The First Integral of the Dissipative Nonlinear Schrödinger Equation with Nucci’s Direct Method and Explicit Wave Profile Formation

Abstract

The propagation of optical soliton profiles in plasma physics and atomic structures is represented by the $(1+1)-$ dimensional Schrödinger dynamical equation, which is the subject of this study. New solitary wave profiles are discovered by using Nucci’s scheme and a new extended direct algebraic method. The new extended direct algebraic approach provides an easy and general mechanism for covering 37 solitonic wave solutions, which roughly corresponds to all soliton families, and Nucci’s direct reduction method is used to develop the first integral and the exact solution of partial differential equations. Thus, there are several new solitonic wave patterns that are obtained, including a plane solution, mixed hyperbolic solution, periodic and mixed periodic solutions, a mixed trigonometric solution, a trigonometric solution, a shock solution, a mixed shock singular solution, a mixed singular solution, a complex solitary shock solution, a singular solution, and shock wave solutions. The first integral of the considered model and the exact solution are obtained by utilizing Nucci’s scheme. We present 2-D, 3-D, and contour graphics of the results obtained to illustrate the pulse propagation characteristics while taking suitable values for the parameters involved, and we observed the influence of parameters on solitary waves. It is noticed that the wave number $\alpha$ and the soliton speed $\mu$ are responsible for controlling the amplitude and periodicity of the propagating wave solution.

1. Introduction

The Schrödinger equation is one of the fundamental equations in quantum theory. In quantum physics, the Schrödinger equation is frequently employed to determine a wave function’s progression over time. In other words, the primary application of the Schrödinger equation is to the prediction of the trajectory of a particle’s motion, and these findings play a significant role in the advancement of modern physics, as well as applied mathematics [1]. It is also characterized as a classical field equation. One of the applications is the comparison of the propagation of light in a planar waveguide with a nonlinear optical fiber [2,3]. The study of wave theory shows that the the nonlinear Schrödinger equation (NLSE) is a feasible tool for the demonstration of self-monochromatic waves in a dispersive medium. The NLSE is more common in the disciplines of plasma physics, matter physics, Heisenberg spin chains, deep water waves, computer networking, and optical communication (Seadawy et al. [4], Liu et al. [5], and Wang et al. [6]). The dissipative NLSE is extremely helpful for the modeling of amplitude in the study of dispersive and dissipative media. Therefore, the d-NLSE is a viable technique for the modeling of dissipative self-modulating monochromatic waves with dispersion. The estimation of quasi-monochromatic waves’ packets with slow alterations in nonlinear systems is provided by the NLSE’s linear and nonlinear dispersive and dissipative effects (Ma et al. [7]; Mo et al. [8]; Jiang et al. [9]).

Meanwhile, numerous researchers have worked on distinct forms of NLSEs. Chen et al. [10] gave interactions and a general periodic solution of the NLSE. The d-NLSE’s asymptotic behaviors were studied by Cazenave et al. [11]. Keller–Segel dynamics were studied by Lopez with the NLSE [12]. Weng et al. [13] also conducted an asymptotic analysis with some semi-rational vector solutions of the n-component NLSE, and some of the optical solutions were given by Savaissou by using a power law [14]. Seadway found soliton solutions of the Korteweg–de-Vries–Zakharov–Kuznetsov equation [15]. Akram worked on the CGL equation and obtained optical solutions, and they also used the Triki–Biswas equation to study ultrashort pulse propagation in optical fiber frameworks [16,17,18]. Kumar studied higher-order equations, such as the Chaffee–Infante equation, BLMP equation, and rdDym equation, with different approaches [19,20,21]. Imran explored an improved perturbed Schrödinger equation with the use of the Ker law [22] and examined the Chen–Lee–Liu dynamical equation with Nucci’s reduction approach [23]. Adil et al. [24] obtained the traveling wave solution of nonlinear directional couplers and studied fusion and fission reactions with soliton solutions [25]. The bright and dark soliton solutions of the $(2+1)-$ NLSE were studied by Wazwaz [26].

The optical solution of the (3 + 1) NLSE was given by Lanre [27]. Kudryashov gave the solutions of periodic and solitary waves by using a reduced form of a higher-order NLSE, and he also vigilantly worked on a resonant NLSE for optical solitons [28,29]. Wang worked on a chiral NLSE and figured out its abundant solutions [30]. Faridi et al. [31] analyzed the isotropic bi-quadratic Heisenberg spin chain phenomenon and found soliton solutions, and they also analyzed propagating wave structures of cold bosonic atoms and obtain soliton solutions [32]. Tahir worked on different models and found their soliton solution with the help of updated techniques [33,34,35]. In parallel to this, there are multiple researchers who have worked on different linear and nonlinear equations, such as Akgül [36], who worked on the nonlinear stochastic Newell–Whitehead–Segel equation, an HIV/AIDS model [37], pseudo-parabolic differential equations [38], the fractal–fractional Klein–Gordon equation [39], and the kernel Hilbert space method [40]. Cubic–quintic nonlinear Schrödinger equations were also studied by Fabio [41], and he obtained the dark and bright solutions of the nonlinear Schrödinger equation [42], etc.

The considered model was studied by Seadawy. He produced the bright and dark soliton solutions by using ansatz function methods [43]. To obtain the soliton solutions in the current work, a novel direct extended algebraic method is applied to the model.

