The small dispersion limit of the Korteweg-de Vries equation. I.

*(English)*Zbl 0532.35067In a series of three papers, the authors have analyzed the bahavior of solutions \(u(x,t;\epsilon)\) of the equation \(u_ t-6uu_ x+\epsilon^ 2u_{xxx}=0\) as \(\epsilon\to 0\) while the initial values are fixed. Only nonpositive initial data were considered; in that case the limiting reflection coefficient vanishes. It is known from computer studies that for t greater than a critical time, independent of \(\epsilon\), dependent only on the initial data, \(u(x,t;\epsilon)\) becomes oscillatory as \(\epsilon\to 0\). The wavelength of these oscillations is of the order \(0(\epsilon)\), and their amplitude is independent of \(\epsilon\). This indicates that \(\lim u(x,t;\epsilon\to 0)\) exists only in the weak sense. This paper represents the first part of the work. The paper was organized as follows: In Section 1, the direct scattering problem for given initial data, assumed for simplicity to have a single local minimum, was solved asymptotically. In Section 2, the Kay-Moses explicit solution of the reflectionless inverse problem was used to carry out the limit \(\epsilon\to 0\). The authors have shown that \(\bar u(x,t)=\lim u(x,t;\epsilon \to 0)\) exists in the sense of weak convergence in \(L_ 2(R)\) with respect to x, and that the weak limit \(\bar u\) can be described as \(\bar u=\partial_{xx}Q^*\). The function \(Q^*(x,t)\) is determined by solving a quadratic programming problem
\[
Q^*(x,t)=\min_{0\leq \psi \leq \phi}Q(\psi;x,t).
\]
Here \(Q(\psi;x,t)\) is a quadratic functional of \(\psi\), which depends linearly on the parameters x and t, while the function \(\phi\) is determined by the initial data. In Section 3 the authors have shown that Q is continuous in a weak sequential topology, and that the space of admissible functions is compact in that topology. They further showed that Q is a strictly convex function; since the admissible functions form a convex set, this implies not only that the minimum of Q is taken on at a unique function, but that this function is the only one which satisfies variational conditions. The variational conditions were then converted to a Riemann-Hilbert problem.

Reviewer: L.-Y.Shih

##### MSC:

35Q99 | Partial differential equations of mathematical physics and other areas of application |

35B40 | Asymptotic behavior of solutions to PDEs |

47A40 | Scattering theory of linear operators |

35P25 | Scattering theory for PDEs |

35A15 | Variational methods applied to PDEs |

##### Keywords:

KdV equation; nonpositive initial data; weak dispersion limit; scattering transform method; minimum problem
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\textit{P. D. Lax} and \textit{C. D. Levermore}, Commun. Pure Appl. Math. 36, 253--290 (1983; Zbl 0532.35067)

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