This paper employs the extended rational sin–cos and sinh–cosh method to construct precise solutions to the nonlinear Schrödinger equation. It is illustrated that the proposed technique provides a foremost and effectual mathematical tool for solving numerous types of partial differential equations applied in mathematics, optics, physics, and chemical engineering. We obtain the consequences of periodic, dark, and bright solutions. Furthermore, we can imagine the acquired solutions by drawing two-dimensional and three-dimensional plots.

Research on nonlinear evolution equations (NLEEs), which portray mathematical models and simulations of complex physical phenomena that emerge in numerous fields of engineering, chemistry, plasma physics, applied physics, mathematics, optics, quantum mechanics, biology, and finance, has become very significant in the literature. A plethora of powerful approaches and effective techniques to obtain precise traveling wave solutions of NLEEs have been proposed: variational iteration method, semi-inverse variational principle, Lie symmetry method, and the other ones.4–21 The nonlinear Schrödinger equation (NLSE) is among the most eminent in physics, chemical, plasma physics, and optical fibers, especially in optics. The NLSE is of specific importance in the elucidation of nonlinear effects in optical fibers, molecular biology, hydrodynamics, and optics. Assume that the nonlinear Schrödinger equation has the following form:

(1)

where ϱ = ϱ(x, t) stands for the function with complex valued, μ ≠ 0 is a real constant, and x and t are real variables. The nonlinear Schrödinger equation (1) illustrates solitary kinds of solutions. The case + referred to as NLS+ and NLS includes the case −. The concept of both of them is structurally different. For instance, NLS has dark soliton solutions although NLS+ has the bright ones. This article is organized as follows: in Sec. II, the algorithms of the extension rational sin–cos and sinh–cosh methods are presented; in Secs. III and IV, these methods are represented with the Schrödinger equation. Furthermore, the representation of some acquired solutions in Sec. V will be considered. In addition, we provide a conclusion in Sec. VI.

Consider the nonlinear partial differential equation of the form

(2)

in which Ω = Ω(x, t) is a traveling wave solution. By employing the next transformation,

(3)

where g is the wave speed, Eq. (2) can be changed into the following ordinary differential equation:

(4)

See Refs. 2 and 3 for more information.

  1. Step 1. Assume that Eq. (4) has the solution of the form
    (5)
    or
    (6)
    where η is the wave number and the parameters of βi (i = 0, 1, 2) will be determined.
  2. Step 2. In this step, we concatenated one of the parameters mentioned earlier in Eq. (4); then, compiling all terms with the identical powers of cos(ηξ)q or sin(ηξ)q and equaling to zero all coefficients of cos(ηξ)q or sin(ηξ)q lead to a set of algebraic equations. By using mathematics software, algebraic equations’ solutions can be found.

  3. Step 3. In this step, by concatenating the values of β0, β1, β2, and η in Eq. (5) or Eq. (6), the solution to Eq. (4) can be solved.2 

  1. Step 1. Assume that Eq. (4) has the solution of the form
    (7)
    or
    (8)
    where η is the wave number and the parameters of βi (i = 0, 1, 2) will be determined.
  2. Step 2. In this step, we concatenated one of the parameters mentioned earlier in Eq. (4); then, compiling all terms with the identical powers of cosh(ηξ)q or sinh(ηξ)q and equaling to zero all coefficients of cosh(ηξ)q or sinh(ηξ)q lead to a set of algebraic equations. By using mathematics software, algebraic equations’ solutions can be found.

  3. Step 3. In this step, by concatenating the values of β0, β1, β2, and η in Eq. (7) or Eq. (8), the solution to Eq. (4) can be solved.2 

The application of the extended rational technique of the sin–cos and sinh–cosh method will be illustrated in Secs. III and IV. These methods will be applied to solve NLS+ and NLS equations, respectively. Suppose that the NLS+ equation is defined in the following form:

(9)

Then, by adopting the following transformations:

(10)

we obtain the relation v = −2λ and carry the equation to the following ODE:1 

(11)

Suppose that Eq. (11) has solutions of the form

(12)

By concatenating Eq. (12) in Eq. (11) and then gathering all terms with the identical powers of cos(ηξ)q and equating to zero all the coefficients of cos(ηξ)q, we obtain the following set of algebraic equations:

These algebraic equations will be solved by using software, and then, we have the following solutions:

  • Part (1):
    (13)
  • Part (2):
    (14)
  • Part (3):
    (15)
    1. Case 1: Taking part 1 into consideration, the solution of (11) can be obtained as
      (16)
      By combining Eqs. (10) and (16), we get
      (17)
    2. Case 2: Likewise, for part 2, the solution of (11) can be obtained as
      (18)
      By combining Eqs. (10) and (18), we get
      (19)
    3. Case 3: Likewise, for part 3, the solutions of (11) can be obtained as
      (20)
      By combining Eqs. (10) and (20), we get
      (21)
    Or assume that Eq. (11) has the solutions of the form
    (22)
    By concatenating Eq. (22) in Eq. (11) and then gathering all terms with the identical powers of sin(ηξ)q and equating to zero all coefficients of sin(ηξ)q, we have the following set of algebraic equations in the following stage:
    These algebraic equations will be solved by using software, and then, we have the following solutions:
  • Part (4):
    (23)
  • Part (5):
    (24)
  • Part (6):
    (25)
    1. Case 4: Taking part 4 into consideration, the solution of (11) can be obtained as
      (26)
      By combining Eqs. (10) and (26), we get
      (27)
    2. Case 5: Likewise, for part 5, the solution of (11) can be obtained as
      (28)
      By combining Eqs. (10) and (28), we get
      (29)
    3. Case 6: Likewise, for part 6, the solutions of (11) can be obtained as
      (30)
      By combining Eqs. (10) and (30), we get
      (31)

Assume that Eq. (11) has solutions of the form

(32)

By concatenating Eq. (32) in Eq. (11) and then gathering all terms with the identical powers of cosh(ηξ)q and equating to zero all coefficients of cosh(ηξ)q, we have the following set of algebraic equations in the following stage:

These algebraic equations will be solved by using software, and then, we have the following solutions:

  • Part (7):
    (33)
  • Part (8):
    (34)
  • Part (9):
    (35)
    1. Case 7: Taking part 7 into consideration, the solution of (11) can be obtained as
      (36)
      By combining Eqs. (10) and (36), we get
      (37)
    2. Case 8: Likewise, for part 8, the solution of (11) can be obtained as
      (38)
      By combining Eqs. (10) and (38), we get
      (39)
    3. Case 9: Likewise, for part 9, the solutions of (11) can be obtained as
      (40)
      By combining Eqs. (10) and (40), we get
      (41)
    Or assume that Eq. (11) has solutions of the form
    (42)
    By concatenating Eq. (42) in Eq. (11) and then gathering all terms with the identical powers of sinh(ηξ)q and equating to zero all coefficients of sinh(ηξ)q, we obtain the following set of algebraic equations in the following stage:

    These algebraic equations will be solved by using software, and then, we have the following solutions:

  • Part (10):
    (43)
  • Part (11):
    (44)
    1. Case 10: Taking part 10 into consideration, the solution of (11) can be obtained as
      (45)
      By combining Eqs. (10) and (45), we get
      (46)
    2. Case 11: Likewise, for part 11, the solution of (11) can be obtained as
      (47)
    By combining Eqs. (47) and (10), we get
    (48)

In this section, we consider another form of Schrödinger equation (NLS) in the form

(49)

By adopting the following transformations:

(50)

we obtain the relation v = −2λ and carry the equation to the following ordinary differential equation (ODE):1 

(51)

Suppose that Eq. (51) has solutions of the form

(52)

By concatenating Eq. (52) in Eq. (51) and then gathering all terms with the identical powers of cos(ηξ)q and equating to zero all coefficients of cos(ηξ)q, we obtain the following set of algebraic equations in the following stage:

These algebraic equations will be solved by using mathematics software, and then, we have the following solutions:

  • Part (12):
    (53)
  • Part (13):
    (54)
  • Part (14):
    (55)
    1. Case 12: Taking part 12 into consideration, the solution of (51) can be obtained as
      (56)
      By combining Eqs. (50) and (56), we get
      (57)
    2. Case 13: Likewise, for part 13, the solution of (51) can be obtained as
      (58)
      By combining Eqs. (50) and (58), we get
      (59)
    3. Case 14: Likewise, for part 14, the solutions of (51) can be obtained as
      (60)
      By combining Eqs. (50) and (60), we get
      (61)
    Or assume that Eq. (51) has the solutions of the form
    (62)
    By concatenating Eq. (62) in Eq. (51) and then gathering all terms with the identical powers of sin(ηξ)q and equating to zero all coefficients of sin(ηξ)q, we obtained the following set of algebraic equations in the following stage:

    These algebraic equations will be solved by using software, and then, we have the following solutions:

  • Part (15):
    (63)
  • Part (16):
    (64)
  • Part (17):
    (65)
    1. Case 15: Taking part 15 into consideration, the solution of (51) can be obtained as
      (66)
      By combining Eqs. (50) and (66), we get
      (67)
    2. Case 16: Likewise, for part 16, the solution of (51) can be obtained as
      (68)
      By combining Eqs. (50) and (68), we get
      (69)
    3. Case 17: Likewise, for part 17, the solutions of (51) can be obtained as
      (70)
      By combining Eqs. (50) and (70), we get
      (71)

Assume that Eq. (51) has solutions of the form

(72)

By concatenating Eq. (72) in Eq. (51) and then gathering all terms with the identical powers of cosh(ηξ)q and equating to zero all coefficients of cosh(ηξ)q, we have the following set of algebraic equations in the following stage:

These algebraic equations will be solved by using mathematics software, and then, we have the following solutions:

  • Part (18):
    (73)
  • Part (19):
    (74)
  • Part (20):
    (75)
    1. Case 18: Taking part 18 into account, the solution of (51) can be obtained as
      (76)
      By combining Eqs. (50) and (76), we get
      (77)
    2. Case 19: Likewise, for part 19, the solution of (51) can be obtained as
      (78)
      By combining Eqs. (50) and (78), we get
      (79)
    3. Case 20: Likewise, for part 20, the solutions of (51) can be obtained as
      (80)
      By combining Eqs. (50) and (80), we get
      (81)
    Or assume that Eq. (51) has solutions of the form
    (82)
    By concatenating Eq. (82) in Eq. (51) and then gathering all terms with the identical powers of sinh(ηξ)q and equating to zero all coefficients of sinh(ηξ)q, we obtain the following set of algebraic equations in the following stage:

    These algebraic equations will be solved by using mathematics software, and then, we have the following solutions:

  • Part (21):
    (83)
  • Part (22):
    (84)
    1. Case 21: Taking part 21 into consideration, the solution of (51) can be obtained as
      (85)
      By combining Eqs. (50) and (85), we get
      (86)
    2. Case 22: Likewise, for part 22, the solution of (51) can be obtained as
      (87)
    By combining Eqs. (50) and (87), we get
    (88)

In this section, we have prepared two-dimensional and three-dimensional graphs based on the proper values of the parameters to some of the acquired conclusions that have been explained in their captions. By employing the extended rational sin–cos and sinh–cosh techniques, the wave behaviors of the NLS+ and NLS have been checked.

Figures 1(a) and 1(b) demonstrate the 2D and 3D surfaces of the bright periodic solution of |ϱ1(x, t)| for the values μ = −1, λ = 2, and ρ = 3, respectively.

FIG. 1.

The 3D surfaces of Eq. (17) by considering the values of μ = −1, λ = 2, and ρ = 3 in (a) and the 2D surface of Eq. (17) by considering the values of μ = −1, λ = 2, and ρ = 3 in (b).

FIG. 1.

The 3D surfaces of Eq. (17) by considering the values of μ = −1, λ = 2, and ρ = 3 in (a) and the 2D surface of Eq. (17) by considering the values of μ = −1, λ = 2, and ρ = 3 in (b).

Close modal

Figures 2(a) and 2(b) illustrate the 2D and 3D surfaces of |ϱ4(x, t)| by the proper values of μ = −1, λ = 2, and ρ = 3, respectively, which has a periodic solution.

FIG. 2.

The 3D surfaces of Eq. (27) by considering the values of μ = -1, λ = 2, and ρ = 3 in (a) and the 2D surface of Eq. (27) by considering the values of μ =-1, λ =2, and ρ = 3 in (b).

FIG. 2.

The 3D surfaces of Eq. (27) by considering the values of μ = -1, λ = 2, and ρ = 3 in (a) and the 2D surface of Eq. (27) by considering the values of μ =-1, λ =2, and ρ = 3 in (b).

Close modal

Figures 3(a) and 3(b) show the 2D and 3D surfaces of the bright periodic solution of |ϱ7(x, t)| by adopting the appropriate values of μ = −1, λ = 2, and ρ = 3, respectively.

FIG. 3.

The 3D surfaces of Eq. (37) by considering the values of μ = -1, λ = 2, and ρ = 3 in (a) and the 2D surface of Eq. (37) by considering the values of μ = -1, λ = 2, and ρ = 3 in (b).

FIG. 3.

The 3D surfaces of Eq. (37) by considering the values of μ = -1, λ = 2, and ρ = 3 in (a) and the 2D surface of Eq. (37) by considering the values of μ = -1, λ = 2, and ρ = 3 in (b).