In the present study, Section 2 presents the model and its structure. Section 3 portrays a detailed description of the proposed technique and solutions of the model with graphs of the solutions. A graphical discussion is included in Section 4. At the end, Section 5 contains the summary as a conclusion.

2. Description of the Model

We consider the dissipative NLSEs

$$i{U}_t+\frac{1}{2}{U}_{xx}+U{\left|U\right|}^2=0,i=\sqrt{-1},$$

(1)

and

$$i{U}_t-\frac{1}{2}{U}_{xx}+U{\left|U\right|}^2=0,i=\sqrt{-1}.$$

(2)

The $(1+1)-$ dNLSE is [43],

$$i{U}_t(x,t)+a{U}_{xx}(x,t)+b{\left|U(x,t)\right|}^{2}U(x,t)+icU(x,t)=0,i=\sqrt{-1},$$

(3)

where the dispersion is represented by a, b is a non-linearity constant, and the dissipative effects are shown by c. If the dissipative effect is weak, $c=0$. This effect can also be shown for various forms of signs of $ab<0$ and $ab>0$. The exact localized traveling wave solutions can be generated for the NLSE [43]. There are also some chances of significant modification of the pulse shape if the absorption is reduced and the nonlinear term is made comparable to the dispersion term. The shape of the stationary solution is affected by the sign of a. In fact, if both a > 0 and b > 0, then one can determine the group of exact solutions, which are localized and stationary.

3. Computation of Soliton Solutions

3.1. Description of the Method

Assume a general NPDE of the type:

$$\begin{array}{c}\hfill Y(U,U_t,U_x,U_{tt},U_{xx},\cdots )=0,\end{array}$$

(4)

which can be transformed:

$$\begin{array}{c}\hfill M(R,R^{\prime },R^{″},\cdots )=0,\end{array}$$

(5)

by using the following transformation:

$$U(x,t)=U\left(\kappa \right)e^{i\Phi },$$

(6)

where $\kappa =k_{1}x+k_{2}t$ and $\Phi =k_{3}x+k_{4}t,$ where the prime sign represents the order of differentiation for different attributes in Equation (5). Assume that Equation (5) has a solution of the form:

$$\begin{array}{c}\hfill U\left(\kappa \right)=\sum _{r=0}^j{a}_r{\left(P\left(\kappa \right)\right)}^r,\end{array}$$

(7)

where

$$\begin{array}{c}\hfill P^{\prime }\left(\kappa \right)=ln\left(\rho \right)\left(\phi +\upsilon P\left(\kappa \right)+\varsigma {\left(P\left(\kappa \right)\right)}^2\right),\rho \ne 0,1.\end{array}$$

(8)

In addition, $\varsigma ,\upsilon ,$ and $\phi$ are real constants. The general forms of the solutions of Equation (8) w.r.t the real parameters $\varsigma ,\upsilon$, and $\phi$ are as follows:

• For ${\upsilon }^2-4\phi \varsigma <0$ and $\varsigma \ne 0,$

  $$\begin{array}{c}\hfill P_1\left(\kappa \right)=-\frac{\upsilon }{2\varsigma }+\frac{\sqrt{-Ꮗ}}{2\varsigma }{tan}_{\rho }\left(\frac{\sqrt{-Ꮗ}}{2}\kappa \right),\end{array}$$

  (9)

  $$\begin{array}{c}\hfill P_2\left(\kappa \right)=-\frac{\upsilon }{2\varsigma }-\frac{\sqrt{-Ꮗ}}{2\varsigma }{cot}_{\rho }\left(\frac{\sqrt{-Ꮗ}}{2}\kappa \right),\end{array}$$

  (10)

  $$\begin{array}{c}\hfill P_3\left(\kappa \right)=-\frac{\upsilon }{2\varsigma }+\frac{\sqrt{-Ꮗ}}{2\varsigma }\left({tan}_{\rho }\left(\sqrt{-Ꮗ}\kappa \right)\pm \sqrt{mn}{sec}_{\rho }\left(\sqrt{-Ꮗ}\kappa \right)\right),\end{array}$$

  (11)

  $$\begin{array}{c}\hfill P_4\left(\kappa \right)=-\frac{\upsilon }{2\varsigma }+\frac{\sqrt{-Ꮗ}}{2\varsigma }\left({cot}_{\rho }\left(\sqrt{-Ꮗ}\kappa \right)\pm \sqrt{mn}{csc}_{\rho }\left(\sqrt{-Ꮗ}\kappa \right)\right),\end{array}$$

  (12)

  $$\begin{array}{c}\hfill P_5\left(\kappa \right)=-\frac{\upsilon }{2\varsigma }+\frac{\sqrt{-Ꮗ}}{4\varsigma }\left({tan}_{\rho }\left(\frac{\sqrt{-Ꮗ}}{4}\kappa \right)-{cot}_{\rho }\left(\frac{\sqrt{-Ꮗ}}{4}\kappa \right)\right).\end{array}$$

  (13)
• For ${\upsilon }^2-4\phi \varsigma >0$ and $\varsigma \ne 0,$

  $$\begin{array}{c}\hfill P_6\left(\kappa \right)=-\frac{\upsilon }{2\varsigma }-\frac{\sqrt{Ꮗ}}{2\varsigma }{tanh}_{\rho }\left(\frac{\sqrt{Ꮗ}}{2}\kappa \right),\end{array}$$

