# Relay-Zone Technique for Numerical Boundary Treatments in Simulating Dark Solitons

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## Abstract

To simulate dark solitons in the defocusing nonlinear Schrödinger equation, we introduce a relay-zone technique, by alternately using a Robin boundary condition to treat the nonzero far field, and a derivative boundary condition to match the dark soliton. Numerical tests and comparisons demonstrate the effectiveness of the proposed boundary treatment. Stability and interaction of dark solitons are also studied.

## Keywords

Artificial boundary condition Nonlinear Schrödinger equation Dark soliton## Introduction

Solitons have drawn attention in many physical subjects such as fluid mechanics [29], plasma physics [45], solid state physics [1, 22], atomic physics [20, 34], and Bose–Einstein condensate [31, 38, 46]. First found by Russell [41], solitons are defined as localized waves that propagate without change of shape and velocity, and are stable against mutual collisions [49]. Different from a solitary wave, when any number of solitons interact they do not change form, and the only outcome of the interaction is a phase change [11, 37].

*x*is space,

*t*is time, \(\psi \) is a complex-valued wave function, and \(\beta \) is a dimensionless constant. With an initial condition

To simulate NLSE (1)–(2) on a bounded computational domain, the numerical boundaries need to be carefully treated. Simple Dirichlet/Neumann/Robin boundary conditions or the periodic boundary condition usually cause large errors on the boundary, particularly when a soliton passes across. To avoid such errors, more involved boundary treatments are required.

There are lots of high-order boundary conditions designed for the linear Schrödinger equation [\(\beta =0\) in (1)], including exact boundary conditions such as DtN map [10] and NtD map [36]. More precisely, based on finite difference schemes, there are two boundary conditions for temporally semi-discrete boundary conditions which are designed via Z transform [8, 43, 44, 51]. Based on Crank–Nicolson finite difference scheme, there is also a fully discrete transparent boundary condition designed by Arnold [10]. They are usually nonlocal in time and contain time-dependent pseudo-different operators. The operator approximations, which involve time fractional derivatives and integrals, are usually used in practical computations. There are also some local boundary conditions such as perfectly matched layer method by Zheng [54], Páde approximation method by Zhang [52], pole conditions [30, 42], and ALmost EXact boundary conditions [39].

Some above treatments have been extended to the focusing nonlinear case. Zheng [53] used exact boundary conditions which are designed by Darboux transform [24]. NLABC methods based on quasi differential operator were designed to compute bright solitons [9], in addition to some local boundary conditions based on linearization [39, 52]. For more information about artificial boundary conditions, please refer to the review papers [6, 7]. When the bounded computational domain and the boundary condition are chosen suitably, the numerical approximation is usually very accurate. Then the truncated problem is usually discretized in space by finite difference methods [5, 25], spectral method [13, 14], or finite element methods [55]; and in time by the Crank–Nicolson method [25], time splitting method [15, 18, 50], semi-implicit method [4], or Runge–Kutta method [4, 5].

For the defocusing case, however, significant difficulties and numerical instabilities arise due to the nonzero background at the far field. Truncated on a bounded domain, the simple boundary conditions and the above transparent boundary conditions are no longer valid. To our knowledge of the state-of-the-art, there is not much work on designing accurate boundary conditions/treatments for solving NLSE with nonzero far field conditions, especially with non-rest phase background such as dark solitons. On the other hand, applications such as the Bose–Einstein condensation, urge to explore theoretically the interaction and stability of dark solitons in NLSE [21, 27, 32, 33, 48]. Modulus-squared Dirichlet (MSD) method [19], Robin boundary conditions [16] were used to compute the nonzero background. But they cause considerable amount of reflection where solitons are near or cross the numerical boundary. The aim of this work is to design effective boundary treatment for solving the defocusing NLSE (1) with nonzero background such as for the dark solitons.

