1 Introduction

Recently, finding numerical solution of the complex nonlinear evolution equations is becoming rapidly attractive and popular [1, 2, 6–8, 10, 13, 15, 17]. One type of the famous complex nonlinear evolution equations with various and important applications in hydrodynamics, plasma physics, nonlinear optics, self-focusing in laser pulses, propagation of heat pulses in crystals, description of the dynamics of Bose–Einstein condensate at extremely low temperature, etc. is the class of Schrödinger equations [9].

The one-dimensional nonlinear Schrödinger equation (NLSE) with a central role in quantum mechanics is one of the most important equations in mathematical physics and physical chemistry with applications in many different fields such as plasma physics, nonlinear optics, water waves, particle-in-a-box, the harmonic oscillator, the hydrogen atom, the rigid rotator, and bimolecular dynamics. This equation plays the role of Newton’s laws and conservation of energy in classical mechanics.

The coupled nonlinear Schrödinger equations (CNLSEs) arise in a great variety of physical situations, for example, propagation of pulses with equal mean frequencies in birefringent nonlinear fiber offers the opportunity to investigate the quasiparticle behavior soliton governed by them; see, e.g., [17] and references therein.

In the sequel, \(i =\sqrt{-1}\) and (x,t)∈(−∞,∞)×(0,T). Considered NLSE is as follows:

$$ i\frac{\partial u}{\partial t}+\frac {\partial^2u}{\partial x^2}+q|u|^2u=0, $$
(1.1)

where q is a real parameter and u(x,t), which governs weakly nonlinear, strongly dispersive, and almost monochromatic wave [23], is a complex-valued function of the spatial coordinate x and the time variable t. Usually, (1.1) is endowed with the initial condition u(x,0)=s(x) and the asymptotic boundary condition u→0 as |x|→∞.

The CNLSEs which arose in a great variety of physical situations [17] are as follows:

(1.2)

where e is the cross phase modulation coefficient, δ is the normalized strength of the linear birefringence, and u 1 and u 2 are the wave amplitude in two polarizations.

The strong coupled nonlinear Schrödinger equations (SCNLSEs) which considered here are as follows:

(1.3)

where the linear coupling parameter Γ accounts for effects that arise from the twisting of the fiber and elliptic deformation of a fiber. It is also referred to the linear birefringence or relative propagation constant. The term proportional to α 1 describes the self-focusing of a signal for pulses in birefringent media. The parameter β describes the group velocity dispersion, and (α 1+2α 2) is the cross phase modulation. Finally, the term γ appears as constant ambient potential called normalized birefringence.

Equations (1.2) and (1.3) are usually equipped with the following initial conditions:

and the following asymptotic boundary conditions

$$u_1,u_2\rightarrow0,\quad|x|\rightarrow\infty. $$

In this study, we aim to solve numerically NLSE (1.1), CNLSEs (1.2), and SCNLSEs (1.3) by using a boundary method of Treffz-type newly developed by Reutskiy et al. [18–21]. When this method applies in a collocation regime, it can be considered as a meshless collocation method similar to the radial basis functions (RBFs) collocation method, which has shown to provide a truly meshless computational approach for solving various partial differential equations (PDEs); see, e.g., [14, 16] and references therein. Such methods have advantage in comparing with most traditional mesh-dependent methods such as finite element and finite difference method since mesh construction is not a trivial work especially for nonlinear, moving boundary, and multidimensional problems [5].

The rest of the paper is organized as follows. In Sect. 2, we describe briefly the Delta-shaped basis functions. Without taking details into account, construction of a semi-discrete method based on the Delta-shaped basis functions is described in Sect. 3. Section 4 is devoted to the numerical results. Section 5 is a brief conclusion.

