here, the fractional derivatives are described in conformable sense. The obtained traveling wave solutions are expressed by the hyperbolic, trigonometric, exponential and rational functions. The graphs for some of these solutions have been presented by choosing suitable values of parameters to visualize the mechanism of the given nonlinear fractional evolution equations.

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.

Around Chaotic Disturbance and Irregularity for Higher Order Traveling Waves

Many unknown features in the theory of wave motion are still captivating the global scientific community. In this paper, we consider a model of seventh order Korteweg–de Vries (KdV) equation with one perturbation level, expressed with the recently introduced derivative with nonsingular kernel, Caputo-Fabrizio derivative (CFFD). Existence and uniqueness of the solution to the model are established and proven to be continuous. The model is solved numerically, to exhibit the shape of related solitary waves and perform some graphical simulations. As expected, the solitary wave solution to the model without higher order perturbation term is shown via its related homoclinic orbit to lie on a curved surface. Unlike models with conventional derivative (γ=1) where regular behaviors are noticed, the wave motions of models with the nonsingular kernel derivative are characterized by irregular behaviors in the pure factional cases (γ<1). Hence, the regularity of a soliton can be perturbed by this nonsingular kernel derivative, which, combined with the perturbation parameter ζ of the seventh order KdV equation, simply causes more accentuated irregularities (close to chaos) due to small irregular deviations.

Stability of solitary and cnoidal traveling wave solutions for a fifth order Korteweg-de Vries equation

We establish the nonlinear stability of solitary waves (solitons) and periodic traveling wave solutions (cnoidal waves) for a Korteweg-de Vries (KdV) equation which includes a fifth order dispersive term. The traveling wave solutions which yield solitons for zero boundary conditions and wave-trains of cnoidal waves for nonzero boundary conditions are analyzed using stability theorems, which rely on the positivity properties of the Fourier transforms. We show that all families of solutions considered here are (orbitally) stable.

Spectral analysis of the stochastic time-fractional-KdV equation

A numerical solution of the stochastic fractional order Korteweg-de Vries equation arising in oceanography and plasma physics is approximated using a novel spectral/finite difference tool. The special case for additive time-dependent noise is discussed by taking into account both the uniform and the Gaussian random inputs. A spectral technique for space and a multi-step finite difference scheme for time are employed. Emphasis is made on the spatial spectral discretization error in conjunction with the stochastic approximation error, on the stability bounds and on the amplitude of the stochastic perturbation.

Shock Waves in Dispersive Hydrodynamics with Nonconvex Dispersion

Dissipationless hydrodynamics regularized by dispersion describe a number of physical media including water waves, nonlinear optics, and Bose-Einstein condensates. As in the classical theory of hyperbolic equations where a non-convex flux leads to non-classical solution structures, a non-convex linear dispersion relation provides an intriguing dispersive hydrodynamic analogue. Here, the fifth order Korteweg-de Vries (KdV) equation, also known as the Kawahara equation, a classical model for shallow water waves, is shown to be a universal model of Eulerian hydrodynamics with higher order dispersive effects. Utilizing asymptotic methods and numerical computations, this work classifies the long-time behavior of solutions for step-like initial data. For convex dispersion, the result is a dispersive shock wave (DSW), qualitatively and quantitatively bearing close resemblance to the KdV DSW. For non-convex dispersion, three distinct dynamic regimes are observed. For small jumps, a perturbed KdV DSW with positive polarity and orientation is generated, accompanied by small amplitude radiation from an embedded solitary wave leading edge, termed a radiating DSW or RDSW. For moderate jumps, a crossover regime is observed with waves propagating forward and backward from the sharp transition region. For jumps exceeding a critical threshold, a new type of DSW is observed we term a translating DSW or TDSW. The TDSW consists of a traveling wave that connects a partial, non-monotonic, negative solitary wave at the trailing edge to an interior nonlinear periodic wave. Its speed, a generalized Rankine-Hugoniot jump condition, is determined by the far-field structure of the traveling wave. The TDSW is resolved at the leading edge by a harmonic wavepacket moving with the linear group velocity. The non-classical TDSW exhibits features common to both dissipative and dispersive shock waves.

Travelling wave solutions of the generalized nonlinear fifth-order KdV water wave equations and its stability

In the present study, by implementing the direct algebraic method, we present the traveling wave solutions for some different kinds of the Korteweg–de Vries (KdV) equations. The exact solutions of the Kawahara, fifth order KdV and generalized fifth order KdV equations are obtained. Solutions for the Kawahara, fifth order KdV and generalized fifth order KdV equations are obtained precisely and efficiency of the method can be demonstrated. The stability of these solutions and the movement role of the waves by making the graphs of the exact solutions are analyzed. All solutions are exact and stable, and have applications in physics.

Exact solutions for the family of third order Korteweg de-Vries equations

In this work we apply an extended hyperbolic function method to solve the nonlinear family of third order Korteweg de-Vries (KdV) equations, namely, the KdV equation, the modified KdV (mKdV) equation, the potential KdV (pKdV) equation, the generalized KdV (gKdV) equation and gKdV with two power nonlinearities equation. New exact travelling wave solutions are obtained for the KdV, mKdV and pKdV equations. The solutions are expressed in terms of hyperbolic functions, trigonometric functions and rational functions. The method used is promising method to solve other nonlinear evaluation equations.

Modified method of simplest equation for obtaining exact analytical solutions of nonlinear partial differential equations: further development of the methodology with applications

We discuss the application of a variant of the method of simplest equation for obtaining exact traveling wave solutions of a class of nonlinear partial differential equations containing polynomial nonlinearities. As simplest equation we use differential equation for a special function that contains as particular cases trigonometric and hyperbolic functions as well as the elliptic function of Weierstrass and Jacobi. We show that for this case the studied class of nonlinear partial differential equations can be reduced to a system of two equations containing polynomials of the unknown functions. This system may be further reduced to a system of nonlinear algebraic equations for the parameters of the solved equation and parameters of the solution. Any nontrivial solution of the last system leads to a traveling wave solution of the solved nonlinear partial differential equation. The methodology is illustrated by obtaining solitary wave solutions for the generalized Korteweg–deVries equation and by obtaining solutions of the higher order Korteweg–deVries equation.

Application of Hyperbola Function Method to the Family of Third Order Korteweg-de Vries Equations

In this work, we apply a hyperbola function method to solve the nonlinear family of third order Korteweg-de Vries equations. Exact travelling wave solutions are obtained and expressed in terms of hyperbolic functions and trigonometric functions. The method used is a promising method to solve other nonlinear evaluation equations.

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.

Discrete Singular Convolution Method for Numerical Solutions of Fifth Order Korteweg-De Vries Equations

A new computational method for solving the fifth order Korteweg-de Vries (fKdV) equation is proposed. The nonlinear partial differential equation is discretized in space using the discrete singular convolution (DSC) scheme and an exponential time integration scheme combined with the best rational approximations based on the Carathéodory-Fejér procedure for time discretization. We check several numerical results of our approach against available analytical solutions. In addition, we computed the conservation laws of the fKdV equation. We find that the DSC approach is a very accurate, efficient and reliable method for solving nonlinear partial differential equations.