  (14)

  $$\begin{array}{c}\hfill P_7\left(\kappa \right)=-\frac{\upsilon }{2\varsigma }-\frac{\sqrt{Ꮗ}}{2\varsigma }{coth}_{\rho }\left(\frac{\sqrt{Ꮗ}}{2}\kappa \right),\end{array}$$

  (15)

  $$\begin{array}{c}\hfill P_8\left(\kappa \right)=-\frac{\upsilon }{2\varsigma }+\frac{\sqrt{Ꮗ}}{2\varsigma }\left(-{tanh}_{\rho }\left(\sqrt{Ꮗ}\kappa \right)\pm i{\sqrt{mn}}_{\rho }\left(\sqrt{Ꮗ}\kappa \right)\right),\end{array}$$

  (16)

  $$\begin{array}{c}\hfill P_9\left(\kappa \right)=-\frac{\upsilon }{2\varsigma }+\frac{\sqrt{Ꮗ}}{2\varsigma }\left(-{coth}_{\rho }\left(\sqrt{Ꮗ}\kappa \right)\pm {\sqrt{mn}}_{\rho }\left(\sqrt{Ꮗ}\kappa \right)\right),\end{array}$$

  (17)

  $$\begin{array}{c}\hfill P_{10}\left(\kappa \right)=-\frac{\upsilon }{2\varsigma }-\frac{\sqrt{Ꮗ}}{4\varsigma }\left({tanh}_{\rho }\left(\frac{\sqrt{Ꮗ}}{4}\kappa \right)+{coth}_{\rho }\left(\frac{\sqrt{Ꮗ}}{4}\kappa \right)\right).\end{array}$$

  (18)
• For $\phi \varsigma >0$ and $\upsilon =0,$

  $$\begin{array}{c}\hfill P_{11}\left(\kappa \right)=\sqrt{\frac{\phi }{\varsigma }}{tan}_{\rho }\left(\sqrt{\phi \varsigma }\kappa \right),\end{array}$$

  (19)

  $$\begin{array}{c}\hfill P_{12}\left(\kappa \right)=-\sqrt{\frac{\phi }{\varsigma }}{cot}_{\rho }\left(\sqrt{\phi \varsigma }\kappa \right),\end{array}$$

  (20)

  $$\begin{array}{c}\hfill P_{13}\left(\kappa \right)=\sqrt{\frac{\phi }{\varsigma }}\left({tan}_{\rho }\left(2\sqrt{\phi \varsigma }\kappa \right)\pm \sqrt{mn}{sec}_{\rho }\left(2\sqrt{\phi \varsigma }\kappa \right)\right),\end{array}$$

  (21)

  $$\begin{array}{c}\hfill P_{14}\left(\kappa \right)=\sqrt{\frac{\phi }{\varsigma }}\left(-{cot}_{\rho }\left(2\sqrt{\phi \varsigma }\kappa \right)\pm \sqrt{mn}{csc}_{\rho }\left(2\sqrt{\phi \varsigma }\kappa \right)\right),\end{array}$$

  (22)

  $$\begin{array}{c}\hfill P_{15}\left(\kappa \right)=\frac{1}{2}\sqrt{\frac{\phi }{\varsigma }}\left({tan}_{\rho }\left(\frac{\sqrt{\phi \varsigma }}{2}\kappa \right)-{cot}_{\rho }\left(\frac{\sqrt{\phi \varsigma }}{2}\kappa \right)\right).\end{array}$$

  (23)
• For $\phi \varsigma <0$ and $\upsilon =0,$

  $$\begin{array}{c}\hfill P_{16}\left(\kappa \right)=-\sqrt{-\frac{\phi }{\varsigma }}{tanh}_{\rho }\left(\sqrt{-\phi \varsigma }\kappa \right),\end{array}$$

  (24)

  $$\begin{array}{c}\hfill P_{17}\left(\kappa \right)=-\sqrt{-\frac{\phi }{\varsigma }}{coth}_{\rho }\left(\sqrt{-\phi \varsigma }\kappa \right),\end{array}$$

  (25)

  $$\begin{array}{c}\hfill P_{18}\left(\kappa \right)=\sqrt{-\frac{\phi }{\varsigma }}\left(-{tanh}_{\rho }\left(2\sqrt{-\phi \varsigma }\kappa \right)\pm i{\sqrt{mn}}_{\rho }\left(2\sqrt{-\phi \varsigma }\kappa \right)\right),\end{array}$$

  (26)

  $$\begin{array}{c}\hfill P_{19}\left(\kappa \right)=\sqrt{-\frac{\phi }{\varsigma }}\left(-{coth}_{\rho }\left(2\sqrt{-\phi \varsigma }\kappa \right)\pm {\sqrt{mn}}_{\rho }\left(2\sqrt{-\phi \varsigma }\kappa \right)\right),\end{array}$$

  (27)

  $$\begin{array}{c}\hfill P_{20}\left(\kappa \right)=-\frac{1}{2}\sqrt{-\frac{\phi }{\varsigma }}\left({tanh}_{\rho }\left(\frac{\sqrt{-\phi \varsigma }}{2}\kappa \right)+{coth}_{\rho }\left(\frac{\sqrt{-\phi \varsigma }}{2}\kappa \right)\right).\end{array}$$