2 Delta-shaped basis functions

New delta-shaped basis function derived by Reutskiy from the Fourier series of the Dirac-delta function can be applied to successfully simulate a set of scattered data in regular or irregular domains [5, 18–21]. In the sequel, we describe briefly these functions. Let (φ n (x),λ n ) be a solution of the following Sturm–Liouville eigenvalue problem

$$ \everymath{\displaystyle} \left \{ \begin{array}{l} -\frac{d^2\varphi}{dx^2}=\lambda\varphi,\quad x\in (-1,1),\\[10pt] \varphi(-1)=\varphi(1)=0. \end{array} \right . $$
(2.1)

Obviously, \(\lambda_{n}=\frac{n\pi}{2}\), \(\varphi_{n}(x)=\sin (n \pi\frac{x+1}{2} )\), and furthermore

$$\int_{-1}^1\varphi_m(x) \varphi_n(x)\,dx=\delta_{mn}= \left \{ \begin{array}{l@{\quad}l} 1,&m=n,\\ 0,&m\neq n. \end{array} \right . $$

In other words, eigenfunctions \(\{\varphi_{n}(x)\}_{n=1}^{\infty}\) form an orthogonal system on [-1,1] and the Dirac’s delta function can be formally written as follows:

$$ \delta(x-\xi)=\sum_{n=1}^\infty \varphi_n(\xi)\varphi_n(x). $$
(2.2)

Since this series diverges at any point in the interval [−1,1], using some kinds of regularization techniques such as Lanczos or Riesz or Abel [19], a smooth delta-shaped function, I M,χ (x,ξ), can be obtained through the formal series expansion (2.2). We consider here the Riesz regularization technique and, therefore, the regularized delta-shaped functions have the form

$$ I_{M,\chi}(x,\xi)=\sum _{n=1}^M \biggl(1-\frac {n^2}{(M+1)^2} \biggr)^\chi\varphi_n(\xi) \varphi_n(x). $$
(2.3)

Here, χ plays the role of regularizing and M plays the role of scaling. The support of the basis function decreases as M increases. The parameters M and χ which can be known shape parameters (because of their close relation with the properties of the functions) should be taken in coupling. It must be pointed out that the optimal choice of these shape parameters is an open problem and can be dealt with experimentally. In general, choosing χ=4,6,8,12,14,16,18 for M=10,20,30,40,50,80,100 is found to be close to the optimal one. More details can be found in [18, 21].

3 Construction of the method

In order to approximate solution to (1.1), we assume that f and g are real and imaginary parts of u, respectively, i.e.,

$$ u(x,t)=f(x,t)+i g(x,t). $$
(3.1)

Substituting (3.1) into (1.1) leads to the following coupled real partial differential equations:

$$ \everymath{\displaystyle} \left \{ \begin{array}{l} g_t=f_{xx}+q\bigl(f^2+g^2\bigr)f,\\ [4pt] f_t=-g_{xx}-q\bigl(f^2+g^2\bigr)g. \end{array} \right . $$
(3.2)

For solving (1.1) on the interval [a,b], artificial boundary conditions u(a,t)=0 and u(b,t)=0 are needed to model the physical boundary condition. Therefore, system (3.2) must be equipped with the following boundary conditions:

$$f(a,t)=f(b,t)=0=g(a,t)=g(b,t),\quad\! t\in[0,T]. $$

Then, we choose some center points

$$\xi_1<\xi_2<\cdots<\xi_N, $$

in the interval [a,b] and approximate f and g as follows:

$$ \everymath{\displaystyle} \left \{ \begin{array}{l} f(x,t)\simeq\sum_{j=1}^Nf_j(t)I_{M,\chi}(x,\xi_j),\\ [14pt] g(x,t)\simeq\sum_{j=1}^Ng_j(t)I_{M,\chi}(x,\xi_j). \end{array} \right . $$
(3.3)

By choosing some collocation points,

$$a=x_1<x_2<\cdots<x_N=b, $$

(often equidistant with step size h) substituting (3.3) into (3.2) and imposing boundary conditions, we get a system of ordinary differential equations which can be solved with the aid of the classical Runge–Kutta method of order four (RK4).

It must be pointed out that center points and collocation points are different but for ease of computation it is better to be identical.