Following [47], we develop a relay zone technique to monitor the dark soliton propagation near the numerical boundary, and to measure the propagation velocity. According to the soliton position, we adopt alternately a derivative boundary condition to deal with the dark soliton passing through the numerical boundary, and a Robin boundary condition to treat the background state when the dark soliton is far away.

The rest of this paper is organized as follows. In next section, we introduce the relay-zone technique for computing dark soliton based on Robin boundary conditions and derivative boundary conditions, respectively. In next section, we show the effectiveness for derivative boundary conditions and the whole relay-zone technique. Also, we present how the boundary conditions work and how they are used. The numerical study of stability and interaction of dark solitons in NLSE (1) using relay-zone technique can be found in next section. Finally, some conclusions are drawn in last section.

## Relay-Zone Technique and Numerical Boundary Conditions

*k*is a given real constant, and \(A_+^0\), \(A_-^0\) are two given complex constants. Under this far field assumption on the initial data, the solution \(\psi \) of NLSE (1) with (2) satisfies

Figures 1 and 2 show dark solitons, with their background \(\psi _{bg}=A_{2}e^{i(kx-\frac{1}{2}(k^2+2a^2+2(v-k)^2)t)}\).

*J*a positive integer, and denote the grid points by \(x_{j} = x_L+jh\), for \(j = 0,1,\ldots ,J\). Let \(\psi _{j}(t)\) be the numerical approximation of \(\psi (x_j,t)\), for \(j = 0,1,\ldots ,J\). Then the semidiscrete form of NLSE (1) reads

To treat both nonzero background solution and dark solitons, we introduce the a relay-zone technique. Then we propose Robin boundary condition to treat the nonzero background. Derivative boundary condition is also proposed to match the solitons. At last, we give the whole flow chart for the relay-zone technique.

Figure 3 gives a schematic plot for relay-zone technique. We assume that the solitons move from left to right. The computational domain *AD* contains a velocity zone *BC* where we measure the propogation speeds of the solitons. We use the standard Crank–Nicolson finite difference(CNFD) scheme inside the whole domain *AD*. We use Robin boundary condition to treat the background solution on the left boundary point *A*. It is also used to treat the background solution on the right boundary point *D* when the soltions are far away from this boundary point. If the soliton has passed the velocity zone *BC* and entered into the domain *CD*, we use at the right boundary a derivative boundary condition with the measured speed of the soliton. After a certain time, when it moves away, we use the Robin boundary condition to treat the background solution again.

Robin boundary conditions (12) are not valid when solitons are near the boundaries. We then need a new boundary condition.

*v*, the frequency

*k*and the depth

*a*. We have the relation

*x*and

*t*, respectively. We find

*v*, defined as the speed of the soliton. We notice that it is not the velocity of background wave. We use the relay-zone technique to measure this parameter

*v*.

- 1.
Use Robin boundary condition (12) on the boundary point \(x_R+2b\).

- 2.
If \(\min _{x_R<x<x_R+b}|\psi (x,t)|^2<\eta ^2 b^2/\beta \) (here \(\eta \in (0,1)\) is an adjustable parameter and we choose \(\eta =0.9\) in simulations), and the minimum modulus of \(\psi \) is located at the points \(x_R\) and \(x_R+b\), we consider that the soliton is located in the velocity zone. Record the current coordinates \(x^{n_1}\) and time step \(t^{n_1}\).

- 3.
If the soliton locates on one of the grid points \(\{ x_R+b-h,x_R+b,x_R+b+h \}\), record the current coordinate \(x^{n_2}\) and the time step \(t^{n_2}\). Calculate the soliton’s speed \(v=\displaystyle \frac{x^{n_2}-x^{n_1}}{t^{n_2}-t^{n_1}}\).

- 4.
Change the boundary condition to derivative boundary condition (18) from \(t=t^{n_2}\) to \(t^{n_2}+2b/v\). Then change back to Robin boundary condition (12) and return to step 1 when \(t>t^{n_2}+2b/v\).