  (28)
• For $\upsilon =0$ and $\phi =\varsigma ,$

  $$\begin{array}{c}\hfill P_{21}\left(\kappa \right)={tan}_{\rho }\left(\phi \kappa \right),\end{array}$$

  (29)

  $$\begin{array}{c}\hfill P_{22}\left(\kappa \right)=-{cot}_{\rho }\left(\phi \kappa \right),\end{array}$$

  (30)

  $$\begin{array}{c}\hfill P_{23}\left(\kappa \right)={tan}_{\rho }\left(2\phi \kappa \right)\pm \sqrt{mn}{sec}_{\rho }\left(2\phi \kappa \right),\end{array}$$

  (31)

  $$P_{24}\left(\kappa \right)=-{cot}_{\rho }\left(2\phi \kappa \right)\pm \sqrt{mn}{csc}_{\rho }\left(2\phi \kappa \right),$$

  (32)

  $$\begin{array}{c}\hfill P_{25}\left(\kappa \right)=\frac{1}{2}\left({tan}_{\rho }\left(\frac{\phi }{2}\kappa \right)-{cot}_{\rho }\left(\frac{\phi }{2}\kappa \right)\right).\end{array}$$

  (33)
• For $\upsilon =0$ and $\varsigma =-\phi ,$

  $$\begin{array}{c}\hfill P_{26}\left(\kappa \right)=-{tanh}_{\rho }\left(\phi \kappa \right),\end{array}$$

  (34)

  $$\begin{array}{c}\hfill P_{27}\left(\kappa \right)=-{coth}_{\rho }\left(\phi \kappa \right),\end{array}$$

  (35)

  $$\begin{array}{c}\hfill P_{28}\left(\kappa \right)=-{tanh}_{\rho }\left(2\phi \kappa \right)\pm i{\sqrt{mn}}_{\rho }\left(2\phi \kappa \right),\end{array}$$

  (36)

  $$\begin{array}{c}\hfill P_{29}\left(\kappa \right)=-{coth}_{\rho }\left(2\phi \kappa \right)\pm {\sqrt{mn}}_{\rho }\left(2\phi \kappa \right),\end{array}$$

  (37)

  $$\begin{array}{c}\hfill P_{30}\left(\kappa \right)=-\frac{1}{2}\left({tanh}_{\rho }\left(\frac{\phi }{2}\kappa \right)+{coth}_{\rho }\left(\frac{\phi }{2}\kappa \right)\right).\end{array}$$

  (38)
• For ${\upsilon }^2=4\phi \varsigma$,

  $$\begin{array}{c}\hfill P_{31}\left(\kappa \right)=\frac{-2\phi (\upsilon \kappa ln(\rho )+2)}{{\upsilon }^2\kappa ln\left(\rho \right)}.\end{array}$$

  (39)
• For $\upsilon =p,\phi =pq,(q\ne 0)$, and $\varsigma =0,$

  $$\begin{array}{c}\hfill P_{32}\left(\kappa \right)={\rho }^{p\kappa }-q.\end{array}$$

  (40)
• For $\upsilon =\varsigma =0,$

  $$\begin{array}{c}\hfill P_{33}\left(\kappa \right)=\phi \kappa ln\left(\rho \right).\end{array}$$

  (41)
• For $\upsilon =\phi =0,$

  $$\begin{array}{c}\hfill P_{34}\left(\kappa \right)=\frac{-1}{\varsigma \kappa ln\left(\rho \right)}.\end{array}$$

  (42)
• For $\phi =0$ and $\upsilon \ne 0,$

  $$\begin{array}{c}\hfill P_{35}\left(\kappa \right)=-\frac{m\upsilon }{\varsigma \left({cosh}_{\rho }\left(\upsilon \kappa \right)-{sinh}_{\rho }\left(\upsilon \kappa \right)+m\right)},\end{array}$$

  (43)

  $$\begin{array}{c}\hfill P_{36}\left(\kappa \right)=-\frac{\upsilon \left({sinh}_{\rho }\left(\upsilon \kappa \right)+{cosh}_{\rho }\left(\upsilon \kappa \right)\right)}{\varsigma \left({sinh}_{\rho }\left(\upsilon \kappa \right)+{cosh}_{\rho }\left(\upsilon \kappa \right)+n\right)}.\end{array}$$

  (44)
• For $\upsilon =p,\varsigma =pq,(q\ne 0$ and $\phi =0),$

  $$\begin{array}{c}\hfill P_{37}\left(\kappa \right)=-\frac{m{\rho }^{p\kappa }}{m-qn{\rho }^{p\kappa }}.\end{array}$$

  (45)

$$\begin{array}{c}\hfill {sinh}_{\rho }\left(\kappa \right)=\frac{m{\rho }^{\kappa }-n{\rho }^{(-\kappa )}}{2},{cosh}_{\rho }\left(\kappa \right)=\frac{m{\rho }^{\kappa }+n{\rho }^{(-\kappa )}}{2},{tanh}_{\rho }\left(\kappa \right)=\frac{m{\rho }^{\kappa }-n{\rho }^{(-\kappa )}}{m{\rho }^{\kappa }+n{\rho }^{(-\kappa )}},\end{array}$$

