Abstract
In this chapter, we shall consider questions of the stability of solitary waves. As it is noted in the Introduction, usually solitary waves of the KdVE and NLSE are unstable with respect to distances of standard spaces of functions as Lebesgue or Sobolev spaces and it is natural to study the stability of solitary waves vanishing as |x| → ∞ with respect to the distance p in the case of the KdVE and to d for the NLSE (for definitions of p and d see Introduction or Section 3.1). We also recall that we named a solitary wave u(x, t), where u(x, t) = φ(w, x − wt) in the case of the KdVE and u(x, t) = e iwtφ (w, x) for the NLSE, a kink if φ′x(w, x) ≠ 0 for all x ∈ R and a soliton-like solution if there is a unique x 0 ∈ R such that φ ′ x (w, x 0) = 0, x 0 is a point of extremum of φ(w, x) as a function of the argument x and φ(− ∈) = φ(+∈).
Keywords
- Cauchy Problem
- Solitary Wave
- Differentiable Function
- Finite Interval
- Solitary Wave Solution
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© 2001 Springer-Verlag Berlin Heidelberg
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(2001). Stability of solutions. In: Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory. Lecture Notes in Mathematics, vol 1756. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45276-1_5
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DOI: https://doi.org/10.1007/3-540-45276-1_5
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-41833-7
Online ISBN: 978-3-540-45276-8
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