_{1}

We develop an exponential spline interpolation method to solve the nonlinear Schrödinger equation. The truncation error and stability analysis of the method are investigated and the method is shown to be unconditionally stable. The conservation quantities are computed to determine the conservation properties of the problem. We will describe the method and present numerical tests by two problems. The numerical simulations results demonstrate the well performance of the proposed method.

Consider the following nonlinear Schrödinger equation

With the boundary conditions

And the initial condition

where

In the past few years a great deal of efforts has been expended to solve NLS equations. It is more difficult to find the analytical solutions of the NLS equation, so the study of the numerical solution of NLS equation in the theory and application is important. Its numerical solutions have been researched by many authors. For example, finite difference method [

The aim of this paper is to give an exponential spline interpolation method for the NLS equation. The paper is organized as follows. In Section 2, construction of the method is presented. The stability analysis of the scheme is investigated in Section 3. In Section 4, the computation of conserved quantities and error norms are given. In Section 5, two numerical examples are presented to demonstrate our theoretical results. The last section is a brief conclusion.

We set up a grid in the

In the interval

where

Since the Taylor series expansions of the hyperbolic functions are

We note that

So the exponential spline defined above share a number of interesting properties:

(1) When

(2) A change of character of the exponential spline function is from linear to third order polynomial on adjacent support intervals.

(3) In the general case the stiffness parameters p are different on every interval which provides the extremely high flexibility of the exponential spline function.

We wish to find

The terms involving the values

The function

The continuity requirement for the first derivative

where

Remark 1.

(1) By expanding Equation (10) in Taylor series, the truncation error for Equation (10) is of the form

where

For

error in space of the relation (10) is of

From Equation (10), we can obtain

Or

Further, when

In order to get the error estimates of Equation (10), we put

Or

At the grid point

From Equation (18), we have

Substituting Equation (19), Equation (20) and Equation (21) into Equation (15) and after some simplifications, we obtain

where

The local truncation error of the relation (22) is of

The boundary conditions (2) and the system given in the Equation (22) consists of

where

Once the vectors

Following the von Neumann technique, we first linearize the nonlinear term in Equation (18) by making the quantity

where

Using Eulers formula, we have

where

Since

Thus this method is unconditionally stable.

The nonlinear Schrödinger equation possesses two conservation quantities:

(1) Mass conservation:

Calculated by

(2) Energy conservation: If

Calculated by

where

The maximum error norm

The relative error of numerical solution is defined as

In the section, we present the results of our numerical experiments for the proposed scheme described in the previous section.

Example 1. Consider the one dimensional Gross-Pitaevskii equation

With the analytical solution

Conserved quantities and error norms at various times are recorded in

The absolute error at different space step sizes h at time

Example 2. Consider the equation (1) with

The exact solution of this problem is

5.0 | 3.14159265358952 | 5.00720563249462 | 1.4158e−004 | 2.5096e−004 | 1.4158e−004 |

10 | 3.14159265358946 | 5.00720563249418 | 2.8317e−004 | 5.0191e−004 | 2.8317e−004 |

20 | 3.14159265358965 | 5.00720563249524 | 5.6635e−004 | 1.0038e−003 | 5.6635e−004 |

30 | 3.14159265358984 | 5.00720563234957 | 8.4953e−004 | 1.5057e−003 | 8.4953e−004 |

Real parts | Imaginary parts | |||||
---|---|---|---|---|---|---|

Exact solution | Approximation | Absolute error | Exact solution | Approximation | Absolute error | |

0.05001875498139 | 0.05001908991577 | 3.35e-007 | −0.70533546922731 | −0.70533544547538 | 2.37e−008 | |

0.07073720166770 | 0.07073767533643 | 4.73e-007 | −0.99749498660405 | −0.99749495301379 | 3.35e−008 | |

0.05001875498139 | 0.05001908991578 | 3.35e-007 | −0.70533546922731 | −0.70533544547537 | 2.37e−008 | |

−0.05001875498139 | −0.05001908991578 | 3.35e-007 | 0.70533546922731 | 0.70533544547538 | 2.37e−008 | |

−0.07073720166770 | −0.07073767533646 | 4.73e-007 | 0.99749498660405 | 0.99749495301376 | 3.36e−008 | |

−0.05001875498139 | −0.05001908991577 | 3.35e-007 | 0.70533546922731 | 0.70533544547537 | 2.37e−008 |

Real parts | Imaginary parts | |||||
---|---|---|---|---|---|---|

Exact solution | Approximation | Absolute error | Exact solution | Approximation | Absolute error | |

0.05001875498139 | 0.05001908991577 | 3.35e-007 | −0.70533546922731 | −0.70533544547538 | 2.37e−008 | |

0.07073720166770 | 0.07073767533643 | 4.73e-007 | −0.99749498660405 | −0.99749495301379 | 3.35e−008 | |

0.05001875498139 | 0.05001908991578 | 3.35e-007 | −0.70533546922731 | −0.70533544547537 | 2.37e−008 | |

−0.05001875498139 | −0.05001908991578 | 3.35e-007 | 0.70533546922731 | 0.70533544547538 | 2.37e−008 | |

−0.07073720166770 | −0.07073767533646 | 4.73e-007 | 0.99749498660405 | 0.99749495301376 | 3.36e−008 | |

−0.05001875498139 | −0.05001908991577 | 3.35e-007 | 0.70533546922731 | 0.70533544547537 | 2.37e−008 |

Conserved quantities and error norms at various times are presented in

The numerical solutions at various times are given in

A numerical method based on exponential spline interpolation function is applied to study a class of nonlinear Schrödinger equation. We use exponential spline collocation method, which results in tri-diagonal systems of

equations that can be solved efficiently by the Thomas algorithm. The numerical simulations confirm and demonstrate the reliability and efficiency of the schemes and tell us that the method is applicable technique, relatively simple and approximates the exact solution very well.

The authors would like to thank the editor and the reviewers for their valuable comments. This work was supported by the Natural Science Foundation of Guangdong (2015A030313827).

Bin Lin, (2016) Spline Solution for the Nonlinear Schrödinger Equation. Journal of Applied Mathematics and Physics,04,1600-1609. doi: 10.4236/jamp.2016.48170