gKdV Equation Research Articles

Weak compactons of nonlinearly dispersive KdV and KP equations

AbstractA weak formulation is devised for the equation, which is a nonlinearly dispersive generalization of the gKdV equation having compacton solutions. With this formulation, explicit weak compacton solutions are derived, including ones that do not exist as classical (strong) solutions. Similar results are obtained for a nonlinearly dispersive generalization of the gKP equation in two dimensions, which possesses line compacton solutions.

Asymptotic stability of a finite sum of solitary waves for the Zakharov–Kuznetsov equation

We prove the asymptotic stability of a finite sum of well-ordered solitary waves for the Zakharov–Kuznetsov equation in dimensions two and three. We also derive a qualitative version of the orbital stability result, which will be useful for studying the collision of two solitary waves in a forthcoming paper.The proof extends the ideas of Martel, Merle and Tsai for the sub-critical gKdV equation in dimension one to the higher-dimensional case. It relies on monotonicity properties on oblique half-spaces and rigidity properties around one solitary wave introduced by Côte, Muñoz, Pilod and Simpson in dimension two, and by Farah, Holmer, Roudenko and Yang in dimension three.

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.

New (3+1)-dimensional integrable generalized KdV equation: Painlevé property, multiple soliton/shock solutions, and a class of lump solutions

The present work aims to examine a newly proposed (3+1)-dimensional integrable generalized Korteweg-de Vries (gKdV) equation. By employing the Weiss- Tabor-Carnevale technique in conjunction with Kruskal ansatz, we establish the com- plete integrability of the suggested model by demonstrating its ability to satisfy the Painlev´e property. The bilinear form of the (3+1)-dimensional gKdV equation is em- ployed to construct multiple soliton solutions. By manipulating the various values of the corresponding parameters, we generate a category of lump solutions that exhibit localization in all dimensions and algebraic decay.

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 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.

Classical Solutions for the Coupled System gKdV Equations

Classical Solutions for the Coupled System gKdV Equations

Invariance of the Gibbs measures for periodic generalized Korteweg-de Vries equations

In this paper, we study the Gibbs measures for periodic generalized Korteweg-de Vries equations (gKdV) with quartic or higher nonlinearities. In order to bypass the analytical ill-posedness of the equation in the Sobolev support of the Gibbs measures, we establish deterministic well-posedness of the gauged gKdV equations within the framework of the Fourier-Lebesgue spaces. Our argument relies on bilinear and trilinear Strichartz estimates adapted to the Fourier-Lebesgue setting. Then, following Bourgain’s invariant measure argument, we construct almost sure global-in-time dynamics and show invariance of the Gibbs measures for the gauged equations. These results can be brought back to the ungauged side by inverting the gauge transformation and exploiting the invariance of the Gibbs measures under spatial translations. We thus complete the program initiated by Bourgain [Comm. Math. Phys. 166 (1994), pp 1–26] on the invariance of the Gibbs measures for periodic gKdV equations.

Local well-posedness for the gKdV equation on the background of a bounded function

We prove the local well-posedness for the generalized Korteweg–de Vries equation in $H^s(\mathbb{R})$, $s>1/2$, under general assumptions on the nonlinearity $f(x)$, on the background of an $L^\infty\_{t,x}$-function $\Psi(t,x)$, with $\Psi(t,x)$ satisfying some suitable conditions. As a consequence of our estimates, we also obtain the unconditional uniqueness of the solution in $H^s(\mathbb{R})$. This result not only gives us a framework to solve the gKdV equation around a Kink, for example, but also around a periodic solution, that is, to consider localized non-periodic perturbations of a periodic solution. As a direct corollary, we obtain the unconditional uniqueness of the gKdV equation in $H^s(\mathbb{R})$ for $s>1/2$. We also prove global existence in the energy space $H^1(\mathbb{R})$, in the case where the nonlinearity satisfies $\vert f''(x)\vert\lesssim 1$.

Exact solitary wave and numerical solutions for geophysical KdV equation

Tsunamis are strong waves, arisen from volcanic eruptions, landslides, or earthquakes sweeping across oceans. The geophysical Korteweg-de Vries (gKdV) equation which governs the tsunami wave propagation in oceans is investigated in this work using an improved exp(-F(η))-expansion method. Shooting and adaptive moving approaches are taken into account. We retrieve several new solitary solutions for the gKdV equation. The obtained solution from implementing shooting method is successfully used as an initial value for the adaptive approach which is utilized to construct the numerical solution of the proposed problem. The constructed exact solutions coincide with the obtained numerical solutions. The accuracy of the presented numerical approximations is discussed. We apply Fourier concept to explore the accuracy and stability of the numerical schemes which is unconditionally stable. A clear comparison between the analytic and numerical outcomes is presented via some 2D and 3D sketches which are depicted under special selections of some parameters. Moreover, we illustrate the relative error and CPU time for the numerical technique. The proposed approaches can be easily utilized to deal with other partial differential equations.

Computational simulation and nonlinear vibration motions of isolated waves localized in small part of space

Nonlinear Schrödinger equations are examined to see if any new solution structures can be found for three different kinds of nonlinear equations. The generalized Korteweg–de Vries (gKdV) equation, the (2 + 1)–Ablowitz–Kaup–Newwell–Segur (AKNS) equation, and the Maccari system are all investigated. These models commonly characterize kinetic energy operators, which are affected by mass location and time dependence and the numerous representations of kinetic energy operators. Soliton wave solutions for the three new models are created by using the analytical Khater II approach. Figures depicting the numerical simulations of the resulting solutions may be seen all over the place. Mathematica 12 software is used to verify the approaches’ accuracy and re-incorporate all new results into the original models.