We end this section by stating the following useful proposition which proved by Hon et al. [5].

Proposition 3.1

The coefficient matrix

$$A=\bigl[I_{M,\chi}(x_k,\xi_j)\bigr]_{N\times N} $$

is positive definite if M≥N.

4 Numerical results

4.1 Numerical tests of NLSE

Accuracy of the numerical results are measured by using the following L ∞ error norm

$$ E_\infty=\big\|u^n-U^n \big\|_\infty=\max_j \big|u^n_j-U^n_j\big|. $$
(4.1)

where u n and U n denote the exact and numerical solution at nth time level, respectively. The reliable soliton solutions of Eq. (1.1) must keep some conservation laws [24]. The following two conserved quantities are considered here:

(4.2)
(4.3)

Relative changes of invariants are defined as \(EC_{1}=\frac{C_{1}-C^{0}_{1}}{C^{0}_{1}}\) and \(EC_{2}=\frac{C_{2}-C^{0}_{2}}{C^{0}_{2}}\) where \(C^{0}_{1}\) and \(C^{0}_{2}\) are the values of the conserved quantities C 1 and C 2 at time t=0, respectively.

4.1.1 Motion of single soliton

Equation (1.1) has been used to model a number of physical situations involving nonlinearity and dispersion. When a certain balance is reached between nonlinearity and dispersion, solitons are formed. This is simply illustrated by the single soliton solution of the equation as follows:

(4.4)

where c represents the speed of the soliton whose magnitude depends on α, which determines its amplitude. Parameters q=2, c=4, α=1 and solution interval −20≤x≤20 are chosen to coincide with earlier works so that comparison of results can be done. We establish the initial condition and boundary conditions from the exact solution (4.4). Analytical values of conserved quantities are C 1=2 and C 2=7.33333333333334. In Table 1, L ∞-error norm and relative errors of conserved quantities are shown for various time and space step size.

Table 1 Comparison of single soliton solution of Eq. (1.1) at time t=1 for M=4000
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For M=4000 and χ=1200, graph of the travelling soliton is presented in Figs. 1, 2, 3 and 4 at different times t=0.0, t=1.0 and t=2.5 in which the real and imaginary components and the modules are displayed. Plot of the error at time t=1.0 for the parameters M=4000, χ=1200, h=0.125 and Δt=0.0025 is shown in Fig. 5. Results are reasonably comparable with the other numerical method [10].

Fig. 1
figure 1

Single soliton with real and imaginary components and the modules at different times 0, 1, and 2.5

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Fig. 2
figure 2

Imaginary part of single soliton at different times

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Fig. 3
figure 3

Real part of single soliton at different times

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Fig. 4
figure 4

Modules of single soliton at different times

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Fig. 5
figure 5

Plot of error at time t=1 with M=4000, χ=1200, h=0.01 and Δt=0.0025

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For obtaining experimental optimal values of M and χ, one can repeat computation once by changing χ for fixed M and then by changing M for fixed χ. Results depicted in Table 2 imply that this problem is not a problem of extreme sensitivity with respect to choosing M and χ.

Table 2 Investigating error of single soliton solution of Eq. (1.1) for h=0.125 and dt=0.001 up to time t=1 for different values of M and χ
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4.1.2 Interaction of two solitons

Interaction of two positive solitary waves is studied by using the initial condition

$$ u(x, 0) = u_1(x, 0) + u_2(x, 0). $$
(4.5)

where

(4.6)

We choose the parameters q=2, h=0.125, Δt=0.0025, α 1=1.0, α 2=0.1, x 1=−10, x 2=10, c 1=4, c 2=10, and −20≤x≤20. These parameters give solitons with the equal amplitudes 9.999999972506763e−001 occurring at x 1=−10 and x 2=10, respectively, and both of them have the velocity of 4 but move in opposite direction, collide and separate. The simulation of interaction is shown at different times in Fig. 6. Results are in good agreement with other numerical methods [2, 4, 10].