(46)

$$\begin{array}{c}\hfill {}_{\rho }\left(\kappa \right)=\frac{2}{m{\rho }^{\kappa }-n{\rho }^{(-\kappa )}}{}_{\rho }\left(\kappa \right)=\frac{2}{m{\rho }^{\kappa }+n{\rho }^{(-\kappa )}},{coth}_{\rho }\left(\kappa \right)=\frac{m{\rho }^{\kappa }+n{\rho }^{-\kappa }}{m{\rho }^{\kappa }-n{\rho }^{-\kappa }},\end{array}$$

(47)

$$\begin{array}{c}\hfill {sin}_{\rho }\left(\kappa \right)=\frac{m{\rho }^{i\kappa }-n{\rho }^{(-i\kappa )}}{2i},{cos}_{\rho }\left(\kappa \right)=\frac{m{\rho }^{i\kappa }+n{\rho }^{(-i\kappa )}}{2},{tan}_{\rho }\left(\kappa \right)=-i\frac{m{\rho }^{i\kappa }-n{\rho }^{(-i\kappa )}}{m{\rho }^{i\kappa }+n{\rho }^{(-i\kappa )}},\end{array}$$

(48)

$$\begin{array}{c}\hfill {csc}_{\rho }\left(\kappa \right)=\frac{2i}{m{\rho }^{\kappa }-n{\rho }^{(-\kappa )}}{sec}_{\rho }\left(\kappa \right)=\frac{2}{m{\rho }^{\kappa }+n{\rho }^{(-\kappa )}},{cot}_{\rho }\left(\kappa \right)=i\frac{m{\rho }^{i\kappa }+n{\rho }^{(-i\kappa )}}{m{\rho }^{i\kappa }-n{\rho }^{(-i\kappa )}},\end{array}$$

(49)

where m and n are arbitrary constants that have a value greater than zero, and they are also known as deformation parameters.

3.2. Imposition of the Technique on Equation (4)

In order to find the solution, we used the ansatz in Equation (3),

$$U(x,t)=U\left(\kappa \right)e^{i\varphi },\kappa =\alpha x+\mu t,\Phi =k_{1}x+\lambda t,$$

(50)

where $\alpha$ is the wave number and $\mu$ is the speed of the soliton. By imposing the considered transformation in Equation (50) on Equation (3), we produce the following real parts:

$$a{\alpha }^2{U}^{″}-(a{k}_1^2+\lambda )U+b{U}^3=0.$$

(51)

The homogeneous balancing constant of Equation (51) is one; thus, the solution is given as follows:

$$U\left(\kappa \right)=a_0+a_{1}P\left(\kappa \right),$$

(52)

where,

$$P^{\prime }{\left(\kappa \right)=ln\left(\rho \right)(\phi +\upsilon P+\varsigma \left(P\left(\kappa \right)\right)}^2.$$

(53)

By plugging in the solution of Equation (52) along with Equations (51)–(53) and by calculating the coefficients of the different powers of P($\kappa$), we get the system of equations. This system is a system of algebraic equations, and it was solved with the help of the Mapple software; the results are as follows:

$$a_o=\Delta \upsilon ,a_1=2\Delta \varsigma ,$$

(54)

where $\Delta =\pm \sqrt{\frac{-a}{2b}}\alpha ln\left(\rho \right)$.

We get the general solution of Equation (3) by plugging Equation (54) into Equation (52):

$$U(x,t)=\Delta \upsilon +2\Delta \varsigma P_i\left(\kappa \right),$$

(55)

Here, $Ꮗ={\upsilon }^2-4\phi \varsigma$. It should be seen that by placing different values of $P_r$ from Equations (9)–(45), we can get many solutions.

(1)

For ${\upsilon }^2$ − 4 $\phi$$\varsigma$ < 0, $\varsigma$ ≠ 0,

$$U_1(x,t)=\left[\Delta \sqrt{-Ꮗ}{tan}_{\rho }\left(\frac{\sqrt{-Ꮗ}}{2}\kappa \right)\right]e^{\iota \Phi },$$

(56)

$$U_2(x,t)=-\left[\Delta \sqrt{-Ꮗ}{cot}_{\rho }\left(\frac{\sqrt{-Ꮗ}}{2}\kappa \right)\right]e^{\iota \Phi },$$

(57)

$$U_3(x,t)=\left[\Delta \sqrt{-Ꮗ}\left({tan}_{\rho }\left(\sqrt{-Ꮗ}\kappa \right)\pm \sqrt{mn}{sec}_{\rho }\left(\sqrt{-Ꮗ}\kappa \right)\right)\right]e^{\iota \Phi },$$

(58)

$$U_4(x,t)=\left[\Delta \sqrt{-Ꮗ}\left({cot}_{\rho }\left(\sqrt{-Ꮗ}\kappa \right)\pm \sqrt{mn}{csc}_{\rho }\left(\sqrt{-Ꮗ}\kappa \right)\right)\right]e^{\iota \Phi },$$

