Korteweg-de Vries Type Research Articles

Integration of the Korteweg–de Vries type equations with a loaded term in the class of periodic functions

In this paper, we study the integrability of a Korteweg–de Vries type equation with a loaded term in the class of periodic functions using direct and inverse spectral problems posed for the Sturm–Liouville operator on the entire axis. We present some information about the Sturm–Liouville operator with a periodic coefficient and its application to solving a loaded Korteweg–de Vries type equation. We show that the function constructed using the obtained system of Dubrovin differential equations and trace formulas is a solution to the problem posed. We present important consequences about the period of the solution with respect to $x$ and about the analyticity of the solution with respect to $x$ for a Korteweg–de Vries type equation with a loaded term.

Impact of relativistic positron beam on ion-acoustic solitary, periodic and breather waves in Earths’ ionospheric region through the framework of KdV and modified KdV equation

Abstract This study explores the propagation characteristics of ion-acoustic periodic, soliton, and breather waves in electron-positron-ion (EPI) plasma with a relativistic positron beam. The Korteweg–de Vries (KdV) equation is obtained by applying the traditional reductive perturbation method (RPM) to the fundamental set of fluid equations. When the KdV model is unable to accurately represent the nonlinear system’s evolution, a modified Korteweg–de Vries (mKdV) equation is constructed. In both models, Jacobi elliptic functions are used to derive periodic solutions, and a connection between periodic waves and soliton solutions is established. Hirota’s bilinear method is used to generate breathers directly from the KdV type framework without utilizing the modified Schrödinger framework inferred from the KdV type framework, which is a prevalent method in studies of nonlinear waves. Numerical knowledge of various physical factors in the ionospheric region is incorporated into the model to elucidate wave propagation in the Earth’s upper atmosphere.

Derivation of Sawada-Kotera and Kaup-Kupershmidt Equations KdV Flow Equations from Modified Nonlinear Schrödinger Equation (MNLS)ns from Modfied Nonlinear Schrödinger Equation (MNLS)

Mathematical models of problems that arise in almost every branch of science are nonlinear evolution equations (NLEE). As a result, nonlinear evolution equations have served as a language for formulating many engineering and scientific problems. For this reason, many different and effective techniques have been developed regarding nonlinear evolution equations and solution methods. The main reason for this situation is that nonlinear evolution equations involve the problem of nonlinear wave propagation. In this study, (1 + 1) dimensional fifth-order nonlinear Korteweg-de Vries (fKdV) type equations were obtained by applying the multi-scale method known as the perturbation method for the modified nonlinear Schrödinger (MNLS) equation. Thus, we showed the relationship between KdV equations and MNLS type equations.

A space-time generalized finite difference scheme for long wave propagation based on high-order Korteweg-de Vries type equations

In this paper, the space-time generalized finite difference scheme is applied to solve the nonlinear high-order Korteweg-de Vries equations in multiple dimensions. The proposed numerical scheme combines the space-time generalized finite difference method, the Levenberg-Marquardt algorithm, and a time-marching approach. The space-time generalized finite difference method treats the temporal axis as a spatial axis, enabling the proposed scheme to discretize all derivatives in the governing equation. This is accomplished through Taylor series expansion and the moving least squares method. Due to the expandability of the Taylor series to any order, the proposed numerical scheme excels in efficiently handling mixed and higher-order derivatives. These capabilities are distinct advantages of the proposed scheme. The resulting system of algebraic equations is sparse but overdetermined. Therefore, the Levenberg-Marquardt algorithm is directly applied to solve this nonlinear algebraic system. During the calculation process, the time-marching approach reduces computational effort and improves efficiency by dividing the space-time domain.

Exact solutions of stochastic Burgers–Korteweg de Vries type equation with variable coefficients

We will present exact solutions for three variations of the stochastic Korteweg de Vries–Burgers (KdV–Burgers) equation featuring variable coefficients. In each variant, white noise exhibits spatial uniformity, and the three categories include additive, multiplicative, and advection noise. Across all cases, the coefficients are time-dependent functions. Our discovery indicates that solving certain deterministic counterparts of KdV–Burgers equations and composing the solution with a solution of stochastic differential equations leads to the exact solution of the stochastic Korteweg de Vries–Burgers (KdV–Burgers) equations.