## Numerical Validation

In this section, we perform several numerical tests to verify the effectiveness of the above boundary conditions. We take the standard fourth order Runge–Kutta scheme for time integration of Eq. (10) and the boundary conditions (12) and (18). First, we test the effectiveness of the derivative boundary condition for defocusing NLSE. Then we show the effectiveness of relay-zone technique for computing single/two dark solitons going across the boundary.

In all our tests, we select the dimensionless constant \(\beta =1.\)

*T*so that the soltions leave the computational domain completely at \(t=T\).

Figure 5 shows the error \(\varepsilon (T)\) with different *a*. The error is less than \(10^{-3}\). It proves that derivative boundary condition matches the dark soliton very well if a correct speed *v* is known in advance.

*h*is shown in Tables 1 and 2.

Error with different mesh size with background parameter \(k=0\)

| 0.1 | 0.05 | 0.025 |
---|---|---|---|

\(\varepsilon (T)\) | \(6.1\times 10^{-3}\) | \(1.8 \times 10^{-3}\) | \(8\times 10^{-4}\) |

\(\varepsilon ^{\infty } (T)\) | \(6.4\times 10^{-3}\) | \(1.9 \times 10^{-3}\) | \(8\times 10^{-4}\) |

Error with different mesh size with background parameter \(k=1\)

| 0.1 | 0.05 | 0.025 |
---|---|---|---|

\(\varepsilon (T)\) | \(7.6\times 10^{-3}\) | \(2.1 \times 10^{-3}\) | \(9\times 10^{-4}\) |

\(\varepsilon ^{\infty } (T)\) | \(7.6\times 10^{-3}\) | \(2.1 \times 10^{-3}\) | \(9\times 10^{-4}\) |

*k*, there are not many differences. The error is less than 1/1000 if we select \(h=0.025\). Thus this boundary condition may serve as a non-reflective boundary condition.

Boundary conditions for the sensitivity of a given speed (\(k=0\), \(h=0.025\))

\(v_{num}/v\) | 0.96 | 0.97 | 0.98 | 0.99 | 1 |
---|---|---|---|---|---|

\(\varepsilon (T)\) | \(1.540\times 10^{-2}\) | \(1.102 \times 10^{-2}\) | \(7.865\times 10^{-3}\) | \(4.215\times 10^{-3}\) | \(8.191\times 10^{-4}\) |

\(\varepsilon ^{\infty } (T)\) | \(1.540\times 10^{-2}\) | \(1.102 \times 10^{-2}\) | \(7.865\times 10^{-3}\) | \(4.215\times 10^{-3}\) | \(1.044\times 10^{-3}\) |

\(v_{num}/v\) | 1.01 | 1.02 | 1.03 | 1.04 | 1.05 |

\(\varepsilon (T)\) | \(3.175\times 10^{-3}\) | \(6.344 \times 10^{-3}\) | \(9.549\times 10^{-3}\) | \(1.235\times 10^{-2}\) | \(1.654\times 10^{-2}\) |

\(\varepsilon ^{\infty } (T)\) | \(3.175\times 10^{-3}\) | \(6.344 \times 10^{-3}\) | \(9.549\times 10^{-3}\) | \(1.235\times 10^{-2}\) | \(1.654\times 10^{-2}\) |

Boundary conditions for the sensitivity of a given speed (\(k=1\), \(h=0.025\))

\(v_{num}/v\) | 0.96 | 0.97 | 0.98 | 0.99 | 1 |
---|---|---|---|---|---|

\(\varepsilon (T)\) | \(1.440\times 10^{-2}\) | \(1.039 \times 10^{-2}\) | \(7.115\times 10^{-3}\) | \(4.402\times 10^{-3}\) | \(9.107\times 10^{-4}\) |

\(\varepsilon ^{\infty } (T)\) | \(1.440\times 10^{-2}\) | \(1.039 \times 10^{-2}\) | \(7.115\times 10^{-3}\) | \(4.402\times 10^{-3}\) | \(9.151\times 10^{-4}\) |