(59)

$$U_5(x,t)=[\Delta \sqrt{\frac{-Ꮗ}{4}}\left({tan}_{\rho }\left(\frac{\sqrt{-Ꮗ}}{4}\kappa )-{cot}_{\rho }\left(\frac{\sqrt{-Ꮗ}}{4}\kappa \right)\right)\right]e^{\iota \Phi }.$$

(60)

(2)

For ${\upsilon }^2$ − 4 $\phi$$\varsigma$ > 0, $\varsigma$ ≠ 0,

$$U_6(x,t)=-\left[\Delta \sqrt{Ꮗ}{tanh}_{\rho }\left(\frac{\sqrt{Ꮗ}}{2}\kappa \right)\right]e^{\iota \Phi },$$

(61)

$$U_7(x,t)=-\left[\Delta \sqrt{Ꮗ}{coth}_{\rho }\left(\frac{\sqrt{Ꮗ}}{2}\kappa \right)\right]e^{\iota \Phi },$$

(62)

$$U_8(x,t)=\left[\Delta \sqrt{Ꮗ}\left(-{tanh}_{\rho }\left(\sqrt{Ꮗ}\kappa \right)\pm i{\sqrt{mn}}_{\rho }\left(\sqrt{Ꮗ}\kappa \right)\right)\right]e^{\iota \Phi },$$

(63)

$$U_9(x,t)=\left[\Delta \sqrt{Ꮗ}\left(-{coth}_{\rho }\left(\sqrt{Ꮗ}\kappa \right)\pm {\sqrt{mn}}_{\rho }\left(\sqrt{Ꮗ}\kappa \right)\right)\right]e^{\iota \Phi },$$

(64)

$$U_{10}(x,t)=\left[\frac{\Delta }{2}\sqrt{Ꮗ}\left({tanh}_{\rho }\left(\frac{\sqrt{Ꮗ}}{4}\kappa \right)+{coth}_{\rho }\left(\frac{\sqrt{Ꮗ}}{4}\kappa \right)\right)\right]e^{\iota \Phi }.$$

(65)

(3)

For $\phi \varsigma >0$ and $\upsilon =0$,

$$U_{11}(x,t)=\left[2\Delta \sqrt{\varsigma \phi }\left({tan}_{\rho }\left(\sqrt{\phi \varsigma }\kappa \right)\right)\right]e^{\iota \Phi },$$

(66)

$$U_{12}(x,t)=-\left[2\Delta \sqrt{\phi \varsigma }\left({cot}_{\rho }\left(\sqrt{\phi \varsigma }\kappa \right)\right)\right]e^{\iota \Phi },$$

(67)

$$U_{13}(x,t)=\left[2\Delta \sqrt{\phi \varsigma }\left({tan}_{\rho }\left(2\sqrt{\phi \varsigma }\kappa \right)\pm \sqrt{mn}{sec}_{\rho }\left(2\sqrt{\phi \varsigma }\kappa \right)\right)\right]e^{\iota \Phi },$$

(68)

$$U_{14}(x,t)=\left[2\Delta \sqrt{\phi \varsigma }\left(-{cot}_{\rho }\left(2\sqrt{\phi \varsigma }\kappa \right)\pm \sqrt{mn}{csc}_{\rho }\left(2\sqrt{\phi \varsigma }\kappa \right)\right)\right]e^{\iota \Phi },$$

(69)

$$U_{15}(x,t)=\left[\Delta \sqrt{\phi \varsigma }\left({tan}_{\rho }\left(\frac{\sqrt{\phi \varsigma }}{2}\kappa \right)-{cot}_{\rho }\left(\frac{\sqrt{\phi \varsigma }}{2}\kappa \right)\right)\right]e^{\iota \Phi }.$$

(70)

(4)

For $\phi \varsigma <0$ and $\upsilon =0$,

$$U_{16}(x,t)=-\left[2\Delta \sqrt{-\phi \varsigma }\left({tanh}_{\rho }\left(\sqrt{-\phi \varsigma }\kappa \right)\right)\right]e^{\iota \Phi },$$

(71)

$$U_{17}(x,t)=-\left[2\Delta \sqrt{-\phi \varsigma }\left({coth}_{\rho }\left(\sqrt{-\phi \varsigma }\kappa \right)\right)\right]e^{\iota \Phi },$$

(72)

$$U_{18}(x,t)=\left[2\Delta \sqrt{-\phi \varsigma }\left(-{tanh}_{\rho }\left(2\sqrt{-\phi \varsigma }\kappa \right)\pm i{\sqrt{mn}}_{\rho }\left(2\sqrt{-\phi \varsigma }\kappa \right)\right)\right]e^{\iota \Phi },$$

(73)

$$U_{19}(x,t)=\left[2\Delta \sqrt{-\phi \varsigma }\left(-{coth}_{\rho }\left(2\sqrt{-\phi \varsigma }\kappa \right)\pm {\sqrt{mn}}_{\rho }\left(2\sqrt{-\phi \varsigma }\kappa \right)\right)\right]e^{\iota \Phi },$$

(74)

$$U_{20}(x,t)=-\left[\Delta \sqrt{-\phi \varsigma }\left(({tanh}_{\rho }\left(\frac{\sqrt{-\phi \varsigma }}{2}\kappa \right)+{coth}_{\rho }\left(\frac{\sqrt{-\phi \varsigma }}{2}\kappa \right)\right)\right]e^{\iota \Phi }.$$

(75)

