Generalized Korteweg-de Vries Equation Research Articles

Superharmonic instability for regularized long-wave models* *JCB is supported by NSF DMS-1615418. VMH is supported by NSF DMS-2009981.

We examine the spectral stability and instability of periodic traveling waves for regularized long-wave models. Examples include the regularized Boussinesq, Benney–Luke, and Benjamin–Bona–Mahony equations. Of particular interest is a striking new instability phenomenon—spectrum off the imaginary axis extending into infinity. The spectrum of the linearized operator of the generalized Korteweg–de Vries equation, for instance, lies along the imaginary axis outside a bounded set. The spectrum for a regularized long-wave model, by contrast, can vary markedly with the parameters of the periodic traveling waves. We perform rigorous spectral asymptotics for short wavelength perturbations to establish conditions under which the spectrum tends to infinity along the imaginary axis or some curve whose real part is nonzero. We conduct numerical experiments which corroborate our analytical findings.

Bright solitons of the model with arbitrary refractive index and unrestricted dispersion

Objective:The generalized Schrödinger equation with arbitrary refractive index and unrestricted order is considered. The objective of this paper is to demonstrate the bright solitons of this mathematical model can be found by the same formula taking into account the order of nonlinear differential equation. Method:The algorithm of constructing optical solitons of the generalized nonlinear Schrödinger equation with unrestricted dispersion using the simplest equation method is presented and applied for finding of optical solitons. Results:The optical solitons of the generalized nonlinear Schrödinger equation of the twelfth order are found at three powers on nonlinearity. The application of the method for finding solution of the generalized Korteweg–de Vries equation with arbitrary power of nonlinearity and unrestricted dispersion is discussed.

Spatial decay of multi-solitons of the generalized Korteweg-de Vries and nonlinear Schrödinger equations

We study pointwise spatial decay of multi-solitons of the generalized Korteweg-de Vries equations. We obtain that, uniformly in time, these solutions and their derivatives decay exponentially in space on the left of and in the solitons region, and prove rapid decay on the right of the solitons. We also prove the corresponding result for multi-solitons of the nonlinear Schrödinger equations, that is, exponential decay in the solitons region and rapid decay outside.

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.

On existence and uniqueness of asymptotic N-soliton-like solutions of the nonlinear Klein–Gordon equation

We are interested in solutions of the nonlinear Klein–Gordon equation (NLKG) in \(\mathbb {R}^{1+d}\), \(d\ge 1\), which behave as a soliton or a sum of solitons in large time. In the spirit of other articles focusing on the supercritical generalized Korteweg-de Vries equations and on the nonlinear Schrödinger equations, we obtain an N-parameter family of solutions of (NLKG) which converges exponentially fast to a sum of N given (unstable) solitons. For \(N=1\), this family completely describes the set of solutions converging to the soliton considered; for \(N\ge 2\), we prove uniqueness in a class with explicit algebraic rate of convergence.

Conservative compact difference scheme for the generalized Korteweg-de Vries equation

In this paper, a linear mass and energy conservative finite difference scheme for the generalized Korteweg-de Vries (GKdV) equation is proposed and analyzed. The scheme is three-level and linear implicit and gives second- and fourth-order accuracy in time and space, respectively. It is rigorously proved by using the discrete energy method that the proposed difference scheme is uniquely solvable, stable and convergent. Numerical examples are given to confirm the stability and convergence of the numerical solution with fourth-order accuracy and the effectiveness of the present scheme for handling the single and multi-solitary waves for a long time.

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 $\vert f''(x)\vert\lesssim 1$.

Numerical Study of the Generalized Korteweg–de Vries Equations with Oscillating Nonlinearities and Boundary Conditions

The focus here is upon the generalized Korteweg–de Vries equation, $$\begin{aligned} u_t + u_x + \frac{1}{p} \left( u^p \right) _x +u_{xxx} \, = \, 0, \end{aligned}$$where \(p = 2, 3, \ldots \). When \(p \ge 5\), it is thought that the equation is not globally well posed in time for \(L_2\)-based Sobolev class data. Various numerical simulations carried out by multiple research groups indicate that solutions can blowup in finite time for large, smooth initial data. This is known to be the case in the critical case \(p = 5\), but remains a conjecture for supercritical values of p. Studied here are methods for controlling this potential blow up. Several candidates are put forward; the addition of dissipation or of higher order dispersion are two obvious candidates. However, these apparently can only work for a limited range of nonlinearities. However, the introduction of high frequency temporal oscillations appear to be more effective. Both temporal oscillation of the nonlinearity and of the boundary condition in an initial-boundary-value configuration are considered. The bulk of the discussion will turn around this prospect in fact.

