Coding of Nonlinear States for NLS-Type Equations with Periodic Potential

Alfimov, G. L.; Avramenko, A. I.

doi:10.1007/978-3-319-02057-0_2

Cited by 2 publications

(2 citation statements)

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“…In the case of the periodic potentials, the blowing up solutions are also generic [16]. This fact gives rise to the classification of nonlinear structures in terms of bi-infinite symbolic sequences [17,16,39].…”

Section: Singular Solutionsmentioning

confidence: 99%

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Global search for localised modes in scalar and vector nonlinear Schrödinger-type equations

Alfimov

Barashenkov

Fedotov

et al. 2019

Physica D: Nonlinear Phenomena

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We present a new approach for search of coexisting classes of localised modes admitted by the repulsive (defocusing) scalar or vector nonlinear Schrödinger-type equations. The approach is based on the observation that generic solutions of the corresponding stationary system have singularities at finite points on the real axis. We start with establishing conditions on the initial data of the associated Cauchy problem that guarantee the formation of a singularity. Making use of these sufficient conditions, we identify the bounded, nonsingular, solutions -and then classify them according to their asymptotic behaviour.To determine the bounded solutions, a properly chosen space of initial data is scanned numerically. Due to asymptotic or symmetry considerations, we can limit ourselves to a one-or two-dimensional space. For each set of initial conditions we compute the distances X ± to the nearest forward and backward singularities; large X + or X − indicate the proximity to a bounded solution. We illustrate our method with the Gross-Pitaevskii equation with a PT -symmetric complex potential, a system of coupled Gross-Pitaevskii equations with real potentials, and the Lugiato-Lefever equation with normal dispersion.

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Section: Singular Solutionsmentioning

confidence: 99%

“…We impose the initial condition at the point x = x 0 , with a large negative x 0 . The value of u(x 0 ) is set to (39) and the value of u x (x 0 ) to the dominant term in the derivative of (39):…”

Section: Asymptotics-based Approachmentioning

confidence: 99%