Fig. 6
figure 6

Interaction of two solitons at different times t=0,2,2.5,5 with M=4000, χ=500, h=0.125 and Δt=0.0025

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Conserved quantities and amplitudes and peak positions of both solitons at various times and comparison of conserved quantities and their relative changes with some earlier works are seen at Fig. 7 and Tables 3 and 4, respectively. The conservation laws are satisfied up to time t=5.0 with a greater degree of accuracy. C 1 and C 2 remain the same to eight digits at time t=5.0, that are better than results in [10].

Fig. 7
figure 7

Values of C 1 and C 2 for interaction of two solitons with M=4000, χ=500, h=0.125 and Δt=0.0025

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Table 3 Values of C 1 and C 2 for interaction of two solitons with M=4000, χ=500, h=0.125 and Δt=0.0025
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Table 4 Comparison of two soliton simulations with some earlier results for M=4000
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4.1.3 Maxwellian initial condition

According to the theory, if \(\int_{-\infty}^{\infty}u(x,0)\,dx \geq\pi\), a soliton is generated, or else it fades out [3]. This is studied in the paper using the Maxwellian initial condition, given by [4],

$$ u(x, 0) = A \exp \bigl(-x^2 \bigr), $$
(4.7)

where h=0.15, Δt=0.002 and q=2 for the domain −45≤x≤45. Numerical experiment of the birth of soliton is carried out for the values A=1 and 1.78. The program is run until the time t=6 to observe the development of the soliton. Figure 8 presents the birth of standing soliton with A=1.78 while the solution decays away with value A=1 as shown in Fig. 9. Maximas of the solutions are drawn in Fig. 10 from which evolution of soliton of magnitude of about 4 and dissolution can visualized clearly. Thus, the theory is verified that if \(A = 1.78 > \sqrt{\pi}= 1.7725\), a soliton is produced, so with choice of the less than \(A < \sqrt{\pi} \) initial soliton decays away. Conserved quantities for both cases are graphed in Figs. 11 and 12. Using the Maxwellian initial condition (4.7), analytical conserved quantities can be computed as following:

$$C_1 = A^2 \sqrt{\pi/2} \quad \mathrm{and} \quad C_2= \frac{1}{4} A^2 \bigl( 2\sqrt{2} -q A^2 \bigr) \sqrt{\pi}, $$

and results of those quantities and their relative errors are recorded in Table 5. Consequently, conserved quantities obtained using numerical method preserved satisfactorily well.

Fig. 8
figure 8

Maxwellian with A=1.78, h=0.15, Δt=0.002, M=4000 and χ=500

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Fig. 9
figure 9

Maxwellian with A=1, h=0.15, Δt=0.002, M=4000 and χ=500

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Fig. 10
figure 10

Maximas, amplitude of wave for A=1 and A=1.78 with h=0.15, Δt=0.002, M=4000, and χ=500

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Fig. 11
figure 11

Conservation quantity for A=1 and A=1.78 with h=0.15, Δt=0.002, M=4000, and χ=500

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Fig. 12
figure 12

Conservation quantity for A=1 and A=1.78 with h=0.15, Δt=0.002, M=4000, and χ=500

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Table 5 Error of conserved quantities for Maxwellian with parameters M=4000, χ=500, h=0.15 and Δt=0.002
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4.1.4 Birth of mobile soliton

The birth of the mobile soliton is studied using the initial condition

$$ u(x, 0) = A \exp \bigl(-x^2 + 2 i x \bigr), $$
(4.8)

where h=0.125 and Δt=0.001 over the domain [−30,80]. Simulation is studied with values A=1 and A=1.78 to see solutions until time t=6.0. Mobile soliton of amplitude 4 is produced with A=1.78 and the formation of soliton can be observed form the Fig. 13. Once again solution is faded out with A=1 and is graphed in Fig. 14. Maximas of two solutions and conserved quantities are depicted in Figs. 15 and 16, respectively. Conserved quantities and their relative errors are presented in Table 6. Both invariants for this simulation are conserved reasonably well.

