Generalized Korteweg-de Vries Equation Research Articles

Generalized KdV equation subject to a stochastic perturbation

We prove global well-posedness of the subcritical generalized Korteweg-de Vries equation (the mKdV and the gKdV with quartic power of nonlinearity) subject to an additive random perturbation. More precisely, we prove that if the driving noise is a cylindrical Wiener process on $L^2(\mathbb{R})$ and the covariance operator is Hilbert-Schmidt in an appropriate Sobolev space, then the solutions with $H^1(\mathbb{R})$ data are globally well-posed in $H^1(\mathbb{R})$. This extends results obtained by A. de Bouard and A. Debussche for the stochastic KdV equation.

Rapidly oscillating solutions of a generalized Korteweg–de Vries equation

For a generalized Korteweg–de Vries equation, the existence of families of rapidly oscillating periodic solutions is proved and their asymptotic representation is found. The asymptotics of tori of different dimensions are examined. Formulas for solutions depending on all parameters of the problem are derived.

Conservation laws and symmetries of time-dependent generalized KdV equations

A complete classification of low-order conservation laws is obtained for time-dependent generalized Korteweg-de Vries equations. Through the Hamiltonian structure of these equations, a corresponding classification of Hamiltonian symmetries is derived. The physical meaning of the conservation laws and the symmetries is discussed.

Existence of a minimal non-scattering solution to the mass-subcritical generalized Korteweg–de Vries equation

In this article, we prove the existence of a non-scattering solution, which is minimal in some sense, to the mass-subcritical generalized Korteweg–de Vries (gKdV) equation in the scale critical Lˆr space where Lˆr={f∈S′(R)|‖f‖Lˆr=‖fˆ‖Lr′<∞}. We construct this solution by a concentration compactness argument. Then, key ingredients are a linear profile decomposition result adopted to Lˆr-framework and approximation of solutions to the gKdV equation which involves rapid linear oscillation by means of solutions to the nonlinear Schrödinger equation.

Global well-posedness for periodic generalized Korteweg-de Vries equation

Global well-posedness for periodic generalized Korteweg-de Vries equation

Construction of a minimal mass blow up solution of the modified Benjamin–Ono equation

We construct a minimal mass blow up solution of the modified Benjamin–Ono equation (mBO) mBO $$\begin{aligned} u_{t}+(u^3-D^1 u)_{x}=0, \end{aligned}$$ which is a standard mass critical dispersive model. Let $$Q\in H^{\frac{1}{2}}$$ , $$Q>0$$ , be the unique ground state solution of $$D^1 Q +Q=Q^3$$ , constructed using variational arguments by Weinstein (Commun. Part. Differ. Equations 12:1133–1173, 1987a; J. Differ. Equations 69:192–203, 1987b) and Albert et al. (Proc. R. Soc. Lond. A 453:1233–1260, 1997), and whose uniqueness was recently proved by Frank and Lenzmann (Acta Math. 210:261–318, 2013). We show the existence of a solution S of (mBO) satisfying $$\Vert S \Vert _{L^2}=\Vert Q\Vert _{L^2}$$ and $$\begin{aligned} S(t)-\frac{1}{\lambda ^{\frac{1}{2}}(t)} Q\left( \frac{\cdot - x(t)}{\lambda (t)}\right) \rightarrow 0\quad \text{ in } H^{\frac{1}{2}}(\mathbb R) \text{ as } t\downarrow 0, \end{aligned}$$ where $$\begin{aligned} \lambda (t)\sim t,\quad x(t) \sim -|\ln t| \quad \hbox {and}\quad \Vert S(t)\Vert _{\dot{H}^{\frac{1}{2}}} \sim t^{-\frac{1}{2}}\Vert Q\Vert _{\dot{H}^{\frac{1}{2}}} \quad \hbox {as} t\downarrow 0. \end{aligned}$$ This existence result is analogous to the one obtained by Martel et al. (J. Eur. Math. Soc. 17:1855–1925, 2015) for the mass critical generalized Korteweg-de Vries equation. However, in contrast with the (gKdV) equation, for which the blow up problem is now well-understood in a neighborhood of the ground state, S is the first example of blow up solution for (mBO). The proof involves the construction of a blow up profile, energy estimates as well as refined localization arguments, developed in the context of Benjamin–Ono type equations by Kenig et al. (Ann. Inst. H. Poincare Anal. Non Lin. 28:853–887, 2011). Due to the lack of information on the (mBO) flow around the ground state, the energy estimates have to be considerably sharpened in the present paper.

