Abstract
In this chapter we present two methods for computing the NLS solitary waves.
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Notes
- 1.
See sample Matlab code in Fig. 28.1.
- 2.
In this section \(R_{(j)}=R_{(j)}(\mathbf{x})\) denotes the \(j\)th iteration. Thus, \(R_{(0)}\) is the initial guess, \(R_{(1)}\) is the first iteration, etc.
- 3.
The above informal argument explains why renormalization prevents the divergence to zero or infinity. It does not explain, however, why these iterations generically converge to the ground state and not to an excited state.
- 4.
The superscript \(ns\) (non-spectral) emphasizes that the integral quantities are different from those in the spectral case.
- 5.
Since \(R_B(r)\) is even, all odd derivatives of \(R_B\) must vanish at \(r=0\).
- 6.
If the solitary wave is unstable, this agreement will only persist for a “short” distance.
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Fibich, G. (2015). Computation of Solitary Waves. In: The Nonlinear Schrödinger Equation. Applied Mathematical Sciences, vol 192. Springer, Cham. https://doi.org/10.1007/978-3-319-12748-4_28
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DOI: https://doi.org/10.1007/978-3-319-12748-4_28
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