Stabilization results for delayed fifth-order KdV-type equation in a bounded domain

Studied here is the Kawahara equation, a fifth-order Korteweg-de Vries type equation, with time-delayed internal feedback. Under suitable assumptions on the time delay coefficients, we prove that the solutions of this system are exponentially stable. First, considering a damping and delayed system, with some restriction of the spatial length of the domain, we prove that the energy of the Kawahara system goes to $ 0 $ exponentially as $ t \rightarrow \infty $. After that, by introducing a more general delayed system, and by introducing suitable energies, we show using the Lyapunov approach, that the energy of the Kawahara equation goes to zero exponentially, considering the initial data small and a restriction in the spatial length of the domain. To remove these hypotheses, we use the compactness-uniqueness argument which reduces our problem to prove an observability inequality, showing a semi-global stabilization result.

On the study the radius of analyticity for Korteweg-de-Vries type systems with a weakly damping

<p>In the present paper, we considered a Korteweg-de Vries type system with weakly damping terms and initial data in the analytic Gevery spaces. The presence of tow functions $ c_1(x), c_2(x) $, called damping coefficients, made the system more interesting from an application point of view due to their great importance in physics. To start, by using the fixed point theorem in Banach space, we investigated the local well-posedness. Additionally, by employing an approximate conservation law, we extended this to be global in time, ensuring that the radius of analyticity of solutions remained uniformly bounded below by a fixed positive number for all time.</p>

Insensitizing control problem for the Hirota–Satsuma system of KdV–KdV type

This paper is concerned with the existence of insensitizing controls for a nonlinear coupled system of two Korteweg–de Vries (KdV) equations, typically known as the Hirota–Satsuma system. The idea is to look for controls such that some functional of the states (the so-called sentinel) is insensitive to the small perturbations of initial data. Since the system is coupled, we consider a sentinel in which we observe both components of the system in a localized observation set. By some classical argument, the insensitizing problem is then reduced to a null-control problem for an extended system where the number of equations is doubled. We study the null-controllability for the linearized model associated to that extended system by means of a suitable Carleman estimate which is proved in this paper. Finally, the local null-controllability of the extended (nonlinear) system is obtained by applying the inverse mapping theorem, and this implies the required insensitizing property for the concerned model.

Differences between two methods to derive a nonlinear Schrödinger equation and their application scopes

The present paper chooses a dusty plasma as an example to numerically and analytically study the differences between two different methods of obtaining nonlinear Schrödinger equation (NLSE). The first method is to derive a Korteweg–de Vries (KdV)-type equation and then derive the NLSE from the KdV-type equation, while the second one is to directly derive the NLSE from the original equation. It is found that the envelope waves from the two methods have different dispersion relations, different group velocities. The results indicate that two envelope wave solutions from two different methods are completely different. The results also show that the application scope of the envelope wave obtained from the second method is wider than that of the first one, though both methods are valuable in the range of their corresponding application scopes. It is suggested that, for other systems, both methods to derive NLSE may be correct, but their nonlinear wave solutions are different and their application scopes are also different.

Four-component integrable hierarchies and their Hamiltonian structures

We aim to construct four-component integrable hierarchies from a kind of matrix spectral problems within the zero curvature formulation. The Liouville integrability of the resulting hierarchies are guaranteed through establishing Hamiltonian structures by the trace identity. Illustrative examples include novel four-component nonlinear Schrödinger type equations and modified Korteweg–de Vries type equations.

On well-posedness for some Korteweg–de Vries type equations with variable coefficients

On well-posedness for some Korteweg–de Vries type equations with variable coefficients

AKNS type reduced integrable bi-Hamiltonian hierarchies with four potentials

The aim of this paper is to derive a kind of integrable hierarchies with four potentials, which possess bi-Hamiltonian structures, from reduced AKNS matrix spectral problems. The associated recursion operators are worked out explicitly. The Lax pair formulation and the trace identity are basic tools in the analysis. Two nonlinear examples in the resulting integrable hierarchies are integrable nonlinear Schrödinger type equations and integrable modified Korteweg–de Vries type equations.

Decay of solitary waves of fractional Korteweg-de Vries type equations

We study the solitary waves of fractional Korteweg-de Vries type equations, that are related to the 1-dimensional semi-linear fractional equations:|D|αu+u−f(u)=0, with α∈(0,2), a prescribed coefficient p⁎(α), and a non-linearity f(u)=|u|p−1u for p∈(1,p⁎(α)), or f(u)=up with an integer p∈[2;p⁎(α)). Asymptotic developments of order 1 at infinity of solutions are given, as well as second order developments for positive solutions, in terms of the coefficient of dispersion α and of the non-linearity p. The main tools are the kernel formulation introduced by Bona and Li, and an accurate description of the kernel by complex analysis theory.