Close modal

By indicating the 2D and 3D surfaces of the bright periodic solution of |ϱ10(x, t)| by the values of μ = −1, λ = 2, and ρ = 3, the visualization of the NLS+ will be completed.

Figure 4(a) shows the 3D surfaces of Eq. (46) by considering the values of μ = −1, λ = 2, and ρ = 3, and Fig. 4(b) displays 2D surfaces of Eq. (46) by considering the values of μ = −1, λ = 2, and ρ = 3. Figures 5(a) and 5(b) display the 2D and 3D dark wave surfaces of |ϱ12(x, t)| for the proper values of μ = 1, λ = 1, and ρ = −3 for NLS, respectively .

FIG. 4.

The 3D surfaces of Eq. (46) by considering the values of μ = −1, λ = 2, and ρ = 3 in (a) and the 2D surface of Eq. (46) by considering the values of μ = −1, λ = 2, and ρ = 3 in (b).

FIG. 4.

The 3D surfaces of Eq. (46) by considering the values of μ = −1, λ = 2, and ρ = 3 in (a) and the 2D surface of Eq. (46) by considering the values of μ = −1, λ = 2, and ρ = 3 in (b).

Close modal
FIG. 5.

The 3D surfaces of Eq. (57) by considering the values of μ = 1, λ = 1, and ρ = −3 in (a) and the 2D surface of Eq. (57) by considering the values of μ = 1, λ = 1, and ρ = −3 in (b).

FIG. 5.

The 3D surfaces of Eq. (57) by considering the values of μ = 1, λ = 1, and ρ = −3 in (a) and the 2D surface of Eq. (57) by considering the values of μ = 1, λ = 1, and ρ = −3 in (b).

Close modal

Figures (6a) and (6b) show the 2D and 3D cusp wave solution surfaces of |ϱ15(x, t)| by adopting the suitable values of μ = 1, λ = 1, and ρ = −3, respectively. Figures 7(a) and 7(b) depict the 2D and 3D dark wave soliton solution of |ϱ18(x, t)| by the proper values of μ = 1, λ = 1, and ρ = −3, respectively.

FIG. 6.

The 3D surfaces of Eq. (67) by considering the values of μ = 1, λ = 1, and ρ = −3 in (a) and the 2D surface of Eq. (67) by considering the values of μ = 1, λ = 1, and ρ = −3 in (b).

FIG. 6.

The 3D surfaces of Eq. (67) by considering the values of μ = 1, λ = 1, and ρ = −3 in (a) and the 2D surface of Eq. (67) by considering the values of μ = 1, λ = 1, and ρ = −3 in (b).

Close modal
FIG. 7.

The 3D surfaces of Eq. (77) by considering the values of μ = 1, λ = 1, and ρ = −3 in (a) and the 2D surface of Eq. (77) by considering the values of μ = 1, λ = 1, and ρ = −3 in (b).

FIG. 7.

The 3D surfaces of Eq. (77) by considering the values of μ = 1, λ = 1, and ρ = −3 in (a) and the 2D surface of Eq. (77) by considering the values of μ = 1, λ = 1, and ρ = −3 in (b).

Close modal

Figures 8(a) and 8(b) demonstrate the 2D and 3D cusp wave soliton solution surfaces of |ϱ21(x, t)| for the values of μ = 1, λ = 1, and ρ = −3, respectively.

FIG. 8.

The 3D surfaces of Eq. (86) by considering the values of μ = 1, λ = 1, and ρ = −3 in (a) and the 2D surface of Eq. (86) by considering the values of μ = 1, λ = 1, and ρ = −3 in (b).

FIG. 8.

The 3D surfaces of Eq. (86) by considering the values of μ = 1, λ = 1, and ρ = −3 in (a) and the 2D surface of Eq. (86) by considering the values of μ = 1, λ = 1, and ρ = −3 in (b).

Close modal

The extended rational sin–cos and sinh–cosh methods have been employed to establish the exact traveling wave solutions of complex nonlinear Schrödinger equations. In this paper, the Schrödinger equations were checked by using these methods. First of all, we assume that Schrödinger has the answer mentioned above. Second, by placing these answers in Schrödinger equations, we obtain a set of equations. By finding the coefficients of the equations by using a pack of mathematics software, we found the exact solutions of the Schrödinger equation.

The authors wish to express their sincere appreciation to the editors and the two anonymous reviewers for valuable comments and suggestions.

The authors have no conflicts to disclose.

Nikan Ahmadi Karchi: Writing – review & editing (equal). Mohammad Bagher Ghaemi: Javad Vahidi: Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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