(5)

For $\upsilon =0$ and $\varsigma =\phi ,$

$$U_{21}(x,t)=\left[2\Delta \phi \left(ta{n}_{\rho }\left(\phi \kappa \right)\right)\right]e^{\iota \Phi },$$

(76)

$$U_{22}(x,t)=-\left[2\Delta \phi \left(co{t}_{\rho }\left(\phi \kappa \right)\right)\right]e^{\iota \Phi },$$

(77)

$$U_{23}(x,t)=\left[2\Delta \phi \left(ta{n}_{\rho }\left(2\phi \kappa \right)\pm \sqrt{mn}se{c}_{\rho }\left(2\phi \kappa \right)\right)\right]e^{\iota \Phi },$$

(78)

$$U_{24}(x,t)=\left[2\Delta \phi \left(-co{t}_{\rho }\left(2\phi \kappa \right)\pm \sqrt{mn}cs{c}_{\rho }\left(2\phi \kappa \right)\right)\right]e^{\iota \Phi },$$

(79)

$$U_{25}(x,t)=\left[\Delta \phi \left(ta{n}_{\rho }\left(\frac{\phi }{2}\kappa \right)-co{t}_{\rho }\left(\frac{\phi }{2}\kappa \right)\right)\right]e^{\iota \Phi }.$$

(80)

(6)

For $\upsilon =0$ and $\varsigma =-\phi$,

$$U_{26}(x,t)=\left[2\Delta \phi \left(tan{h}_{\rho }\left(\phi \kappa \right)\right)\right]e^{\iota \Phi },$$

(81)

$$U_{27}(x,t)=\left[2\Delta \phi \left(cot{h}_{\rho }\left(\phi \kappa \right)\right)\right]e^{\iota \Phi },$$

(82)

$$U_{28}(x,t)=-\left[2\Delta \phi \left(-tan{h}_{\rho }\left(2\phi \kappa \right)\pm i\sqrt{mn}sec{h}_{\rho }\left(2\phi \kappa \right)\right)\right]e^{\iota \Phi },$$

(83)

$$U_{29}(x,t)=-\left[2\Delta \phi \left(-cot{h}_{\rho }\left(2\phi \kappa \right)\pm \sqrt{mn}csc{h}_{\rho }\left(2\phi \kappa \right)\right)\right]e^{\iota \Phi },$$

(84)

$$U_{30}(x,t)=\left[\Delta \phi \left(tan{h}_{\rho }\left(\frac{\phi }{2}\kappa \right)+cot{h}_{\rho }\left(\frac{\phi }{2}\kappa \right)\right)\right]e^{\iota \Phi }.$$

(85)

(7)

For ${\upsilon }^2=4\phi \varsigma$,

$$U_{31}(x,t)=\left[\frac{-2\Delta }{\kappa ln\left(\rho \right)}\right]e^{\iota \Phi }.$$

(86)

(8)

For $\upsilon =p$, $\phi =pq$, and $\varsigma =0$,

$$U_{32}(x,t)=[\Delta p]e^{\iota \Phi }.$$

(87)

(9)

For $\upsilon =\varsigma =0$,

$$U_{33}(x,t)=0.$$

(88)

(10)

For $\upsilon =\phi =0$,

$$U_{34}(x,t)=-\left[\frac{2\Delta }{\kappa ln\left(\rho \right)}\right]e^{\iota \Phi }.$$

(89)

(11)

For $\phi =0$ and $\upsilon$ ≠ 0,

$$U_{35}(x,t)=\pm \Delta \upsilon \left[1-\frac{2m}{cos{h}_{\rho }\left(\upsilon \kappa \right)-sin{h}_p\left(\upsilon \kappa \right)+m}\right]e^{\iota \Phi }.$$

(90)

$$U_{36}(x,t)=\pm \Delta \upsilon \left[1-2\left[\frac{cos{h}_{\rho }\left(\upsilon \kappa \right)+sin{h}_p\left(\upsilon \kappa \right)}{cos{h}_{\rho }\left(\upsilon \kappa \right)+sin{h}_p\left(\upsilon \kappa \right)+n}\right]\right]e^{\iota \Phi }$$

(91)

(12)

For $\upsilon =p$, $\varsigma =pq$, where q ≠ 0 and $\varsigma =0$,

$$U_{37}(x,t)=\Delta p\left[1-\frac{2mq{\rho }^{p\kappa }}{m-nq{\rho }^{p\kappa }}\right]e^{\iota \Phi }$$

(92)

3.3. The First Integral and Exact Solution with Nucci’s Reduction Method

This portion contains the application of Nucci’s reduction method, along with the implications of this technique. Let us assume the change of variables ${\Psi }_1\left(\kappa \right)=U\left(\kappa \right)$, ${\Psi }_2\left(\kappa \right)=U^{\prime }\left(\kappa \right)$, and we get a dynamic system of ODEs of the first order by using the following Galilean transformation:

$$\begin{array}{c}\hfill \begin{array}{cc}& \frac{d{\Psi }_1}{d\kappa }={\Psi }_2,\hfill \\ & \frac{d{\Psi }_2}{d\kappa }=\frac{a{k}_1^2+\lambda }{a{\alpha }^2}{\Psi }_1-\frac{b}{a{\alpha }^2}{\Psi }_1^2.\hfill \end{array}\end{array}$$