Fig. 13
figure 13

Birth of a soliton for A=1.78, h=0.125, Δt=0.001, M=4000, χ=150

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Fig. 14
figure 14

Birth of a soliton for A=1, h=0.125, Δt=0.001, M=4000, χ=150

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Fig. 15
figure 15

Amplitude of birth of a soliton for h=0.125, Δt=0.001, M=4000, χ=150

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Fig. 16
figure 16

Birth of soliton conservation quantity for h=0.125, Δt=0.001, M=4000, χ=150

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Table 6 Error of birth of a mobile soliton with A=1 and A=1.78, h=0.125, Δt=0.001, M=4000, χ=150 on domain [−30,80]
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4.1.5 Bound state of solitons

For the soliton solutions of the Schrödinger equation, the speed and amplitude can be selected separately. This allows solitons to move at same speeds all the time interacting with the other one. Precise results are obtained by Miles [12] by using initial condition

$$ u(x, 0) = \operatorname{sech} (x), $$
(4.9)

which produces a bound state of λ solutions if

$$ q = 2 \lambda^2, \quad \lambda=1,2,\ldots. $$

However, the solution does not have usable form if λ≥3. Numerical studies are conducted for bound state of solitons with the initial conditions using the parameters q=2,8,18,32. Solution of this problem includes extremely large space and time gradients thus producing a numerical method to model these behavior have quite importance in terms of the numerical analysts. Conserved quantities with initial condition (4.9) can be computed analytically by

$$ C_1 = 2,\quad\quad C_2 = \frac{2}{3}(1 - q). $$
(4.10)

With considering rounding errors and computational costs, some numerical methods can be applied in practice to conserve quantities to a fixed number of digits. Parameters h=0.1, Δt=0.0005, M=4000, χ=700, and q=18, 32 over a region −20≤x≤20 are used for the sake of the comparison with studies. When q=18, graphs are depicted in Fig. 17 at early times of simulation in which shapes of 3 bound solitons are in complete agreement with the papers [4, 10]. The graph of modules of the numerical solution from the discrete set of data is also produced successfully for q=32 shown in Fig. 18. Both invariants for both cases remain almost constant and reflect the satisfactory as illustrated in Table 7 with q=18, C 1 and C 2 are converged up to 6 and 5 digits, respectively, at time t=5. Situation deteriorated little with q=32 since changes of C 1 and C 2 are happened at fourth digit.

Fig. 17
figure 17

Bound state for λ=3

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Fig. 18
figure 18

Bound state for λ=4

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Table 7 Error of bound state solitons for h=0.1, Δt=0.0005, M=4000 and χ=700 over a region −20≤x≤20
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4.2 Numerical tests of CNLSEs

Consider the CNLS equations (1.2), and let

(4.11)

where f j (x,t) and g j (x,t) for j=1,2 are the real functions. Substituting Eq. (4.11) into Eq. (1.2) leads to the associated two coupled pair of real partial differential equations

(4.12)

where \(p_{1}= (f_{1}^{2}+g_{1}^{2}+e(f_{2}^{2}+g_{2}^{2}) )\) and \(p_{2}= (e(f_{1}^{2}+g_{1}^{2})+f_{2}^{2}+g_{2}^{2} )\). For solving (1.2) on the interval [a,b], artificial boundary conditions u j (a,t)=0 and u j (b,t)=0 for j=1,2 are needed to model the physical boundary condition. Therefore, system (4.12) must be equipped with the following boundary conditions:

where t∈[0,T]. Then we choose some center points

$$\xi_1<\xi_2<\cdots<\xi_N, $$

in the interval [a,b] and approximate f 1, f 2, g 1 and g 2 as follows:

$$ \everymath{\displaystyle} \left \{ \begin{array}{l} f_1(x,t)\simeq\sum_{j=1}^Nf_{1j}(t)I_{M,\chi}(x,\xi _j),\\[14pt] f_2(x,t)\simeq\sum_{j=1}^Nf_{2j}(t)I_{M,\chi}(x,\xi_j),\\[14pt] g_1(x,t)\simeq\sum_{j=1}^Ng_{1j}(t)I_{M,\chi}(x,\xi _j),\\[14pt] g_2(x,t)\simeq\sum_{j=1}^Ng_{2j}(t)I_{M,\chi}(x,\xi_j). \end{array} \right . $$
(4.13)

