Stochastic Korteweg De Vries Equation Research Articles

A structure-preserving local discontinuous Galerkin method for the stochastic KdV equation

This paper proposes a local discontinuous Galerkin (LDG) method for the stochastic Korteweg-de Vries (KdV) equation with multi-dimensional multiplicative noise. In the mean square sense, we show that the numerical method is L2 stable and it preserves energy conservation and energy dissipation. If the degree of the polynomial is n, the optimal error estimate in the mean square sense can reach as n+1. Finally, structure-preserving and convergence are verified by numerical experiments.

Global dynamics for the stochastic KdV equation with white noise as initial data

We study the stochastic Korteweg-de Vries equation (SKdV) with an additive space-time white noise forcing, posed on the one-dimensional torus. In particular, we construct global-in-time solutions to SKdV with spatial white noise initial data. Due to the lack of an invariant measure, Bourgain’s invariant measure argument is not applicable to this problem. In order to overcome this difficulty, we implement a variant of Bourgain’s argument in the context of an evolution system of measures and construct global-in-time dynamics. Moreover, we show that the white noise measure with variance 1 + t 1+t is an evolution system of measures for SKdV with the white noise initial data.

The stochastic Korteweg–de Vries equation with loss and non-uniformity terms

The stochastic Korteweg–de Vries equation with loss and non-uniformity terms

Numerical conservation issues for the stochastic Korteweg–de Vries equation

In this paper, we focus on structure-preserving issues for the numerical solution of the stochastic Korteweg–de Vries equation, via stochastic ϑ-methods. It is well-known that the aforementioned model exhibits invariant laws along its exact dynamics. Here, our goal is to analyze whether such invariant laws are also reproduced along the numerical dynamics provided by stochastic ϑ-methods. Furthermore, we are also interested in rigorously studying the characterization of such invariant laws along numerical solutions of this model, with respect to the growth of the stochasticity parameter ɛ. At this purpose, the so-called ɛ-expansion of the exact solution to the aforementioned equation will be performed. Numerical results confirming the effectiveness of our analysis are also provided.

Exact solutions of stochastic KdV equation with conformable derivatives in white noise environment

In this article, we have considered Wick-type stochastic Korteweg de Vries (KdV) equation with conformable derivatives. By the help of white noise analysis, Hermit transform and extended G?/G- expansion method, we have obtained exact travelling wave solutions of KdV equation with conformable derivatives. We have applied the inverse Hermit transform for stochastic soliton solutions and then we have shown how stochastic solutions can be presented as Brownian motion functional solutions by an application example.

Exact solutions of stochastic fractional Korteweg de–Vries equation with conformable derivatives

We deal with the Wick-type stochastic fractional Korteweg de–Vries (KdV) equation with conformable derivatives. With the aid of the Exp-function method, white noise theory, and Hermite transform, we produce a novel set of exact soliton and periodic wave solutions to the fractional KdV equation with conformable derivatives. With the help of inverse Hermite transform, we get stochastic soliton and periodic wave solutions of the Wick-type stochastic fractional KdV equation with conformable derivatives. Eventually, by an application example, we show how the stochastic solutions can be given as Brownian motion functional solutions.

Numerical Solution of Time Fractional Stochastic Korteweg–de Vries Equation via Implicit Meshless Approach

In this article, a numerical scheme is implemented to approximate the solution of time fractional stochastic Korteweg–de Vries (KdV) equation. The structure of the proposed method is such that it first employ finite difference technique to transform the time fractional stochastic KdV equation into elliptic stochastic differential equations (SDEs). Then resulting elliptic SDEs has been estimated via a meshless method based on radial basis functions (RBFs). It should be mentioned that every RBFs with sufficient smoothness can be applied. In this paper, we use Gaussian RBFs which are infinitely smoothness to approximate the functions in the resulting elliptic SDEs. The most important advantage of proposed method respect to traditional numerical method is that the noise terms are directly simulated at the collocation points in each step time. Finally, some test problems are presented to investigate the performance and accuracy of the new method.

Averaging principle for stochastic Korteweg-de Vries equation

Averaging principle is a powerful tool for studying qualitative analysis of nonlinear dynamical systems. In this paper, we will establish an averaging principle for stochastic Korteweg-de Vries equation under a general averaging condition. With the help of this averaging principle, we can establish an effective approximation for the solution of stochastic Korteweg-de Vries equation, this can tell us the asymptotic behavior of the solution and make the interaction between nonlinearity, uncertainty and multiple scales more clear. In order to obtain this averaging principle, we need to establish the smoothing effect for the third order operator.

Weak convergence rates for stochastic evolution equations and applications to nonlinear stochastic wave, HJMM, stochastic Schrödinger and linearized stochastic Korteweg–de Vries equations

We establish weak convergence rates for noise discretizations of a wide class of stochastic evolution equations with non-regularizing semigroups and additive or multiplicative noise. This class covers the nonlinear stochastic wave, HJMM, stochastic Schr\"odinger and linearized stochastic Korteweg-de Vries equation. For several important equations, including the stochastic wave equation, previous methods give only suboptimal rates, whereas our rates are essentially sharp.

