Slow forward scattering. More...

#include "fnft__akns_discretization.h"
#include <stdio.h>
#include <string.h>
#include "fnft__errwarn.h"
#include "fnft__misc.h"

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Functions
FNFT_INT	fnft__akns_scatter_matrix (FNFT_UINT const D, FNFT_COMPLEX const const q, FNFT_COMPLEX const const r, FNFT_REAL const eps_t, FNFT_UINT const K, FNFT_COMPLEX const const lambda, FNFT_COMPLEX const result, FNFT_INT *const W, fnft__akns_discretization_t discretization, fnft__akns_pde_t const PDE, FNFT_UINT const vanilla_flag, FNFT_UINT const derivative_flag)
	Computes the scattering matrix and its derivative.

FNFT_INT	akns_scatter_bound_states (FNFT_UINT const D, FNFT_COMPLEX const const q, FNFT_COMPLEX const const r, FNFT_REAL const const T, FNFT_UINT const K, FNFT_COMPLEX const const bound_states, FNFT_COMPLEX const a_vals, FNFT_COMPLEX const aprime_vals, FNFT_COMPLEX const b_vals, FNFT_INT Ws, fnft__akns_discretization_t const discretization, fnft__akns_pde_t const PDE, FNFT_UINT const vanilla_flag, FNFT_UINT const skip_b_flag)
	Computes \(a(\lambda)\), \( a'(\lambda) = \frac{\partial a(\lambda)}{\partial \lambda}\) and \(b(\lambda)\) for complex values \(\lambda\) assuming that they are very close to the true bound-states.

Detailed Description

Slow forward scattering.

Function Documentation

◆ akns_scatter_bound_states()

FNFT_INT akns_scatter_bound_states	(	FNFT_UINT const	D,
		FNFT_COMPLEX const *const	q,
		FNFT_COMPLEX const *const	r,
		FNFT_REAL const *const	T,
		FNFT_UINT const	K,
		FNFT_COMPLEX const *const	bound_states,
		FNFT_COMPLEX *const	a_vals,
		FNFT_COMPLEX *const	aprime_vals,
		FNFT_COMPLEX *const	b_vals,
		FNFT_INT *	Ws,
		fnft__akns_discretization_t const	discretization,
		fnft__akns_pde_t const	PDE,
		FNFT_UINT const	vanilla_flag,
		FNFT_UINT const	skip_b_flag
	)

Computes \(a(\lambda)\), \( a'(\lambda) = \frac{\partial a(\lambda)}{\partial \lambda}\) and \(b(\lambda)\) for complex values \(\lambda\) assuming that they are very close to the true bound-states.

The function performs slow direct scattering and is primarily based on the references

Boffetta and Osborne, " Computation of the direct scattering transform for the nonlinear Schroedinger equation," J. Comput. Phys. 102(2), 1992.
Chimmalgi, Prins and Wahls, " Fast Nonlinear Fourier Transform Algorithms Using Higher Order Exponential Integrators," IEEE Access 7, 2019.
Medvedev, Vaseva, Chekhovskoy and Fedoruk, " Exponential fourth order schemes for direct Zakharov-Shabat problem," Optics Express, vol. 28, pp. 20–39, 2020.
Prins and Wahls, " Soliton Phase Shift Calculation for the Korteweg–De Vries Equation," IEEE Access, vol. 7, pp. 122914–122930, July 2019.

Parameters

[in]	D	Number of samples
[in]	q	Array of length D, contains samples \( q_n\) for \(n=0,1,\dots,D-1\) in ascending order (i.e., \( q_0, q_1, \dots, q_{D-1} \)). The values should be specifically precalculated based on the chosen discretization.
[in]	r	Array of length D, contains samples \( r_n\) for \(n=0,1,\dots,D-1\) in ascending order (i.e., \( r_0, r_1, \dots, r_{D-1} \)). The values should be specifically precalculated based on the chosen discretization.
[in]	T	Array of length 2, contains the position in time of the first and of the last sample. It should be T[0]<T[1].
[in]	K	Number of bound-states.
[in]	bound_states	Array of length K, contains the bound-states \(\lambda\).
[out]	a_vals	Array of length K, contains the values of \(a(\lambda)\).
[out]	aprime_vals	Array of length K, contains the values of \( a'(\lambda) = \frac{\partial a(\lambda)}{\partial \lambda}\).
[out]	b_vals	Array of length K, contains the values of \(b(\lambda)\). The \(b(\lambda)\) are calculated using the criterion from Prins and Wahls, " Soliton Phase Shift Calculation for the Korteweg–De Vries Equation,".
[in,out]	Ws	Pass an array of size K. Upon exit, it contains scaling factors that arise due to an internal normalization of the scattering process (to deal with potential overflow issues). The returned values for a and a_prime still have to be multiplied with corresponding power of two (i.e. POW(2, Ws[i])) to obtain the final values. Note that this is not required for the values of b. No normalization is carried out if NULL is passed.
[in]	discretization	The type of discretization to be used. Should be of type fnft__akns_discretization_t. Not all akns_discretization_t discretizations are supported. Check fnft_nse_discretization_t for list of supported types.
[in]	PDE	The partial differential equation for which the calculation has to be done. Should be of type fnft__akns_pde_t.
[in]	vanilla_flag	For calculations for the KdV equation, pass 1 for the original mapping to the AKNS framework with r=-1. Pass 0 for the alternative mapping with q=-1. Unused for NSE.
[in]	skip_b_flag	If set to 1 the routine will not compute \(b(\lambda)\).

