Cauchy problem for the Korteweg-de Vries equation in the case of a nonsmooth unbounded initial function |
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Authors: | A V Faminskii |
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Institution: | 1.Russian Peoples’ Friendship University,Moscow,Russia |
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Abstract: | In the strip П = (?1, 0) × ?, we establish the existence of solutions of the Cauchy problem for the Korteweg-de Vries equation u t + u xxx + uu x = 0 with initial condition either 1) u(?1, x) = ?xθ(x), or 2) u(?1, x) = ?xθ(?x), where θ is the Heaviside function. The solutions constructed in this paper are infinitely smooth for t ∈ (?1, 0) and rapidly decreasing as x → +∞. For the case of the first initial condition, we also establish uniqueness in a certain class. Similar special solutions of the KdV equation arise in the study of the asymptotic behavior with respect to small dispersion of the solutions of certain model problems in a neighborhood of lines of weak discontinuity. |
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