Korteweg-de Vries Type Equations Research Articles

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.

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.

Analytical three-periodic solutions of Korteweg–de Vries-type equations

Based on the direct method of calculating the periodic wave solution proposed by Nakamura, we give an approximate analytical three-periodic solutions of Korteweg–de Vries (KdV)-type equations by perturbation method for the first time. Limit methods have been used to establish the asymptotic relationships between the three-periodic solution separately and another three solutions, the soliton solution, the one- and the two-periodic solutions. Furthermore, it is found that the asymptotic three-soliton solution presents the same repulsive phenomenon as the asymptotic three-soliton solution during the interaction.

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.

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>

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.

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.

Propagation and energy of the dressed solitons in the Thomas–Fermi magnetoplasma

A theoretical investigation is presented for dust-acoustic (DA) waves in a collisionless Thomas–Fermi magnetoplasma. The plasma system consists of electrons, ions, and negatively charged dust grains, all existing in a quantizing magnetic field. The Korteweg–de Vries (KdV) and KdV type equations are derived by using the reductive perturbation method. The solutions of these evolved equations are obtained. The contribution of higher-order corrections to the DA is investigated. The electric field and the soliton energy were also derived. The K-dV and dressed soliton energies are depleted as the dust temperature and magnetic field increase. But they magnify as obliqueness increases. The present results are beneficial in understanding the waves propagating in Thomas–Fermi magnetoplasma that are applicable for high-intensity laser–solid matter interaction experiments and astrophysical compact objects such as white dwarfs.

Dust-acoustic solitary and periodic waves in a plasma with ion distribution with trapped particles

The nonlinear propagation of electrostatic dust-acoustic waves is studied in an unmagnetized three-component dusty plasma consisting of negatively charged dust, free as well as a small percentage of trapped ions and Maxwellian electrons. By using the reductive perturbation technique, a Korteweg de Vries type equation, which governs the dynamics of the small-amplitude solitary and periodic waves of the dusty plasma under consideration, is derived. The traveling soliton (shown to be only of the rarefactive type) and periodic solutions of the KdV type equation are obtained. Bifurcation analysis for the obtained nonlinear KdV type equation is used to describe the qualitative phase portrait for the corresponding dynamical system, which is found to be controlled by the plasma system parameters. The effects of trapped ions and free electrons on the dust-acoustic solitons and periodic waves are discussed. The present investigation may be useful in understanding some nonlinear features of the dust-acoustic waves, which have been observed in astrophysical plasmas such as Saturn's moon Enceladus.

Meshless method of approximate particular solution for an initial and boundary value problem of the Korteweg–de Vries type equation and eventual periodicity

In this work, we approximate the solution of an IBVP of the Korteweg–de Vries type equation with periodic forcing at the boundary using the meshless method of approximate particular solution (MMAPS). This method is based on radial basis functions (RBFs). Further, it is shown that the solution exhibits the eventual periodicity and method approximates the solution accurately and efficiently. To test the accuracy of the method the conserved quantities are also computed for the KdV equation. The accuracy of the method depends on the shape parameter and the optimal value of the shape parameter is also approximated.

Spatial Analyticity of Solutions to Korteweg–de Vries Type Equations

The Korteweg–de Vries equation (KdV) is a mathematical model of waves on shallow water surfaces. It is given as third-order nonlinear partial differential equation and plays a very important role in the theory of nonlinear waves. It was obtained by Boussinesq in 1877, and a detailed analysis was performed by Korteweg and de Vries in 1895. In this article, by using multi-linear estimates in Bourgain type spaces, we prove the local well-posedness of the initial value problem associated with the Korteweg–de Vries equations. The solution is established online for analytic initial data w0 that can be extended as holomorphic functions in a strip around the x-axis. A procedure for constructing a global solution is proposed, which improves upon earlier results.

Dispersive Behaviours in Complex Structures

Abstract In the framework of the Scale Relativity Theory, a fractal Korteweg-de Vries-type equation for describing dispersive behaviours of any complex structure is obtained. In the one-dimensional case, the general solution of this equation show that the dispersive behaviours of any complex structure are given by means of cnoidal oscillation modes. Through these modes degeneration, dispersive behaviours of soliton, soliton-package, harmonic etc. result.

Nonlinear excitations of magnetosonic solitary waves and their chaotic behavior in spin-polarized degenerate quantum magnetoplasma.

The quantum hydrodynamic model is used to study the nonlinear propagation of small amplitude magnetosonic solitons and their chaotic motions in quantum plasma with degenerate inertialess spin-up electrons, spin-down electrons, and classical inertial ions. Spin effects are considered via spin pressure and macroscopic spin magnetization current, whereas the exchange effects are considered via adiabatic local density approximation. By applying the reductive perturbation method, the Korteweg-de Vries type equation is derived for small amplitude magnetosonic solitary waves. We present the numerical predictions about the conservative system's total energy in spin-polarized and usual electron-ion plasma and observed low energy in spin-polarized plasma. We also observe numerically that the soliton characteristics are significantly affected by different plasma parameters such as soliton phase velocity increases by increasing quantum statistics, magnetization energy, exchange effects, and spin polarization density ratio. Moreover, it is independent of the quantum diffraction effects. We have analyzed the dynamic system numerically and found that the magnetosonic solitary wave amplitude and width are getting larger as the quantum statistics and spin magnetization energy increase, whereas their amplitude and width decrease with increasing spin concentration. The wave width increases for high values of quantum statistic and exchange effects, while their amplitude remains constant. Most importantly, in the presence of external periodic perturbations, the periodic solitonic behavior is transformed to quasiperiodic and chaotic oscillations. It is found that a weakly chaotic system is transformed to heavy chaos by a small variation in plasma parameters of the perturbed spin magnetosonic solitary waves. The work presented is related to studying collective phenomena related to magnetosonic solitary waves, vital in dense astrophysical environments such as pulsar magnetosphere and neutron stars.

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.

Excitation of electrostatic plasma waves by a moving charged source in a quantum plasma

Within a quantum hydrodynamic model and using the reductive perturbation technique, the nonlinear ion-acoustic wave (IAW) excitations due to a moving charged object in an electron-pair-ion quantum plasma are studied both analytically and numerically. In such quantum plasmas we have derived forced Korteweg-de Vries (fKdV) type equation for finite amplitude nonlinear IAWs. The effect of relevant plasma parameters on solitonic excitations is investigated. Numerical simulation shows the generation of advancing solitons ahead of the forcing term traveling at a faster rate with trailing wakes behind the forcing disturbance. It is found that propagation characteristics of nonlinear excitations are significantly affected by quantum parameter. Additionally, we have pursued our analysis by extending it to account for arbitrary amplitude IA solitons, and derived a system of nonlinear differential equations which are analyzed numerically to study the dynamics. Nonlinear analysis predicts the existence of periodic and quasiperiodic nature of the nonlinear system and reveals that the transition from quasiperiodic to periodic behavior occurs due to the variation of quantum diffraction.