Returns: FNFT_SUCCESS or one of the FNFT_EC_... error codes defined in fnft_errwarn.h.

◆ fnft__akns_scatter_matrix()

FNFT_INT fnft__akns_scatter_matrix	(	FNFT_UINT const	D,
		FNFT_COMPLEX const *const	q,
		FNFT_COMPLEX const *const	r,
		FNFT_REAL const	eps_t,
		FNFT_UINT const	K,
		FNFT_COMPLEX const *const	lambda,
		FNFT_COMPLEX *const	result,
		FNFT_INT *const	W,
		fnft__akns_discretization_t	discretization,
		fnft__akns_pde_t const	PDE,
		FNFT_UINT const	vanilla_flag,
		FNFT_UINT const	derivative_flag
	)

Computes the scattering matrix and its derivative.

The function computes the scattering matrix and the derivative of the scattering matrix with respect to \(\lambda\). The function performs slow direct scattering and is primarily based on the reference Boffetta and Osborne (J. Comput. Physics 1992 ).

Parameters

[in]	D	Number of samples
[in]	q	Array of length D, contains samples \( q_n\) for \(n=0,1,\dots,D-1\) in ascending order (i.e., \( q_0, q_1, \dots, q_{D-1} \)). The values should be specifically precalculated based on the chosen discretization.
[in,out]	r	Array of length D, contains samples \( r_n\) for \(n=0,1,\dots,D-1\) in ascending order (i.e., \( r_0, r_1, \dots, r_{D-1} \)). The values should be specifically precalculated based on the chosen discretization.
[in]	eps_t	Step-size, eps_t \(= (T[1]-T[0])/(D-1) \).
[in]	K	Number of values of \(\lambda\).
[in]	lambda	Array of length K, contains the values of \(\lambda\).
[out]	result	Array of length K8 or K4, If derivative_flag=0 returns [S11(lambda[0]) S12(lambda[0]) S21(lambda[0]) S22(lambda[0]) S11(lambda[1]) ... S22(lambda[K-1]] in result where S = [S11, S12; S21, S22] is the scattering matrix computed using the chosen discretization. If derivative_flag=1 returns [S11 S12 S21 S22 S11' S12' S21' S22'] in result where S11' is the derivative of S11 w.r.t to \(\lambda\). Should be preallocated with size 4K or 8K accordingly.
[in,out]	W	Pass an array of size K. Upon exit, it contains scaling factors that arise due to an internal normalization of the scattering matrix (to deal with potential overflow issues). The result has to be scaled by the power of 2 to obtain the final values. No normalization is carried out if NULL is passed.
[in]	discretization	The type of discretization to be used. Should be of type fnft__akns_discretization_t. Not all akns_discretization_t discretizations are supported. Check fnft__akns_discretization_t for list of supported types.
[in]	PDE	Partial differential equation of type fnft__akns_pde_t.
[in]	vanilla_flag	For calculations for the KdV equation, pass 1 for the original mapping to the AKNS framework with r=-1. Pass 0 for the alternative mapping with q=-1. Unused for NSE.
[in]	derivative_flag	Should be set to either 0 or 1. If set to 1 the derivatives [S11' S12' S21' S22'] are calculated. result should be preallocated with size 8*K if flag is set to 1.

Returns: FNFT_SUCCESS or one of the FNFT_EC_... error codes defined in fnft_errwarn.h.

Functions

Detailed Description

Function Documentation

◆ akns_scatter_bound_states()

◆ fnft__akns_scatter_matrix()