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Register a new userPower series solution of the modified KdV equation
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Authors
Tu Nguyen
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We prove local-wellposedness of the mKdV equation in $mathcal{F}L^{s,p}$ spaces using the new method of M. Christ.
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We study well-posedness of the complex-valued modified KdV equation (mKdV) on the real line. In particular, we prove local well-posedness of mKdV in modulation spaces $M^{2,p}_{s}(mathbb{R})$ for $s ge frac14$ and $2leq p < infty$. For $s < frac 14$, we show that the solution map for mKdV is not locally uniformly continuous in $M^{2,p}_{s}(mathbb{R})$. By combining this local well-posedness with our previous work (2018) on an a priori global-in-time bound for mKdV in modulation spaces, we also establish global well-posedness of mKdV in $M^{2,p}_{s}(mathbb{R})$ for $s ge frac14$ and $2leq p < infty$.
The linearized Korteweg-De Vries equation can be written as a Hamilton-like system. However, the Hamilton energy depends on the time, and is a nonsymmetric operator on $L^2({bf R})$. By performing suitable unitary transforms on the Hamilton energy, we can reduce this operator into one that is not independent on the time but nonsymmetric. In this study, we consider the $L^2$-stability issues and smoothing estimates for this operator, and prove that it has no eigenvalues.
In this study we examine the energy transfer mechanism during the nonlinear stage of the Modulational Instability (MI) in the modified Korteweg-de Vries equation. The particularity of this study consists in considering the problem essentially in the Fourier space. A dynamical energy cascade model of this process originally proposed for the focusing NLS-type equations is transposed to the mKdV setting using the existing connections between the KdV-type and NLS-type equations. The main predictions of the D-cascade model are outlined and thoroughly discussed. Finally, the obtained theoretical results are validated by direct numerical simulations of the mKdV equation using the pseudo-spectral methods. A general good agreement is reported in this study. The nonlinear stages of the MI evolution are also investigated for the mKdV equation.
We study the spectral (linear) stability and orbital (nonlinear) stability of the elliptic solutions for the focusing modified Korteweg-de Vries (mKdV) equation with respect to subharmonic perturbations and construct the corresponding breather solutions to exhibit the unstable or stable dynamic behavior. The elliptic function solutions of mKdV equation and the fundamental solutions of Lax pair are exactly represented by using the theta function. Based on the `modified squared wavefunction (MSW) method, we construct all linear independent solutions of the linearized KdV equation, and then provide a necessary and sufficient condition of the spectral stability for the elliptic function solutions with respect to subharmonic perturbations. In the case of spectrum stable, the orbital stability of the elliptic function solutions with respect to subharmonic perturbations is established under a suitable Hilbert space. Using Darboux-Backlund transformation, we construct the breather solutions to exhibit the unstable or stable dynamic behavior. Through analyzing the asymptotical behavior, we find the breather solution under the $mathrm{cn}$-background is equivalent to the elliptic function solution adding a small perturbation as $ttopminfty$.
We address the long time behavior of solutions of the stochastic Korteweg-de Vries equation $ du + (partial^3_x u +upartial_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.
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