Time Fractional Generalized Korteweg-de Vries Equation: Explicit Series Solutions and Exact Solutions

In this article, an attempt is made to achieve the series solution of the time fractional generalized Korteweg-de Vries equation which leads to a conditionally convergent series solution. We have also resorted to another technique involving conversion of the given fractional partial differential equations to ordinary differential equations by using fractional complex transform. This technique is discussed separately for modified Riemann-Liouville and conformable derivatives. Convergence analysis and graphical view of the obtained solution are demonstrated in this work.

Smoothing and growth bound of periodic generalized Korteweg–De Vries equation

For generalized Korteweg–De Vries (KdV) models with polynomial nonlinearity, we establish a local smoothing property in [Formula: see text] for [Formula: see text]. Such smoothing effect persists globally, provided that the [Formula: see text] norm does not blow up in finite time. More specifically, we show that a translate of the nonlinear part of the solution gains [Formula: see text] derivatives for [Formula: see text]. Following a new simple method, which is of independent interest, we establish that, for [Formula: see text], [Formula: see text] norm of a solution grows at most by [Formula: see text] if [Formula: see text] norm is a priori controlled.

Effects of viscosity and surface tension on soliton dynamics in the generalized KdV equation for shallow water waves

Various theories have been formulated for the study of weakly damped free-surface flows. These theories have been essentially focused on the forces relatively perpendicular to the fluid particle such as pressure forces, while neglecting forces relatively parallel to the fluid particle such as viscosity forces. In this work, with the help of linear approximation applying on the Navier-Stokes equations, we obtain a system of equations for potential flow which includes dissipative effect due to viscosity. The correction due to the viscosity is applied not only to the kinematics boundary condition on the surface, but also to the dynamics condition modeled by Bernoulli’s equation. We show that, in the context of wave motion in shallow water, an expansion of the Boussinesq system can be decomposed into a set of coupled equations. The first equation depends only on the surface elevation for the right-moving, while the other equation depends simultaneous on the surface elevation for the right- and left-moving waves. The wave equation corresponding to the pure right-moving has the form of a generalized inhomogeneous Korteweg de Vries (KdV) equation with higher-order nonlinear and dissipative terms. We then investigate the soliton solutions of this equation by using the Hirota’s bilinear method. The results show that, both group and phase velocities are a decreasing functions of the viscosity and surface tension parameters, δ and τ, respectively. The width of the soliton increases with the parameters δ and τ. The effects of viscosity on the soliton dynamics are more pronounced and are amplified by the surface tension effects.

Bilinear Bäcklund transformation, N‐soliton, and infinite conservation laws for Lax–Kadomtsev–Petviashvili and generalized Korteweg–de Vries equations

In this paper, we obtain the bilinear form for the Lax–Kadomtsev–Petviashvili (Lax–KP) and the generalized (3 + 1)‐dimensional Korteweg–de Vries equations based on the binary Bell polynomials. Accordingly, N‐soliton solutions, bilinear Bäcklund transformation, Lax pair, and infinite conservation laws will be constructed to Lax–KP and the generalized (2 + 1)‐dimensional Korteweg–de Vries equation . At the same time, we get another bilinear Bäcklund transformation. Finally, exact solutions were obtained by using the exchange formulas for Hirota's bilinear operators.

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.

Numerical Study of Zakharov–Kuznetsov Equations in Two Dimensions

We present a detailed numerical study of solutions to the (generalized) Zakharov-Kuznetsov equation in two spatial dimensions with various power nonlinearities. In the $L^{2}$-subcritical case, numerical evidence is presented for the stability of solitons and the soliton resolution for generic initial data. In the $L^2$-critical and supercritical cases, solitons appear to be unstable against both dispersion and blow-up. It is conjectured that blow-up happens in finite time and that blow-up solutions have some resemblance of being self-similar, i.e., the blow-up core forms a rightward moving self-similar type rescaled profile with the blow-up happening at infinity in the critical case and at a finite location in the supercritical case. In the $L^{2}$-critical case, the blow-up appears to be similar to the one in the $L^{2}$-critical generalized Korteweg-de Vries equation with the profile being a dynamically rescaled soliton.

