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On the uniqueness of traveling waves in perturbed Korteweg-de Vries equations

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Abstract

We consider the perturbed Hamiltonian system

$$u_t = \partial _x \delta H(u) - \varepsilon P(u)$$

with\(H(u) = \int {\left( {\frac{1}{2}u_x^2 + \frac{1}{3}u^3 } \right)} dx\). We prove for various perturbationsP(u) that there is a unique bifurcation point of traveling wave solutions on the curve of relative equilibriau γ such that

$$H(u_\gamma ) = \mathop {\min }\limits_u \left\{ {H(u)\left| { \int {u^2 } = \gamma } \right.} \right\}.$$

As an additional result, the curve γ→H(u γ) is proven to be concave.

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Authors and Affiliations

  1. Faculty of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE, Enschede, The Netherlands

    Gianne Derks & Stephan van Gils

Authors
  1. Gianne Derks
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  2. Stephan van Gils
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Derks, G., van Gils, S. On the uniqueness of traveling waves in perturbed Korteweg-de Vries equations. Japan J. Indust. Appl. Math. 10, 413–430 (1993). https://doi.org/10.1007/BF03167282

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  • DOI: https://doi.org/10.1007/BF03167282

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