Kink dynamics in the $\phi ^4$ model: Asymptotic stability for odd perturbations in the energy space

We consider a classical equation known as the $\phi^4$ model in one space dimension. The kink, defined by $H(x)=\tanh(x/{\sqrt{2}})$, is an explicit stationary solution of this model. From a result of Henry, Perez and Wreszinski it is known that the kink is orbitally stable with respect to small perturbations of the initial data in the energy space. In this paper we show asymptotic stability of the kink for odd perturbations in the energy space. The proof is based on Virial-type estimates partly inspired from previous works of Martel and Merle on asymptotic stability of solitons for the generalized Korteweg-de Vries equations. However, this approach has to be adapted to additional difficulties, pointed out by Soffer and Weinstein in the case of general Klein-Gordon equations with potential: the interactions of the so-called internal oscillation mode with the radiation, and the different rates of decay of these two components of the solution in large time.

Co-periodic stability of periodic waves in some Hamiltonian PDEs

The stability of periodic traveling wave solutions to dispersive PDEs with respect to ‘arbitrary’ perturbations is still widely open. The focus is put here on stability with respect to perturbations of the same period as the wave, for KdV-like systems of one-dimensional Hamiltonian PDEs. Stability criteria are derived and investigated first in a general abstract framework, and then applied to three basic examples that are very closely related, and ubiquitous in mathematical physics, namely, a quasilinear version of the generalized Korteweg–de Vries equation (qKdV), and the Euler–Korteweg system in both Eulerian coordinates (EKE) and in mass Lagrangian coordinates (EKL). Those criteria consist of a necessary condition for spectral stability, and of a sufficient condition for orbital stability. Both are expressed in terms of a single function, the abbreviated action integral along the orbits of waves in the phase plane, which is the counterpart of the solitary waves moment of instability introduced by Boussinesq. Regarding solitary waves, the celebrated Grillakis–Shatah–Strauss stability criteria amount to looking for the sign of the second derivative of the moment of instability with respect to the wave speed. For periodic waves, the most striking results obtained here can be summarized as: an odd value for the difference between N—the size of the PDE system—and the negative signature of the Hessian of the action implies spectral instability, whereas a negative signature of the same Hessian being equal to N implies orbital stability. Since these stability criteria are merely encoded by the negative signature of matrices, they can at least be checked numerically. Various numerical experiments are presented, which clearly discriminate between stable cases and unstable cases for (qKdV), (EKE) and (EKL).

The generalized Korteweg-de Vries equation with time oscillating nonlinearity in scale critical Sobolev space

We consider the generalized Korteweg-de Vries (gKdV) equation with the time oscillating nonlinearity: $${\partial}_t u+{\partial}_x^3 u+ g(\omega t) {\partial}_x (|u|^{p-1}u)= 0, \quad (t, x) \in \mathbb{R} \times \mathbb{R}.$$ Under the suitable assumption on g, we show that if the nonlinear term is mass critical or supercritical i.e., \({p \geq 5}\) and \({u(0) \in \dot{H}^{s_{p}}}\), where \({s_{p} = 1/2 - 2/(p-1)}\) is a scale critical exponent, then there exists a unique global solution to (gKdV) provided that \({|\omega|}\) is sufficiently large. We also obtain the behavior of the solution to (gKdV) as \({|\omega| \to \infty}\).