Assuming that N collocation points are distributed uniformly in the domain [a,b] as a=x 1<⋯<x N =b and substituting (4.13) into (4.12) and imposing boundary conditions, a system of ordinary differential equations will be obtained which can be solved with the aid of RK4. In order to obtain a reliable solution, discrete conservation quantities are important features in computing smooth soliton solutions of the CNLS equations. The mass, momentum, and energy conservation quantities of the solution to Eq. (1.2) are computed numerically by employing the composite rectangle rules as follows:

For k=1,2,3,4, the relative error EI k is defined as \(EI_{k}=\frac{I_{k}-I_{k}^{0}}{I_{k}^{0}}\) where \(I_{k}^{0}\) is the value of conserved quantity I k at time t=0. Furthermore, the accuracy of the method is measured by using the L ∞ error norm defined by

$$E_\infty= \max_{1\leq x\leq N} \bigl( \big|u_1(x_j,t)\big|- |f_{1j}+i g_{1j} | \bigr). $$

The exact solution of Eq. (1.2) is

where α and ν are real parameters. Consider the CNLS equations (1.2), subject to the initial conditions

(4.14)

where α,e, and ν are constants [7, 17]. By setting

$$ a=-20,\ \ b=60, \ \ \nu=1 ,\ \ \alpha=1,\ \ e=1,\ \ \delta=0, $$
(4.15)

numerical results are compared with the methods of CDQ [15], Galerkin (GM) [7], DRK4 [6], and DIMPR [6] and by setting

$$ a=-20,\ \ b=80,\ \ \nu=1 ,\ \ \alpha=1,\ \ e=2/3,\ \ \delta=0.5, $$
(4.16)

numerical results are compared with the methods of CDQ [15], central difference (CD) [8] and Chebyshev spectral collocation (CSC) [17] in Table 8. In Table 9, we present relative errors of the conserved quantities obtained from the proposed scheme for parameters setting (4.15) with h=0.2 and Δt=0.002. Considering parameters setting (4.16), Fig. 19 shows the evolution of single soliton moving to right with velocity ν=1.

Fig. 19
figure 19

Motion of single soliton with M=4000, χ=600, h=0.2, and Δt=0.002

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Table 8 Error of a single soliton solution at t=12 for M=4000
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Table 9 Relative errors of conserved quantities for a single soliton with M=4000, χ=600, h=0.2 and Δt=0.002
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4.2.1 Interaction of two solitary waves

Interaction of two solitary waves are studied by using the initial conditions

(4.17)

We choose parameters x1=x, x2=x−25, α11=1, α 2=0.5, ν 1=1, ν 2=0.1, δ=0.5, and e=2/3 to compare our results with those in the literature [8, 17]. These parameters give amplitude 1.095445114992280 and 0.7745966692414844 for larger and smaller solitary wave, respectively. Computations are carried out up to time t=50 with time step Δt=0.002 and space step h=0.2 on the interval −20≤x≤70. From the initial conditions (4.17), the solitary wave is propagated rightwards. In this process, the larger and smaller wave unite and separate while preserving their original shapes. Plots of both waves during the interaction from time t=0 to time t=50 are shown in Fig. 20. Three conserved quantities I 1, I 3 and I 4 are presented in Table 10. We used values M=4000, χ=600, h=0.2, Δt=0.002, and domain [−20,70] to obtain the value of the conserved quantities.

