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Instability of solitons for the critical generalized Korteweg-de Vries equation. (English) Zbl 0985.35071

The paper considers stability of soliton solutions to the generalized KdV equation of the form \[ u_t + u_{xxx} +(u^5)_x = 0. \] This model is called a critical one, as, with power 5 of the nonlinear term in the equation, it exactly falls on the boundary between equations of the KdV type with weaker nonlinearities, which give rise to stable solitons, and equations with stronger nonlinearities, in which strong collapse (formation of a singularity in finite time) takes place, and solitons are unstable, so that a small perturbation provokes the onset of collapse of a soliton. Numerical simulations, reported in earlier works, demonstrated instability of solitons in the critical equation. In the critical case, the equation gives rise to weak collapse (i.e., the onset of the collapse depends on the initial configuration). In accordance with this, numerical simulations show that the instability of solitons in the critical equation leads to formation of a singularity. In this paper, the instability of solitons in the critical KdV equation is proved rigorously by means of variational estimates of the virial type.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
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