Order Korteweg-de Vries Research Articles

The Nonlinear Schrödinger Equation Derived from the Third Order Korteweg-de Vries Equation Using Multiple Scales Method

The Nonlinear Schrödinger Equation Derived from the Third Order Korteweg-de Vries Equation Using Multiple Scales Method

A gradient reproducing kernel based stabilized collocation method for the 5th order Korteweg–de Vries equations

This paper presents a gradient reproducing kernel based stabilized collocation method (GRK-SCM) to solve the generalized nonlinear 5th order Korteweg–de Vries (KdV) equations. By introducing gradient reproducing kernel (GRK) approximations, one can circumvent the intricacy of high-order derivatives in RK approximations, while still satisfying high-order consistency requirements. This leads to the high accuracy and efficiency. GRKs are very good approximation candidates for addressing high order partial differential equations that require higher order derivatives formulated as strong form. The stabilized collocation method (SCM) effectively meets high-order integration constraints which can achieve accurate integration in the subdomains. This ensures both high precision and optimal convergence. Von Neumann analysis is utilized to establish the stability criteria for GRK-SCM when combined with forward difference temporal discretization. Numerical solutions for Sawada–Kotera (SK) equation and Kaup-Kupershmidt (KK) equation are studied where the solitary wave migration and collision, and periodic waves are represented. A fifth order forced KdV equation is also considered. The presented method's high accuracy and convergence are demonstrated through comparative studies with analytical solutions, and investigations of invariants and corresponding errors determine the good conservation properties of this algorithm.

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.

Integration of the negative order Korteweg-de Vries equation by the inverse scattering method

In this paper, we consider the integration of the negative order Korteweg-de Vries equation by the inverse scattering method. The evolution of the spectral data of the Sturm-Liouville operator with a potential associated with the solution of the negative order Korteweg-de Vries equation is determined. The obtained results make it possible to apply the method of inverse scattering problem to solve the negative order Korteweg-de Vries equation in the class of rapidly decreasing functions.

Higher order corrections to beyond-all-order effects in a fifth order Korteweg–de Vries equation

Higher order corrections to beyond-all-order effects in a fifth order Korteweg–de Vries equation

Integration of the negative order Korteweg-de Vries equation with self-consistent source

Integration of the negative order Korteweg-de Vries equation with self-consistent source

Unique Continuation and Time Decay for a Higher-Order Water Wave Model

This work is devoted to prove the exponential decay for the energy of solutions of a higher order Korteweg–de Vries (KdV)–Benjamin–Bona–Mahony (BBM) equation on a periodic domain with a localized damping mechanism. Following the method in [L. Rosier and B.-Y. Zhang, J. Diff. Equ. 254 (2013) 141–178], which combines energy estimates, multipliers and compactness arguments, the problem is reduced to prove the Unique Continuation Property (UCP) for weak solutions of the model. Then, this is done by deriving Carleman estimates for a system of coupled elliptic-hyperbolic equations.

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.

Интегрирование уравнения Кортевега – де Фриза отрицательного порядка с источником специального вида

In this paper, we consider the negative order Korteweg-de Vries equation with a self-consistent source corresponding to the eigenvalues of the corresponding spectral problem. It is shown that the considered equation can be integrated by the method of the inverse spectral problem. The evolution of the spectral data of the Sturm-Liouville operator with a periodic potential associated with the solution of the considered equation is determined. The results obtained make it possible to apply the inverse problem method for solving the negative order Korteweg-de Vries equation with a self-consistent source corresponding to the eigenvalues of the corresponding spectral problem.

Cell-average based neural network method for third order and fifth order KdV type equations

In this paper, we develop the cell-average based neural network (CANN) method to solve third order and fifth order Korteweg-de Vries (KdV) type equations. The CANN method is based on the weak or integral formulation of the partial differential equations. A simple feedforward network is forced to learn the cell average difference between two consecutive time steps. One solution trajectory corresponding to a generic initial value is used to generate the data set to train the network parameters, which uniquely determine a one-step explicit finite volume based network method. Once well-trained, the CANN method can be generalized to a suitable family of initial value problems. Comparing with conventional explicit methods, where the time step size is restricted as Δt = O(Δx3) or Δt = O(Δx5), the CANN method is able to evolve the solution forward accurately with a much larger time step size of Δt = O(Δx). A large group of numerical tests are carried out to verify the effectiveness, stability and accuracy of the CANN method. Wave propagation is well resolved with indistinguishable dispersion and dissipation errors. The CANN approximations agree well with the exact solution for long time simulation.