The structure of algebraic solitons and compactons in the generalized Korteweg–de Vries equation

We analyze the main properties of soliton solutions to the generalized KdV equation ut+[F(u)]x+uxxx=0, where the leading term F(u)∼quα, α>0, q∈R. The far field of such solitons may have three options. For q>0 and α>1 the analysis re-confirmed that all traveling solitons have “light” exponentially decaying tails and propagate to the right. If q<0 and α<1, the traveling solitons (compactons) have a compact support (and thus vanishing tails) and propagate to the left. For more complicated Fu and α>1 (e.g., the Gardner equation) standing algebraic solitons with “heavy” power-law tails may appear. If the leading term of Fu is negative, the set of solutions may include wide or table-top solitons (similar to the solutions of the Gardner equation), including algebraic solitons and compactons with any of the three types of tails. The solutions usually have a single-hump structure but if Fu represents a higher-order polynomial, the generalized KdV equation may support multi-humped pyramidal solitons.

Non dispersive solutions of the generalized Korteweg-de Vries equations are typically multi-solitons

We consider solutions of the generalized Korteweg-de Vries equations (gKdV) which are non dispersive in some sense and which remain close to multi-solitons. We show that these solutions are necessarily pure multi-solitons. For the Korteweg-de Vries equation (KdV) and the modified Korteweg-de Vries equation (mKdV) in particular, we obtain a characterization of multi-solitons and multi-breathers in terms of non dispersion.

Интегрирование общего нагруженного уравнения Кортевега-де Фриза с интегральным источником в классе быстроубывающих комплекснозначных функций

Интегрирование общего нагруженного уравнения Кортевега-де Фриза с интегральным источником в классе быстроубывающих комплекснозначных функций

О симметриях обобщенного уравнения Кортевега–де Фриза

The Korteweg–de Vries equation is a third–order nonlinear partial differential equation that plays an important role in the theory of nonlinear waves, mainly of hydrodynamic origin. It was first obtained by Joseph Boussinesq in 1877, but a detailed analysis was already carried out by Diederik Korteweg and Gustav de Vries in 1895. The Korteweg–de Vries equation, its analogues and generalizations arise in mathematical models in a variety of subject areas. There are at least several thousand publications in which the Korteweg–de Vries equation is considered from one side or another, and the authors are aware of the impossibility of writing at least a simple list listing these publications. Nevertheless, it is possible to mention, for example, The paper considers some generalization of the Korteweg–de Vries equation obtained by introducing an arbitrary function with respect to which a group classification is performed. The admissible operators of the generalized Korteweg–de Vries equation, the basic algebra of admissible operators and possible cases of extension of the algebra of admissible operators for a coefficient function of a special kind are found. Tables of commutators of the obtained algebras are also calculated. At the same time, the fact of isomorphism of admissible algebras in two different cases of specification of the function-coefficient of the generalized Korteweg–de Vries equation is noted. It is shown that the maximum possible dimension of the algebra of admissible operators is equal to four and corresponds to the almost classical Korteweg–de Vries equation with a possible additional linear term. In all other cases, if the resulting algebra of admissible operators and extends, then to a three-dimensional Lie algebra.

Abundant rogue wave solutions for the (2 + 1)-dimensional generalized Korteweg–de Vries equation

Abstract In this paper, we analyse two types of rogue wave solutions generated from two improved ansatzs, to the (2 + 1)-dimensional generalized Korteweg–de Vries equation. With symbolic computation, the first-order rogue waves, second-order rogue waves, third-order rogue waves are generated directly from the first ansatz. Based on the Hirota bilinear formulation, another type of one-rogue waves and two-rogue waves can be obtained from the second ansatz. In addition, the dynamic behaviours of obtained rogue wave solutions are illustrated graphically.

Existence of two-solitary waves with logarithmic distance for the nonlinear Klein–Gordon equation

We consider the focusing nonlinear Klein–Gordon (NLKG) equation [Formula: see text] for [Formula: see text] and [Formula: see text] subcritical for the [Formula: see text] norm. In this paper, we show the existence of a solution [Formula: see text] of the equation such that [Formula: see text] where [Formula: see text] are two solitary waves of the equation with translations [Formula: see text] satisfying [Formula: see text] This behavior is due to the strong interactions between solitary waves which is in contrast with the previous work [R. Côte and Y. Martel, Multi-travelling waves for the nonlinear Klein–Gordon equation, Trans. Amer. Math. Soc. 370(10) (2018) 7461–7487] on multi-solitary waves of the (NLKG), devoted to the case of solitary waves with different speeds. This work is motivated by previous similar existence results for the nonlinear Schrödinger and generalized Korteweg–de Vries equations.