(93)

Since this is an autonomous system of equations, we can assume that ${\Psi }_1$ is a new independent variable. At the same time, Equation (93) can be simplified to

$$\begin{array}{c}\hfill \frac{d{\Psi }_2}{d{\Psi }_1}=\frac{a{k}_1^2+\lambda }{a{\Psi }_2{\alpha }^2}{\Psi }_1-\frac{b}{a{\Psi }_2{\alpha }^2}{\Psi }_1^2.\end{array}$$

(94)

Equation (94) is a separable ODE, and exact solution is acquired through an integration process:

$$\begin{array}{c}\hfill {\Psi }_2\left({\Psi }_1\right)=\frac{\sqrt{2}\sqrt{a\alpha \left(2a{\Psi }_1^2{k}_1^2-b{\Psi }_1^4+2\theta a\alpha +2\lambda {\Psi }_1^2\right)}}{\sqrt{2}a\alpha },\end{array}$$

(95)

with an arbitrary constant $\theta$. The first integral can be obtained with a constant value of $\theta$:

$$\begin{array}{c}\hfill \theta =\frac{{\Psi }_1^2\left(2a{k}_1^2-b{\Psi }_1^2+2\lambda \right)}{a\alpha }.\end{array}$$

(96)

Substituting Equation (95) into Equation (93) yields

$$\begin{array}{c}\hfill \frac{d{\Psi }_1\left(\kappa \right)}{d\kappa }=\frac{1}{2}\sqrt{4a{\Psi }_1^2{k}_1^2-2b{\Psi }_1^4+4\lambda {\Psi }_1^2+4\theta },\end{array}$$

(97)

Equation (97) is also a first-order separable equation, and with same pattern, we can get the general solution as follows:

$$\begin{array}{c}\hfill U(x,t)=2a\alpha \Omega \sqrt{\frac{4a\lambda k_1^2+2{\lambda }^2+a^2{k}_1^2+a^2{k}_1^4}{4a^4{\alpha }^4{b}^2(a^2{k}_1^4+2a\lambda k_1^2+{\lambda }^2)+4{\Omega }^2{a}^2{\alpha }^{2}b(a{k}_1^2+\lambda )+{\Omega }^4}},\end{array}$$

(98)

whenever $\theta =0$ and $\Omega ={\mathrm{e}}^{-\frac{ln\left(2\right)a\alpha -\sqrt{a^2\alpha {\kappa }^2{k_1}^2+a\alpha {\kappa }^2\lambda }}{a\alpha }}$.

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill U(x,t)=& \frac{\sqrt{2\theta a\alpha }}{\sqrt{-a{k_1}^2+\sqrt{a^2{k_1}^4+2a\alpha b\theta +2a\lambda {k_1}^2+{\lambda }^2}-\lambda }}\hfill \\ & \mathit{JacobiSN}\left(\frac{\sqrt{\left(-a{k_1}^2+\Xi -\lambda \right)}\kappa }{\sqrt{2a\alpha }},\sqrt{-\frac{a^2{k_1}^4+\Xi a{k_1}^2+a\alpha b\theta +2a\lambda {k_1}^2+\Xi \lambda +{\lambda }^2}{a\alpha b\theta }}\right),\hfill \end{array}\end{array}$$

(99)

whenever $\theta \ne 0$ and $\Xi =\sqrt{a^2{k_1}^4+2a\alpha b\theta +2a\lambda {k_1}^2+{\lambda }^2}$.

4. Graphical Discussion

In this section, we present a detailed graphical analysis of the model under discussion. The complex solution $U_3(x,t)$ is graphically presented by taking appropriate values of $\rho =5,m=7,n=2,a=5,b=0.4,\varrho =3,\zeta =2,\nu =4,k_1=2,\mu =0.2$, and $\lambda =5$. Figure 1 and Figure 2 presented the graphical impact of the wave number on the propagating wave profile for the real part of the solution $U_3(x,t)$. Figure 3 and Figure 4 presented the graphical impact of the wave number on the propagating wave profile for the imaginary part of the solution $U_3(x,t)$. Figure 5 and Figure 6 presented the graphical impact of the soliton speed on the propagating wave profile for the real and imaginary parts of the solution in Equation (98) at $\alpha =0.09,\rho =5,a=0.5,b=0.4,\varrho =3,k_1=0.1,$ and $\lambda =5$.

5. Conclusions

In this research, we utilized generalized approaches to examine the nonlinear dissipative Schrödinger equation. The established solitonic structures and exact solutions were addressed with a new extended direct algebraic equation technique and Nucci’s reduction, respectively. As a result, we obtained:

• The mixed complex solitary shock solution, singular solution, mixed shock singular solution, mixed trigonometric solution, mixed singular solution, exact solution, mixed periodic solution, and mixed hyperbolic solution, as well as the periodic solution.
• The first integral was developed for the nonlinear dissipative Schrödinger equation.
• A 2D, 3D, and contour visualization was presented, and it was observed that the dissipative parameter and velocity of the soliton were responsible for controlling the amplitude of the propagating wave.

It is hoped that this study will be important for analysts and researchers for improving experimental work, and that it can be expanded in a way that includes multiple solitons, lump interactions, and rogue wave breather types.