Wave solutions to an integrable negative order KdV equation

In this paper, we study the traveling wave solutions to the second representative equation of the negative-order Korteweg–de Vries hierarchy. The corresponding traveling wave system is a singular planar dynamical system with a singular straight line. By using the method of dynamical systems, explicit solutions are given. But this integrable negative order Korteweg–de Vries flow has no traveling solitary wave solution.

The lie symmetry analysis of third order Korteweg-de Vries equation

This study sought to analyse the Lie symmetry of third order Korteweg-de Vries equation. Solving nonlinear partial differential equations is of great importance in the world of dynamics. Korteweg-de Vries equations are partial differential equations arising from the theory of long waves, modelling of shallow water waves, fluid mechanics, plasma fluids and many other nonlinear physical systems, and their effects are relevant in real life. In this study, Lie symmetry analysis is demonstrated in finding the symmetry solutions of the third-order KdV equation of the form. The study systematically showed the formula to find the specific solution attained by developing prolongations, infinitesimal transformations and generators, adjoint symmetries, variation symmetries, invariant transformation and integrating factors to obtain all the lie groups presented by the equation. In conclusion, infinitesimal generators, group transformations and symmetry solutions of third-order KdV equation are acquired using a method of Lie symmetry analysis. This was achieved by generating infinitesimal generators which act on the KdV equation to form infinitesimal transformations. It can be seen from the solutions of this paper that the Lie symmetry analysis method is an effective and best mathematical technique for studying linear and nonlinear PDEs and ODEs.

Integration of the negative order Korteweg–de Vries equation with a self-consistent source in the class of periodic functions

In this paper, we consider the negative order Korteweg–de Vries equation with a self-consistent integral source. It is shown that the negative-order Korteweg–de Vries equation with a self-consistent integral source can be integrated by the method of the inverse spectral problem. The evolution of the spectral data of the Sturm–Liouville operator with a periodic potential associated with the solution of the negative order Korteweg–de Vries equation with a self-consistent integral source is determined. The obtained results make it possible to apply the inverse problem method to solve the negative order Korteweg–de Vries equation with a self-consistent source in the class of periodic functions.

New soliton solutions for the space-time fractional modified third order Korteweg–de Vries equation

In present research work, the soliton solutions of the space-time fractional modified third order Korteweg–de Vries (KdV) equation are explored by Sardar-subequation method (SSM). The (KdV) equation has an established version based on the shallow water waves in the oceans, oceanography theory and the ion-acoustic waves in plasma. The fractional order partial differential equation (PDE) is transform to a non-linear ordinary differential equation (ODE) by using traveling wave transformation. It is noted that in this work the fractional differential equation is solved in the sense of conformable derivative. Then soliton solutions of reduced equation are established by SSM. The present technique provides bright, dark, singular, periodic singular, combined bright-dark and combined dark-singular soliton solutions.

New generalized fuzzy transform computations for solving fractional partial differential equations arising in oceanography

This paper presents a study of nonlinear waves in shallow water. The Korteweg-de Vries (KdV) equation has a canonical version based on oceanography theory, the shallow water waves in the oceans, and the internal ion-acoustic waves in plasma. Indeed, the main goal of this investigation is to employ a semi-analytical method based on the homotopy perturbation transform method (HPTM) to obtain the numerical findings of nonlinear dispersive and fifth order KdV models for investigating the behaviour of magneto-acoustic waves in plasma via fuzziness. This approach is connected with the fuzzy generalized integral transform and HPTM. Besides that, two novel results for fuzzy generalized integral transformation concerning fuzzy partial gH-derivatives are presented. Several illustrative examples are illustrated to show the effectiveness and supremacy of the proposed method. Furthermore, 2D and 3D simulations depict the comparison analysis between two fractional derivative operators (Caputo and Atangana-Baleanu fractional derivative operators in the Caputo sense) under generalized gH-differentiability. The projected method (GHPTM) demonstrates a diverse spectrum of applications for dealing with nonlinear wave equations in scientific domains. The current work, as a novel use of GHPTM, demonstrates some key differences from existing similar methods.