A compact finite difference scheme with absorbing boundary condition for forced KdV equation

Studying the long-time solution behavior of the Korteweg-de Vries (KdV) type equation with a periodic force acting at one end of the long channel is important for simulating the blood flow in artery driven by heart pulses. It is of great interest to develop an accurate numerical method for solving the forced KdV problem. In this article, we present the following methods to obtain an accurate approximation to the solution of KdV problem.•An accurate compact finite difference scheme is proposed for solving the above forced KdV problem with fourth-order accuracy.•An absorbing boundary condition at the right end of the interval is used to avoid the wave reflection.•The stability of scheme is proved by the von Neumann method and then tested by three examples. Results show that the method provides an accurate solution, and the wave propagates without reflection.

Global control aspects for long waves in nonlinear dispersive media

A class of models of long waves in dispersive media with coupled quadratic nonlinearities on a periodic domain T are studied. We used two distributed controls, supported in ω ⊂ T and assumed to be generated by a linear feedback law conserving the “mass” (or “volume”), to prove global control results. The first result, using spectral analysis, guarantees that the system in consideration is locally controllable in Hs(T), for s ≥ 0. After that, by certain properties of Bourgain spaces, we show a property of global exponential stability. This property together with the local exact controllability ensures for the first time in the literature that long waves in nonlinear dispersive media are globally exactly controllable in large time. Precisely, our analysis relies strongly on the bilinear estimates using the Fourier restriction spaces in two different dispersions that will guarantee a global control result for coupled systems of the Korteweg—de Vries type. This result, of independent interest in the area of control of coupled dispersive systems, provides a necessary first step for the study of global control properties to the coupled dispersive systems in periodic domains.

Symmetry analysis and conservation laws of time fractional Airy type and other KdV type equations

<abstract><p>We study the invariance properties of the fractional time version of the nonlinear class of equations $ u_{t}^{\alpha}-g(u)\; u_{x}-f(u)\; u_{xxx} = 0 $, where $ 0 < \alpha < 1 $ using some recently developed symmetry-based techniques. The equations reduce to ordinary fractional Airy type, Korteweg-de Vries (KdV) and modified KdV equations through the change of variables provided by the symmetries. Furthermore, we utilize the symmetries to construct conservation laws for the fractional partial differential equations.</p></abstract>

Integrable reductions of a soliton hierarchy associated with so(3,R)

We construct integrable reductions of a soliton hierarchy associated with the special orthogonal Lie algebra so ( 3 , R ) . The resulting reduced integrable equations include a nonlinear Schrödinger type equation and a modified Korteweg–de Vries type equation. There are two kinds of integrable reductions in our analysis, but they lead to essentially the same scalar integrable equations. This is a particular phenomenon for soliton equations associated with so ( 3 , R ) , which is different from the one for soliton equations associated with sl ( 2 , R ) . • Scalar integrable equations associated with so(3,R). • Innovative reductions, which keep the existence of infinitely many symmetries and conservation laws.

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.

Dissipative solitons in magnetized anisotropic plasma

The propagation of ion-acoustic (IA) dissipative solitons (DSs) is investigated in magnetized plasma with the presence of pressure anisotropy and Maxwellian distributed electrons. A nonlinear damped Korteweg-de Vries (d-KdV) type equation is derived for the potential by employing the reductive perturbation method (RPM) and its dissipative soliton solution is analyzed. The effect of plasma parameters like parallel anisotropic ion pressure p 1, perpendicular anisotropic ion pressure p 2, obliqueness l 3, ion-neutral collisional parameter γ and magnetic field strength Ω on the characteristics of DSs are investigated. The present investigation could be useful in space and astrophysical plasma systems.

Bilinear form and soliton solutions for a higher order wave equation

Under investigation in this paper is a higher order wave equation of the Korteweg–de Vries type, which describes the propagation of more complex wave in the shallow water. Bilinear form is derived based on the binary Bell polynomials. One-soliton solutions are constructed via the Hirota method. Two-soliton solutions are tried to be constructed, but it degenerates to the one-soliton solutions. Influences of the coefficients α and β on the soliton velocity and amplitude are analyzed through the characteristic line.