Existence of invariant measures for the stochastic damped KdV equation

We address the long time behavior of solutions of the stochastic Korteweg-de Vries equation $ du + (\partial^3_x u +u\partial_x u +\lambda u)dt = f dt+\Phi dW_t$ on ${\mathbb R}$ where $f$ is a deterministic force. We prove that the Feller property holds and establish the existence of an invariant measure. The tightness is established with the help of the asymptotic compactness, which is carried out using the Aldous criterion.

The stochastic Korteweg–de Vries equation on a bounded domain

This paper considers the stochastic Korteweg–de Vries equation on a bounded domain. The existence of a weak martingale solution is discussed by the Galerkin’s approximation and compactness method. Based on this result, we establish the Unique Continuation Property for the stochastic Korteweg–de Vries equation.

Periodic solutions of Wick-type stochastic Korteweg–de Vries equations

Nonlinear stochastic partial differential equations have a wide range of applications in science and engineering. Finding exact solutions of the Wick-type stochastic equation will be helpful in the theories and numerical studies of such equations. In this paper, Kudrayshov method together with Hermite transform is implemented to obtain exact solutions of Wick-type stochastic Korteweg–de Vries equation. Further, graphical illustrations in two- and three-dimensional plots of the obtained solutions depending on time and space are also given with white noise functionals.

-Insensitizing controls for the linear stochastic Korteweg-de Vries equation

This paper is addressed to showing the existence of -insensitizing controls for the linear stochastic Korteweg–de Vries equation. We solve this problem by reducing the original problem to a controllability problem.

Stochastic Korteweg-de Vries equation driven by fractional Brownian motion

We consider the Cauchy problem for the Korteweg-de Vries equation driven by a cylindrical fractional Brownian motion (fBm) in this paper. With Hurst parameter $H\geq\frac{7}{16}$ of the fBm, we obtain the local existence results with initial value in classical Sobolev spaces $H^s$ with $s\geq -\frac{9}{16}$. Furthermore, we give the relation between the Hurst parameter $H$ and the index $s$ to the Sobolev spaces $H^s$, which finds out the regularity between the driven term fBm and the initial value for the stochastic Korteweg-de Vries equation.

CARLEMAN ESTIMATE AND UNIQUE CONTINUATION PROPERTY FOR THE LINEAR STOCHASTIC KORTEWEG–DE VRIES EQUATION

AbstractIn this paper, we obtain the well posedness of the linear stochastic Korteweg–de Vries equation by the Galerkin method, and then establish the Carleman estimate, leading to the unique continuation property (UCP) for the linear stochastic Korteweg–de Vries equation. This UCP cannot be obtained from the classical Holmgren uniqueness theorem.

Dependence of polynomial chaos on random types of forces of KdV equations

In this study, one-dimensional stochastic Korteweg–de Vries equation with uncertainty in its forcing term is considered. Extending the Wiener chaos expansion, a numerical algorithm based on orthonormal polynomials from the Askey scheme is derived. Then dependence of polynomial chaos on the distribution type of the random forcing term is inspected. It is numerically shown that when Hermite (Laguerre or Jacobi) polynomial chaos is chosen as a basis in the Gaussian (Gamma or Beta, respectively) random space for uncertainty, the solution to the KdV equation converges exponentially. If a proper polynomial chaos is not used, however, the solution converges with slower rate.

Periodic stochastic Korteweg–de Vries equation with additive space-time white noise

We prove the local well-posedness of the periodic stochastic Korteweg-de Vries equation with the additive space-time white noise. In order to treat low regularity of the white noise in space, we consider the Cauchy problem in the Besov-type space bb s∞(T) for s = 1 +, p = 2+ such that sp < 1. In establishing the local well-posedness, we use a variant of the Bourgain space adapted to b s∞(T) and establish a nonlinear estimate on the second iteration on the integral formulation. The deterministic part of the nonlinear estimate also yields the local well-posedness of the deterministic KdV in M(T), the space of finite Borel measures on T.

Explicit solutions of a generalized Wick-type stochastic Korteweg–de Vries equation

With the help of Hermite transformation, we apply the exp-function method to solve a generalized Wick-type stochastic Korteweg–de Vries equation in the white noise environment. Some Wick-type stochastic exact solutions are obtained.

Three types of exact solutions to Wick-type generalized stochastic Korteweg-de Vries equation

A Wick-type generalized stochastic Korteweg-de Vries equation is researched. By means of Hermite transformation, white noise theory and Riccati equation mapping method, three types of exact solutions to the generalized stochastic Korteweg-de Vries equation, which include the functional solutions of hyperbolic-exponential type, trigonometric-exponential type and exponential type, are derived.

Numerical realizations of solutions of the stochastic KdV equation

We investigate simulations of exact solutions of the stochastic Korteweg–deVries equation under additive noise. We compare the expectation values of the exact solutions to theoretical expectation values and to the numerical simulations of the stochastic Korteweg–deVries equation with and without damping. We find on average the diffused soliton vanishes long before the typically reported asymptotic limit.