Asymptotic stability in energy space for dark solitons of the Landau–Lifshitz equation

We prove the asymptotic stability in the energy space of non-zero speed solitons for the one-dimensional Landau-Lifshitz equation with an easy-plane anisotropy. More precisely, we show that any solution corresponding to an initial datum close to a soliton with non-zero speed, is weakly convergent in the energy space as time goes to infinity, to a soliton with a possible different non-zero speed, up to the invariances of the equation. Our analysis relies on the ideas developed by Martel and Merle for the generalized Korteweg-de Vries equations. We use the Madelung transform to study the problem in the hydrodynamical framework. In this framework, we rely on the orbital stability of the solitons and the weak continuity of the flow in order to construct a limit profile. We next derive a monotonicity formula for the momentum, which gives the localization of the limit profile. Its smoothness and exponential decay then follow from a smoothing result for the localized solutions of the Schr\"odinger equations. Finally, we prove a Liouville type theorem, which shows that only the solitons enjoy these properties in their neighbourhoods.

A Hamiltonian preserving discontinuous Galerkin method for the generalized Korteweg–de Vries equation

The invariant preserving property is one of the guiding principles for numerical algorithms in solving wave equations, in order to minimize phase and amplitude errors after long time simulation. In this paper, we design, analyze and numerically validate a Hamiltonian preserving discontinuous Galerkin method for solving the Korteweg–de Vries (KdV) equation. For the generalized KdV equation, the semi-discrete formulation is shown to preserve both the first and the third conserved integrals, and approximately preserve the second conserved integral; for the linearized KdV equation, all the first three conserved integrals are preserved, and optimal error estimates are obtained for polynomials of even degree. The preservation properties are also maintained by the fully discrete DG scheme. Our numerical experiments demonstrate both high accuracy of convergence and preservation of all three conserved integrals for the generalized KdV equation. We also show that the shape of the solution, after long time simulation, is well preserved due to the Hamiltonian preserving property.

CONVERGENCE OF THE SOLUTIONS ON THE GENERALIZED KORTEWEG–DE VRIES EQUATION∗

We consider the generalized Korteweg-de Vries equation, which contains nonlinear dispersive effects. We prove that as the diffusion parameter tends to zero, the solutions of the dispersive equation converge to discontinuous weak solutions of the scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the Lp setting.

Existence of solitary-wave solutions to nonlocal equations

We prove existence and conditional energetic stability of solitary-wave solutions for the two classes of pseudodifferential equations \begin{equation*} u_t+\left(f(u)\right)_x-\left(L u\right)_x=0 \end{equation*} and \begin{equation*} u_t+\left(f(u)\right)_x+\left(L u\right)_t=0, \end{equation*} where $f$ is a nonlinear term, typically of the form $c|u|^p$ or $cu|u|^{p-1}$, and $L$ is a Fourier multiplier operator of positive order. The former class includes for instance the Whitham equation with capillary effects and the generalized Korteweg-de Vries equation, and the latter the Benjamin-Bona-Mahony equation. Existence and conditional energetic stability results have earlier been established using the method of concentration-compactness for a class of operators with symbol of order $s\geq 1$. We extend these results to symbols of order $0 < s < 1$, thereby improving upon the results for general operators with symbol of order $s\geq 1$ by enlarging both the class of linear operators and nonlinearities admitting existence of solitary waves. Instead of using abstract operator theory, the new results are obtained by direct calculations involving the nonlocal operator $L$, something that gives us the bounds and estimates needed for the method of concentration-compactness.

Simulation of blow-up solutions to the generalized KdV equations by moving collocation methods

The aim of this paper is to simulate the blow-up solutions to generalized Korteweg-de Vries (KdV) equations. The KdV equations are discretized with the use of a quintic Hermite collocation method based on the moving meshes generated by solving moving mesh partial differential equations (MMPDEs). Theoretical analyses are, respectively, conducted to determine the critical parameters in MMPDEs such that the generated meshes can catch up with the blow-up profiles and to show the effectiveness of the generated moving meshes. Lastly, a variety of examples are implemented to confirm our analysis and show the efficiency of the method.

