A direct proof shows the maximal amplitude of finite-gap focusing mKdV solutions equals the sum of imaginary parts of upper-half-plane square roots of the invariant polynomial roots, with an analogous result for a bounded class of defocusing solutions.
Maximum amplitudes of finite-gap solutions for the focusing Nonlinear Schr\"odinger Equation
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abstract
In this paper we prove that the maximum amplitude of a finite-gap solution to the focusing Nonlinear Schr\"{o}dinger equation with given spectral bands does not exceed half of the sum of the length of all the bands. This maximum will be attained for certain choices of the initial phases. A similar result is also true for the defocusing Nonlinear Schr\"{o}dinger equation.
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2026 1verdicts
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Sharp Upper Bound for Amplitudes of Finite-Gap Solutions of the Modified Korteweg-de Vries Equation
A direct proof shows the maximal amplitude of finite-gap focusing mKdV solutions equals the sum of imaginary parts of upper-half-plane square roots of the invariant polynomial roots, with an analogous result for a bounded class of defocusing solutions.