Fig. 20
figure 20

Interaction of two solitons with M=4000, χ=600, h=0.2 and Δt=0.002

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Table 10 Conserved quantities for interaction of two solitons with M=4000, χ=600, h=0.2, and Δt=0.002
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4.2.2 Interaction of three solitary waves

Interaction of three solitary waves are studied by using the initial conditions

We choose the following parameters x 1=x, x 2=x−25, x 3=x−50, α 1=1.2, α 2=0.72, α 3=0.36, ν 1=1.0, ν 2=0.1, ν 3=−1.0, δ=0.5, and e=2/3 on domain [−20,70]. We conclude that among the interaction of three solitary waves, two of them are moving in the same direction with different velocities, while the third one is moving in the opposite direction. It is observed that larger, medium, and solitary waves unite and separate while preserving their original shape (see Fig. 21). The values of the three conserved quantities are shown in Table 11.

Fig. 21
figure 21

Interaction of three solitons with h=0.02, Δt=0.002, χ=600, M=4000, and δ=0.5

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Table 11 Conserved quantities for interaction of three solitons with M=4000, χ=600, h=0.2, and Δt=0.002
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4.3 Numerical tests of SCNLSEs

For testing elastic collisions, inelastic collisions and fusion properties of two solitons, we consider Eq. (1.3) with initial conditions

where D 0 and ν 0 are constants. First, we consider the elastic collision of two solitons by choosing the parameters β=1, α 1=1, α 2=−1/6, γ=1, Γ=1, ν 0=1, D 0=25. Figure 22 shows that the proposed scheme simulates the solitary waves well. The two waves emerge without any changes in their shapes. This phenomenon shows that the interaction is elastic. It must be pointed out that the elastic collisions are those from which the newly formed shapes reemerge under deformation.

Fig. 22
figure 22

Elastic collision. β=1, α 1=1, α 2=−1/6, γ=1, Γ=1, ν 0=1, D 0=25, and M=4000

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Now we consider the inelastic-transitive collision of two solitons. To do so, we choose the parameters β=1, α 1=1, α 2=−1/6, γ=1, Γ=0.0175, ν 0=1, D 0=25. Figure 23 shows inelastic collision of two solitons. From this figure, we see that, after the interaction, the solitary waves leave dispersive oscillations and their amplitudes are altered. Both waves change their shape and the interaction is inelastic.

Fig. 23
figure 23

Inelastic collision. β=1, α 1=1, α 2=−1/6, γ=1, Γ=0.0175, ν 0=1, D 0=25, and M=4000

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We consider now the fusion of two solitons by choosing the parameters β=1, α 1=1, α 2=−1/3, γ=1, Γ=0.0175, and ν 0=0.4, D 0=20. From Fig. 24, we see that the two solitons collapse into one soliton.

Fig. 24
figure 24

Fusion of two solitons. β=1, α 1=1, α 2=−1/3, γ=1, Γ=0.0175 and ν 0=0.4, D 0=20, and M=4000

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5 Conclusion

Applicability and efficiency of a semidiscrete method based on the newly developed Delta-shaped basis functions have been tested by solving some Schrödinger-type equations. The fourth-order Runge–Kutta method for the time integration of the resulting systems of nonlinear ordinary differential equations has been applied. Some satisfactory numerical simulations such as motion of single solitary wave and double and triple solitary waves, generation of solitary waves using the Maxwellian initial condition, birth of mobile soliton, bound state of solitons, elastic and inelastic collision of two solitons, and fusion of two solitons have been carried out.

Without taking computational costs into account, the first two test problems, the motion of single soliton and interaction of two solitons, have been simulated with more accurate results in comparison with earlier works and the proposed method can compete very well with the successful method of differential quadrature [10, 11].

Furthermore, the performance of the method has been monitored by computing some conserved quantities. We have found that for almost all of our experiments, invariant quantities are conserved satisfactorily.

Interaction of double and triple solitary waves, generation of solitary waves using the Maxwellian initial condition, birth of mobile soliton, bound state of solitons, and other numerical simulations have been carried out as well as some earlier successful studies.

Finally, our scheme is sensibly conservative and can generate reasonable numerical results.