Generalized Korteweg-de Vries Research Articles

Convergence rate toward shock wave under periodic perturbation for generalized Korteweg–de Vries–Burgers equation

In this paper, a viscous shock wave under space-periodic perturbation of generalized Korteweg–de Vries–Burgers equation is investigated. It is shown that if the initial periodic perturbation around the viscous shock wave is small, then the solution time asymptotically tends to a viscous shock wave with a shift partially determined by the periodic oscillations. Moreover the exponential time decay rate toward the viscous shock wave is also obtained for some certain perturbations.

Linear asymptotic stability of small-amplitude periodic waves of the generalized Korteweg–de Vries equations

We extend the detailed study of the linearized dynamics obtained for cnoidal waves of the Korteweg–de Vries equation by Rodrigues [J. Funct. Anal. 274 (2018), pp. 2553–2605] to small-amplitude periodic traveling waves of the generalized Korteweg–de Vries equations that are not subject to Benjamin–Feir instability. With the adapted notion of stability, this provides for such waves, global-in-time bounded stability in any Sobolev space, and asymptotic stability of dispersive type. When doing so, we actually prove that such results also hold for waves of arbitrary amplitude satisfying a form of spectral stability designated here as dispersive spectral stability.

Lie symmetry scheme to the generalized Korteweg–de Vries equation with Riemann–Liouville fractional derivative

The Korteweg–de Vries (KdV) equation is an essential model to characterize shallow water waves in fluid mechanics. Here, we investigated the generalized time and time-space fractional KdV equation with fractional derivative of Riemann–Liouville. At the beginning of, we applied the fractional Lie symmetry scheme to derive their symmetry, respectively. We found that the vector fields of these considered equations decrease as the independent variables fractionalize. Subsequently, the one-parameter Lie transformation groups of these concerned models were yielded. At the same time, they can be reduced into fractional order ordinary differential equations with the Erdélyi–Kober fractional operators. Finally, by obtaining the nonlinear self-adjointness, conservation laws of the generalized time-space fractional KdV equation were also found. These good results provide a basis for us to further understand the phenomenon of shallow water waves.

A semilinear non-homogeneous problem related to Korteweg-de Vries Equation

In this paper, we consider a non-homogeneous generalized Korteweg-de Vries problem with some hypotheses on the right-hand side, and we give a new regularity result of the solution in an anisotropic Sobolev space. Then we apply the obtained result to a non-homogeneous KdV problem. This work is an extension of solvability results for a right-hand side f in Lebesgue space.

Finite point blowup for the critical generalized Korteweg-de Vries equation

In the last twenty years, there have been significant advances in the study of the blow-up phenomenon for the critical generalized Korteweg-de Vries equation, including the determination of sufficient conditions for blowup, the stability of blowup in a refined topology and the classification of minimal mass blowup. Exotic blow-up solutions with a continuum of blow-up rates and multi-point blow-up solutions were also constructed. However, all these results, as well as numerical simulations, involve the bubbling of a solitary wave going at infinity at the blow-up time, which means that the blow-up dynamics and the residue are eventually uncoupled. Even at the formal level, there was no indication whether blowup at a finite point could occur for this equation. In this article, we answer this question by constructing solutions that blow up in finite time under the form of a single-bubble concentrating the ground state at a finite point with an unforeseen blow-up rate. Finding a blow-up rate intermediate between the self-similar rate and other rates previously known also reopens the question of which blow-up rates are actually possible for this equation.

Comparison of different discontinuous Galerkin methods based on various reformulations for gKdV equation: Soliton dynamics and blowup

In this work, we explore the use of several discontinuous Galerkin (DG) methods for simulating soliton dynamics in the generalized Korteweg-de Vries (gKdV) equation. The presence of high-order nonlinearity in this model can lead to the finite-time blowup phenomenon, which challenges every aspect of a numerical scheme. The considered DG schemes encompass popular methods derived from the energy and/or Hamiltonian conservations of the gKdV equation. To evaluate the performance of these DG schemes from various angles, we present a series of numerical experiments. Through comprehensive comparisons, we aim to identify the scheme that exhibits the best performance. To further enhance the accuracy and efficiency of DG methods, particularly in the context of blowup simulations, we incorporate the arbitrary Lagrangian-Eulerian method for adaptive mesh movement.