Nonlinear ion-acoustic double-layers in electronegative plasmas with electrons featuring Tsallis distribution

Weakly nonlinear ion-acoustic (IA) double-layers (DLs), which accompany electronegative plasmas composed of positive ions, negative ions, and nonextensive electrons are investigated. A generalized Korteweg–de Vries equation with a cubic nonlinearity is derived using a reductive perturbation method. Different types of electronegative plasmas inspired from the experimental studies of Ichiki et al. (2001) are discussed. It is shown that the IA wave phase velocity, in different mixtures of negative and positive ions, decreases as the nonextensive parameter q increases, before levelling-off at a constant value for larger q. Moreover, a relative increase of Q involves an enhancement of the IA phase velocity. Existence domains of either solitary waves or double-layers are then presented and their parametric dependence is determined. Owing to the electron nonextensivity, our present plasma model can admit compressive as well as rarefactive IA–DLs.

On mass concentration for the critical generalized Korteweg–de Vries equation

AbstractWe show that blow-up solutions of the critical generalized Korteweg–de Vries equation in H1() concentrate at least the mass of the ground state at the blow-up time. The I-method is used to prove a slightly weaker result in Hs() with 16/17 &lt; s &lt; 1. Under an assumption on the precise blow-up rate, we are able to use similar arguments to prove a more precise analogue of the H1() concentration result over the same range of s.

Whitham theory for perturbed Korteweg–de Vries equation

Original Whitham’s method of derivation of modulation equations is applied to systems whose dynamics is described by a perturbed Korteweg–de Vries equation. Two situations are distinguished: (i) the perturbation leads to appearance of right-hand sides in the modulation equations so that they become non-uniform; (ii) the perturbation leads to modification of the matrix of Whitham velocities. General form of Whitham modulation equations is obtained in both cases. The essential difference between them is illustrated by an example of so-called ‘generalized Korteweg–de Vries equation’. Method of finding steady-state solutions of perturbed Whitham equations in the case of dissipative perturbations is considered.

Asymptotic Stability of High-dimensional Zakharov–Kuznetsov Solitons

We prove that solitons (or solitary waves) of the Zakharov-Kuznetsov (ZK) equation, a physically relevant high dimensional generalization of the Korteweg-de Vries (KdV) equation appearing in Plasma Physics, and having mixed KdV and nonlinear Schr\"odinger (NLS) dynamics, are strongly asymptotically stable in the energy space in the physical region. We also prove that the sum of well-arranged solitons is stable in the same space. Orbital stability of ZK solitons is well-known since the work of de Bouard. Our proofs follow the ideas by Martel and Martel and Merle, applied for generalized KdV equations in one dimension. In particular, we extend to the high dimensional case several monotonicity properties for suitable half-portions of mass and energy; we also prove a new Liouville type property that characterizes ZK solitons, and a key Virial identity for the linear and nonlinear part of the ZK dynamics, obtained independently of the mixed KdV-NLS dynamics. This last Virial identity relies on a simple sign condition, which is numerically tested for the two and three dimensional cases, with no additional spectral assumptions required. Possible extensions to higher dimensions and different nonlinearities could be obtained after a suitable local well-posedness theory in the energy space, and the verification of a corresponding sign condition.

Painleve analysis and exact solutions for the modified Korteweg–de Vries equation with polynomial source

The modified Korteweg–de Vries equation with polynomial source is considered. Using the Painleve test we show that the generalized Korteweg–de Vries equation is not integrable by the inverse scattering transform. However there are some expansions of solution in the Laurent series and some exact solutions can exist. Some traveling wave solutions of the modified Korteweg–de Vries equation with polynomial source are found.

Modified method of simplest equation for obtaining exact analytical solutions of nonlinear partial differential equations: further development of the methodology with applications

We discuss the application of a variant of the method of simplest equation for obtaining exact traveling wave solutions of a class of nonlinear partial differential equations containing polynomial nonlinearities. As simplest equation we use differential equation for a special function that contains as particular cases trigonometric and hyperbolic functions as well as the elliptic function of Weierstrass and Jacobi. We show that for this case the studied class of nonlinear partial differential equations can be reduced to a system of two equations containing polynomials of the unknown functions. This system may be further reduced to a system of nonlinear algebraic equations for the parameters of the solved equation and parameters of the solution. Any nontrivial solution of the last system leads to a traveling wave solution of the solved nonlinear partial differential equation. The methodology is illustrated by obtaining solitary wave solutions for the generalized Korteweg–deVries equation and by obtaining solutions of the higher order Korteweg–deVries equation.