On the proximal dynamics between integrable and non-integrable members of a generalized Korteweg-de Vries family of equations

TL;DR

This work addresses how closely non-integrable generalized KdV (gKdV) dynamics track the integrable KdV flow. By proving a size estimate for local gKdV solutions in Sobolev spaces and deriving L^2 and $H^s$ proximity bounds to KdV, the authors show that small-amplitude initial data yield dynamics that remain within $\mathcal{O}(\max\{\varepsilon^2,\varepsilon^{k+1}\})$ over long times, with linear-in-time growth of the deviation. In the power nonlinearity setting, the analysis simplifies when $A_k=0$, and numerical simulations for one- and two-soliton data corroborate the theory, including a caustic-rotation effect that aligns gKdV solitons with a rescaled KdV soliton, extending near-integrable behavior to longer times. The results provide a rigorous basis for the persistence of coherent, soliton-like structures under broad non-integrable perturbations and suggest directions for extending these insights to periodic data, other nonlinearities, and soliton-gas regimes.

Abstract

The distance between the solutions to the integrable Korteweg-de Vries (KdV) equation and a broad class of non-integrable generalized KdV (gKdV) equations is estimated in appropriate Sobolev spaces. Special cases of the latter family of equations include the standard gKdV equation with a power nonlinearity, as well as weakly nonlinear perturbations of the KdV equation. The distance estimates are established for initial data and nonlinearity parameters of arbitrary size. A crucial step for their derivation is the proof of a size estimate for local solutions to the gKdV family of equations, which is linear in the norm of the initial data. As a result, the distance estimates themselves predict that the dynamics between the gKdV and KdV equations remains close over long times for initial amplitudes even close to unity, while for larger amplitudes they predict an explicit rate of deviation of the dynamics between the compared equations. These theoretical results are illustrated via numerical simulations in the case of one-soliton and two-soliton initial data, which are in an excellent agreement with the theoretical predictions. Importantly, in the case of a power nonlinearity and large solitonic initial data, the deviation between the integrable and non-integrable dynamics can be drastically reduced upon incorporating suitable rotation effects by means of a rescaled KdV equation. As a result, the integrable dynamics stemming from the rescaled KdV equation may persist in the gKdV family of equations over remarkably long timescales.

Figures (7)

• Figure 1: The solution $U(x, t)$ of the gKdV equation \ref{['gkdv']} with monomial nonlinearity according to \ref{['gkdvp']}. Top left: The supercritical case $k=5$ supplemented with KdV one-soliton initial data $u_{\text{s}}(x)$ with $c=0.01$ and $x_0=0$. Top right: The critical case $k=4$ supplemented with mKdV one-soliton initial data $U_{\text{s}}(x)$ with $c=0.004$ and $x_0=0$. Bottom: The corresponding time evolutions of the norms $\left\| \Delta(t) \right\|_{\mathcal{X}}$ of the difference between the above gKdV solutions and the corresponding KdV and mKdV one-solitons for $\mathcal{X}= H_x^1$, $L_x^2$ and $L_x^{\infty}$.
• Figure 2: Left and middle: Snapshots comparing the profile of the solution of gKdV with power nonlinearity \ref{['kdvk']} for $k=3$ (solid blue) against the profile of the solution of the integrable KdV \ref{['kdv']} with (red dashed). Right: The top panel shows the contour plot of the evolution of the initial datum $u_{\text{s}}(x)$ in \ref{['e:kdvsoliton']} with $c=0.4$ and $x_0=0$ under gKdV. The dashed black line is the caustic of the analytical KdV soliton. The bottom panel presents the evolution of the distance norms between the solutions $\mathcal{X}= H_x^s$, $s=0,1,2$, and $L_x^{\infty}$.
• Figure 3: Left and middle: Snapshots comparing the profiles of the solution of the gKdV equation with power nonlinearity \ref{['kdvk']} for $k=3$ (solid blue) against the profiles of the soliton \ref{['e:rsoliton']} of the rescaled integrable KdV \ref{['resckdv']} with $\nu=0.47$ (red dashed). Right: The top panel shows the contour plot of the evolution of the initial datum \ref{['e:kdvsoliton']} with $c=0.4$ and $x_0=0$ under gKdV. The dashed black line is the caustic of the rescaled soliton \ref{['e:rsoliton']} with $\nu=0.47$. The bottom panel presents the evolution of the distance measured in the spaces $\mathcal{X}= H_x^s$, $s=0,1,2$, and $L_x^{\infty}$.
• Figure 4: Left and middle: Snapshots comparing the profiles of the gKdV solution in the case of the power nonlinearity \ref{['kdvk']} with $k=4$ (solid blue) against the profiles of the soliton \ref{['e:rsoliton']} of the rescaled integrable KdV \ref{['resckdv']} with $\nu=0.41$ (red dashed). Right: The top panel shows the contour plot of the evolution of the initial condition \ref{['e:kdvsoliton']} with $c=0.4$, $x_0=0$ under gKdV. The dashed black line is the caustic of the rescaled soliton \ref{['e:rsoliton']} with $\nu=0.41$. The bottom panel presents the evolution of norms of the distance between the solutions $\mathcal{X}= H_x^s$, $s=0,1,2$, and $L_x^{\infty}$.
• Figure 5: Top: The solution to the gKdV equation \ref{['gkdv']} with $F(U)=U+\delta U^3$ and $\delta=0.02$, supplemented with the KdV one-soliton initial datum $u_{\text{s}}(x)$ in \ref{['e:kdvsoliton']} with $c=0.3$ and $x_0=0$. Bottom: Spatiotemporal evolution of the field $U(x,t)$ for the gKdV equation with $F(U)=U^2+\delta U^4$ and$\delta=0.01$, supplemented with the mKdV one-soliton initial datum $U_{\text{s}}(x)$ in \ref{['e:kdvsoliton']} with $c=0.2$ and $x_0=0$. Left: The initial profiles. Middle: The solutions at $t=500$. Right: The time evolution of the norms $\left\| \Delta(t) \right\|_{\mathcal{X}}$ with $\mathcal{X}=H_x^1$, $L_x^2$ and $L_x^{\infty}$.

Theorems & Definitions (5)

• Theorem 1
• Corollary 1: Power nonlinearity proximity
• Remark 1
• Theorem 2
• Corollary 2: Power nonlinearity size estimate