On the fractal dynamics for higher order traveling waves

Auto replication processes remain fascinating in sciences, engineering and technology as their applications in machining/biological systems have been widely used to solve number of outstanding problems in communities’ every day lives. Finding innovative techniques capable of generating auto replication processes in various fields has then become the priority for number of scientists. One of those fields includes wave motion. In this paper, we use the 7th order Korteweg–de Vries (KdV) model, combined with the fractal-fractional operator to artificially (numerically) generate auto replication processes characterizing the evolution of higher order traveling waves. The well-posedness for the combined model is first studied with the establishment of its existence and uniqueness results. Numerical simulations then follow and prove that the higher order traveling wave can be involved in a self replication process. There is generation of the exact or approximately exact copies of the initial traveling wave in different scales and where the fractal process produces other multiple traveling waves that look like the preceding ones. Furthermore, the fractal dynamics expand as the model’s parameter γ changes. We are on the right track for the artificially-formed fractal process of higher order traveling waves applicable in wave motion’s domain.

Well-posedness and controllability of Kawahara equation in weighted Sobolev spaces

We consider the Kawahara equation, a fifth order Korteweg–de Vries type equation, posed on a bounded interval. The first result of the article is related to the well-posedness in weighted Sobolev spaces, which one was shown using a general version of the Lax–Milgram Theorem. With respect to the control problems, we will prove two results. First, if the control region is a neighborhood of the right endpoint, an exact controllability result in weighted Sobolev spaces is established. Lastly, we show that the Kawahara equation is controllable by regions on L2 Sobolev space, the so-called regional controllability, that is, the state function is exact controlled on the left part of the complement of the control region and null controlled on the right part of the complement of the control region.

Double Elzaki Transform Decomposition Method for Solving Third Order Korteweg-De-Vries Equations

In this study, we used Double Elzaki Transform (DET) coupled with Adomian polynomial to produce a new method to solve Third Order Korteweg-De Vries Equations (KdV) equations. We will provide the necessary explanation for this method with addition some examples to demonstrate the effectiveness of this method.

New algorithm for the approximate solution of generalized seventh order Korteweg-Devries equation arising in shallow water waves

In the present article, we present for the first time optimal auxiliary function method (OAFM) for partial differential equation (PDEs). To find efficient and precision the proposed method, we take Lax’s seventh order korteweg-de Vries (KdV) and seventh order Sawada Kotera (SK) equations as test examples. The beauty of the planned method lies in auxiliary functions Ai and some parameters Ci which ensure a very rapid convergence of the solution. We compare the approximate solutions got by the proposed method with the homotopy perturbation method (HPM), the Optimal Homotopy asymptotic method. It should be emphasized that very good approximation is obtained at the first iteration. It has been shown, that OAFM is a simple and convergent method for the solution of nonlinear equations. The numerical results rendering that the applied method is explicit, efficacious and facile to utilize, for handling more general nonlinear equations.

Well-posedness of a higher dispersion KdV equation on the half-line

The initial-boundary value problem (ibvp) for the mth order Korteweg–de Vries (KdVm) equation on the half-line is studied by extending a novel approach recently developed for the well-posedness of KdV on the half-line, which is based on the solution formula produced via the Fokas unified transform method for the associated forced linear ibvp. Replacing in this formula the forcing by the nonlinearity and using data in Sobolev spaces suggested by the space–time regularity of the Cauchy problem of the linear KdVm equation give an iteration map for the ibvp, which is shown to be a contraction in an appropriately chosen solution space. The proof relies on key linear estimates and a bilinear estimate similar to the one used for the KdV equation on the line by Kenig, Ponce, and Vega [J. Am. Math. Soc. 4, 323–347 (1991)].