\(v_{num}/v\) | 1.01 | 1.02 | 1.03 | 1.04 | 1.05 |

\(\varepsilon (T)\) | \(4.141\times 10^{-3}\) | \(8.175\times 10^{-3}\) | \(1.181\times 10^{-2}\) | \(1.427\times 10^{-2}\) | \(1.861\times 10^{-2}\) |

\(\varepsilon ^{\infty } (T)\) | \(4.141\times 10^{-3}\) | \(8.175\times 10^{-3}\) | \(1.181\times 10^{-2}\) | \(1.427\times 10^{-2}\) | \(1.861\times 10^{-2}\) |

In practice, the speed obtained from relay-zone technique introduces a certain deviation from the exact value. If the parameter *v* kas a little deviation, can the derivative boundary condition work? We choose the same example, and replace the “numerical” \(v_{num}\) by \(v_{num}=\rho v\). \(\rho \) changes between 0.95 and 1.05. The mesh size is \(h=0.025\). Tables 3 and 4 show their errors. The error increases with the deviation of \(\rho \) from 1. In fact, in this numerical discretization, if \(\rho \in (0.98,1.02)\), the error at the boundary can be controlled within 1/100.

Then we test another dark soliton. The initial center of the soliton locates at \(x_0=0\). We set \(a=2\), \(v=2\) and \(k=1\). The initial computational domain is \([-5,5]\). The mesh size is \(h=0.025\). Since \(2b=2\sqrt{a^2+(v-k)^2}\approx 4.4721\), we approximately set the width of the relay zone as 5. The whole domain is \([-5,12]\). Figure 8 shows the numerical result. The soliton goes across the boundary \(x=12\). The numerical reflection is not seen in the later process.

## Stability and Interaction of Dark Solitons

Now we apply the numerical boundary treatment to study the stability of dark soliton and their interactions. Similarly, we take \(\beta = 1\) in (1).

*k*is the ’velocity’ of phase background. \(v_1\) and \(v_2\) are the velocities of the left-going soliton and right-going soliton, respectively. It holds that \(v_1+v_2=2k\). The problem is solved numerically on \([-15,15]\), with mesh size \(h = 0.02\) and time step size \(\varDelta t=10^{-4}\).

Figure 13 shows time evolution of the density \(\psi \) for the interaction of two dark solitons of NLSE under different parameter values of \(x_0\), *k*, *a*, \(v_1\) and \(v_2\). As shown in these figures, the solitons cross each other and the peak value of the amplitude is equal to the sum of the peak values of the two solitons initially. The solitons separate completely after interaction. Then two solitons go across the numerical boundaries. The proposed boundary treatment is capable to numerically study the interactions of dark solitons.

## Conclusion

In this work, we propose the relay-zone technique for dark soliton simulation in defocusing nonlinear Schrödinger equation. It is based on the Robin boundary condition and the derivative boundary condition. We numerically demonstrate that the derivative boundary condition perfectly match the soliton solution. The solitons go across the boundary perfectly using relay-zone technique. Numerical results demonstrate the effectiveness of the proposed treatment. The algorithm is numerically effectient. We also numerically prove this new method is dynamically stable in numerical NLSE (1). The study of interaction of dark solitons demonstrate the effectiveness again. To our knowledge, this gives for the first time an effective numerical boundary treatment for computing dark soliton in NLSE.

For the future work, please note that it is more difficult to design the artificial boundary conditions for higher dimensional problems. First, we should solve the asymptotic analysis of the solution as one dimensional case. Secondly, the law of motion dynamic of the higher dimensional dark soliton should also be solved analytically. Otherwise, we have no chance to design an effective boundary condition for higher dimensional defocusing nonlinear Schrödinger equation.

## Notes

### Acknowledgements

The research is partially supported by NSFC under Contract no. 11521202 and China Postdoctoral Science Foundation Funded Project no. 2016M600902.

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