Computational wave solutions of some nonlinear evolution equations

This paper aims to realize some of the vital nonlinear evolution of partial differential equations in mathematical physics. We implement an extended simple equation (ESE) method to get exact and solitary traveling wave solutions of the Calogero-Degasperis (CD) equation, potential Kadomtsev–Petviashvili (pKP) equation, generalized Korteweg–de Vries (gKdV) equation, and the (2 + 1)–dimensional Zoomeron (Z) equation. We obtain a new form of solutions for each model as mentioned above. This method is considered a new, robust, practical, and direct method. As we see, the remarkable ability of this method to apply on the fractional and integer order of nonlinear partial differential equations.

Классические решения для системы обобщенных уравнений Кортевега–де Фриза

Исследуется система обобщенных уравнений Кортевега-де Фриза. Применен новый топологический подход для доказательства существования по крайней мере одного классического решения и по крайней мере двух неотрицательных классических решений. Рассуждения опираются на последние теоретические результаты.

Solitary wave solutions and global well-posedness for a coupled system of gKdV equations

In this work, we consider the initial-value problem associated with a coupled system of generalized Korteweg–de Vries equations. We present a relationship between the best constant for a Gagliardo–Nirenberg type inequality and a criterion for the existence of global solutions in the energy space. We prove that such a constant is directly related to the existence problem of solitary wave solutions with minimal mass, the so-called ground state solutions. A characterization of the ground states and the orbital instability of the solitary waves are also established.

Analytical Method for Solving the Fractional Order Generalized KdV Equation by a Beta-Fractional Derivative

The present work is related to solving the fractional generalized Korteweg-de Vries (gKdV) equation in fractional time derivative form of order α . Some exact solutions of the fractional-order gKdV equation are attained by employing the new powerful expansion approach by using the beta-fractional derivative which is used to get many solitary wave solutions by changing various parameters. The obtained solutions include three classes of soliton wave solutions in terms of hyperbolic function, trigonometric function, and rational function solutions. The obtained solutions and the exact solutions are shown graphically, highlighting the effects of nonlinearity. Some of the nonlinear equations arise in fluid dynamics and nonlinear phenomena.

On continuation properties after blow-up time for $L^2$-critical gKdV equations

In this paper, we consider a blow-up solution u(t) (close to the soliton manifold) to the L^2 -critical gKdV equation \partial_tu+(u_{xx}+u^5)_x=0 , with finite blow-up time T < +\infty . We expect to construct a natural extension of u(t) after the blow-up time. To do this, we consider the solution u_{\gamma}(t) to the saturated L^2 -critical gKdV equation \partial_tu+(u_{xx}+u^5-\gamma u|u|^{q-1})_x=0 with the same initial data, where \gamma > 0 and q > 5 . A standard argument shows that u_{\gamma}(t) is always global in time. Moreover, for all t < T , u_{\gamma}(t) converges to u(t) in H^1 as \gamma\rightarrow0 . We prove in this paper that for all t\geq T , u_{\gamma}(t)\rightarrow v(t) as \gamma\rightarrow0 , in a certain sense. This limiting function v(t) is a weak solution to the unperturbed L^2 -critical gKdV equations, hence can be viewed as a natural extension of u(t) after the blow-up time.

Near critical, self-similar, blow-up solutions of the generalised Korteweg–de Vries equation: Asymptotics and computations

In this article we give a detailed asymptotic analysis of the near critical self-similar blowup solutions to the Generalised Korteweg–de Vries equation (GKdV). We compare this analysis to some careful numerical calculations. It has been known that for a nonlinearity that has a power larger than the critical value p=5, solitary waves of the GKdV can become unstable and become infinite in finite time, in other words they blow up. Numerical simulations presented in Klein and Peter (2015) indicate that if p>5 the solitary waves travel to the right with an increasing speed, and simultaneously, form a similarity structure as they approach the blow-up time. This structure breaks down at p=5. Based on these observations, we rescale the GKdV equation to give an equation that will be analysed by using asymptotic methods as p→5+. By doing this we resolve the complete structure of these self-similar blow-up solutions and study the singular nature of the solutions in the critical limit. In both the numerics and the asymptotics, we find that the solution has sech-like behaviour near the peak. Moreover, it becomes asymmetric with slow algebraic decay to the left of the peak and much more rapid algebraic decay to the right. The asymptotic expressions agree to high accuracy with the numerical results, performed by adaptive high-order solvers based on collocation or finite difference methods.

Dynamics near the solitary waves of the supercritical gKDV equations

This work is devoted to study the dynamics of the supercritical gKDV equations near solitary waves in the energy space H1. We construct smooth local center-stable, center-unstable and center manifolds near the manifold of solitary waves and give a detailed description of the local dynamics near solitary waves. In particular, the instability is characterized as follows: any forward flow not starting from the center-stable manifold will leave a neighborhood of the manifold of solitary waves exponentially fast. Moreover, orbital stability is proved on the center manifold, which implies the uniqueness of the center manifold and the solutions on it exist globally and asymptotically approach the solitary waves.

Effect of Coriolis constant on Geophysical Korteweg-de Vries equation

The present article investigates the effect of Coriolis constant on the solution of the Geophysical Korteweg-de Vries (gKdV) equation. As such, the Homotopy Perturbation Method (HPM) has been applied here for solving the nonlinear gKdV equation. Present results are compared with existing results that are available in the literature and they are found to be in good agreement. Then the Coriolis term has been considered in terms of the interval to form interval Geophysical Korteweg-de Vries (IgKdV) equation. IgKdV equation has been solved by HPM to analyse the effect of Coriolis. From this analysis, it has been concluded that the constant of Coriolis is directly proportional to wave height and inversely proportional to wavelength. The presence of the Coriolis term in gKdV equation has a remarkable change in the shape of the solution.