Derivation of anomalously interacting lumps for the (2+1)-dimensional generalized Korteweg–de Vries equation via degeneracy of lump chains

Derivation of anomalously interacting lumps for the (2+1)-dimensional generalized Korteweg–de Vries equation via degeneracy of lump chains

Fifth-order evolution equation for a liquid film with odd viscosity

A fifth-order nonlinear wave equation (generalized Korteweg–deVries equation) that arises in the theory of thin liquid films is introduced with the odd viscosity by the long-wave approximation regime. The presence of the odd viscosity is considered to be a non-dissipative factor in the evolution of the film flow. Sinusoidal and travelling waves arise in systems with non dissipation and instability. The introduced wave equation includes quadratic and cubic nonlinearities. The linear instability of sinusoidal waves is analyzed. The stability and bifurcation of travelling waves are discussed.

Improved algebraic lower bound for the radius of spatial analyticity for the generalized KdV equation

We consider the initial value problem (IVP) for the generalized Korteweg–de Vries (gKdV) equation ∂tu+∂x3u+μuk∂xu=0,x∈R,t∈R,u(x,0)=u0(x),where u(x,t) is a real valued function, u0(x) is a real analytic function, μ=±1 and k≥4. We prove that if the initial data u0 has radius of analyticity σ0, then there exists T0>0 such that the radius of spatial analyticity of the solution remains the same in the time interval [−T0,T0]. In the defocusing case, for k∈2N, k≥4, we prove that when the local solution extends globally in time, then for any T≥T0, the radius of analyticity cannot decay faster than cT−2kk+4+ϵ, ϵ>0 arbitrarily small and c>0 a constant. The result of this work improves the one by Bona et al. (2005) where the authors proved the decay rate is no faster than cT−(k2+3k+2).

Propagation of generalized Korteweg-de Vries solitons along large-scale waves.

We consider propagation of solitons along large-scale background waves in the generalized Korteweg-de Vries (gKdV) equationtheory when the width of the soliton is much smaller than the characteristic size of the background wave. Due to this difference in scales, the soliton's motion does not affect the dispersionless evolution of the background wave. We obtained the Hamilton equationsfor soliton's motion and derived simple relationships which express the soliton's velocity in terms of a local value of the background wave. Solitons' paths obtained by integration of these relationships agree very well with the exact numerical solutions of the gKdV equation.

A note on the quartic generalized Korteweg–de Vries equation in weighted Sobolev spaces

In this paper we establish the persistence property for solutions of the quartic generalized Korteweg–de Vries equation with initial data in weighted Sobolev spaces Hs(R)∩L2(|x|2rdx) for s=1/12+ɛ and any r∈(0,R), for some 0<ɛ<1/4 and 0<R<s/2.

Exploration of solitary wave solutions of highly nonlinear KDV–KP equation arise in water wave and stability analysis

For intellectual curiosity, in this manuscript, we study the sixth-order highly nonlinear generalized Korteweg–de Vries–Kadomtsev–Petviashvili dynamical model. The Kadomtsev–Petviashvili equation is used in conjunction with the popular fifth-order KDV model. The objective of this study is twofold: Firstly, we formulated the two analytical methods, namely, the unified method and the auxiliary method, to derive the differential families of soliton solutions under different conditions that have never been studied before. The scientific community interested in wave phenomena will find these results useful in understanding the dynamics of nonlinear phenomena in wave theory, optics, engineering, and plasma physics. Secondly, we examine the stability analysis of the selected model. For better understanding, we plotted 2D, 3D, contours and density graphs of these acquired solutions. We anticipate that this work will be a significant step forward in the analysis of highly nonlinear models.

Structures of Classical and Special Discontinuities for the Generalized Korteweg–de Vries–Burgers Equation in the Case of a Flux Function with Four Inflection Points

Structures of Classical and Special Discontinuities for the Generalized Korteweg–de Vries–Burgers Equation in the Case of a Flux Function with Four Inflection Points

Classical Solutions for the Generalized Korteweg-de Vries Equation

The Korteweg-de Vries equation models the formation of solitary waves in the context of shallow water in a channel. In our system, f or p=2 and p=3 (Korteweg-de Vries equations (KdV)) and (modified Korteweg-de Vries (mKdV) respectively), these equations have many applications in Physics. (gKdV) is a Hamiltonian system. In this article we investigate the generalized Korteweg-de Vries (gKdV) equation. A new topological approach is applied to prove the existence of at least one classical solution. The arguments are based upon recent theoretical results.

Numerical study of the transverse stability of line solitons of the Zakharov–Kuznetsov equations

We present a detailed numerical study of the stability under periodic perturbations of line solitons of two-dimensional, generalized Zakharov–Kuznetsov equations with various power nonlinearities. In the L2-subcritical case, in accordance with a theorem due to Yamazaki we find a critical speed, below which the line soliton is stable. For higher velocities, the numerical results indicate an instability against the formation of lumps, solitons localized in both spatial directions. In the L2-critical and supercritical cases but subcritical for the 1D generalized Korteweg–de Vries equation), the line solitons are shown to be numerically stable for small velocities, and strongly unstable for large velocities, with a blow-up observed in finite time.

Novel constructed dark, bright and rogue waves of three models of the well-known nonlinear Schrödinger equation

In this work, three models of the nonlinear Schrödinger equation are looked at to see if traveling wave solutions have unique structures. “The generalized Korteweg–de Vries, (2+1)-dimensional Ablowitz–Kaup–Newell–Segur, and the Maccari models” are among the examined systems. The kinetic energy operator is affected by where the mass is, how it changes over time, and how the kinetic energy operator is shown. Using the generalized Riccati expansion method, new soliton wave solutions are constructed for the models that have been looked at. Several different graphics depict the numerical simulations of the deduced answers. Verifying and improving these methods (Mathematica 13.1) means checking to see if they are right and re-entering the results into the models.

A new superposition between lump, breather and line waves of the (2+1)-dimensional generalized Korteweg–de Vries equation in fluid mechanics

This paper focuses on a new superposition between a single lump wave, breather waves and line solitons of the [Formula: see text]-dimensional generalized Korteweg–de Vries (gKdV) equation. After deriving the N-soliton solutions, a new constraint, the long-wave limit and mode resonance method are used to construct a new hybrid solution. Further, a new velocity resonance condition is proposed to obtain a velocity resonance solution consisting of a single lump wave, a line soliton and breather waves, which remains relatively stationary and form a new bound state. It is worth pointing out that in most of the previous literature, lump wave and other waves will separate after they collide, but this paper derives a different result.

Nonlocal symmetries and interaction solutions for the (n + 1)-dimensional generalized Korteweg–de Vries equation

The (n + 1)-dimensional generalized KdV equation is presented in this paper, and we further investigate its nonlocal symmetries by different methods. It can be seen that the symmetrical transformations obtained by different nonlocal symmetries are usually equivalent. Based on the obtained Lie point symmetry as well as the mth finite symmetrical transformations, we obtain its soliton molecules and multiple soliton solutions. Additionally, for the case of n = 4 various types of interaction solutions among solitons and periodic waves are obtained by the similarity reduction method.

Traveling wave solutions of generalized seventh-order time-fractional KdV models through He-Laplace algorithm

Non-linear evolution equations play a prominent role in describing a wide range of phenomena in optical fibers, fluid dynamics, electromagnetic radiation, plasma and solid state physics. An important category of non-linear evolution models that characterizes shallow wave phenomena are the Korteweg-de Vries (KdV) models. In this regard, time-fractional Korteweg-de Vries models of seventh order are the main focus of this research. A general KdV seventh-order equation is considered with different coefficients to form Lax, Kaup–Kuperschimdt and Sawada–Kotera–Ito KdV models. An efficient semi-analytical algorithm named as He-Laplace (HLM) is applied for the solution of these models. In this algorithm, Laplace transform is hybrid with homotopy perturbation method (HPM). This study provides important results as non-linear evolution seventh-order models in fractional sense have not been captured through HLM in current literature. Absolute errors are computed and compared with already existing results to confirm the superiority of proposed algorithm over other existing techniques. Numerical and graphical investigations are conducted to evaluate the approximate series form solutions. The dynamic behavior of fractional parameter is observed by calculating residual errors and plotting two dimensional diagrams throughout the fractional-domain. Analysis confirms that the proposed methodology provides an effective and convenient way for solving fractional KdV models.

Well‐posedness of generalized KdV and one‐dimensional fourth‐order derivative nonlinear Schrödinger equations for data with an infinite L 2 norm

We study the Cauchy problem for the generalized Korteweg–de Vries (KdV) and one-dimensional fourth-order derivative nonlinear Schrödinger equations, for which the global well-posedness of solutions with small rough data in certain scaling limit of modulation spaces M 2 , 1 μ $\mathcal {M}_{2,1}^{\mu }$ is shown, which contain some data with infinite L2 norm.