Navigating waves: Advancing ocean dynamics through the nonlinear Schrödinger equation

Ifrah Iqbal; Salah Mahmoud Boulaaras; Hamood Ur Rehman; Muhammad Shoaib Saleem; Dean Chou

doi:10.1515/nleng-2024-0025

Article Open Access

Navigating waves: Advancing ocean dynamics through the nonlinear Schrödinger equation

Ifrah Iqbal , Salah Mahmoud Boulaaras , Hamood Ur Rehman , Muhammad Shoaib Saleem and Dean Chou

Published/Copyright: September 27, 2024

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From the journal Nonlinear Engineering Volume 13 Issue 1

Abstract

The nonlinear Schrödinger equation, held in high regard in the realms of plasma physics, fluid mechanics, and nonlinear optics, reverberates deeply within the field of ocean engineering, imparting profound insights across a plethora of phenomena. This article endeavours to establish a connection between the equation’s theoretical framework and its practical applications in ocean engineering, presenting a range of solutions tailored to grasp the intricacies of water wave propagation. By employing three methodologies, namely, the simplest equation method, the ratio technique, and the modified extended tanh-function method, we delineate various wave typologies, encompassing solitons and periodic manifestations. Enhanced by visual representations, our findings have the potential to deepen the comprehension of wave dynamics, with promising implications for the advancement of ocean engineering technologies and the refinement of marine architectural design.

Keywords: nonlinear Schrödinger equation; multisoliton; ocean engineering; simplest equation method; [1/φ(ς), φ′(ς)/φ(ς)] method; modified extended tanh-function method; nonlinear equations

1 Introduction

In recent times, researchers and scholars have increasingly derived precise solutions for nonlinear partial differential equations (NLPDEs). NLPDEs play an important role in elucidating the underlying physical mechanisms of various natural phenomena and dynamic processes across multiple scientific domains such as geochemistry, ocean engineering, fluid mechanics, physics, geophysics, plasma physics, and optical fibres [1–10]. In recent years, substantial progress has been made in exploring analytical solutions for NLPDEs. These nonlinear wave equations introduce diverse solution types that significantly differ from their linear counterparts. Notably, solutions such as water waves, shock waves, and solitary waves have emerged as prominent examples. Solitary wave solutions, in particular, hold significance due to their potential applications in various physical domains, including neural physics, chaos theory, diffusion processes, and reaction kinetics. Many methodologies have been employed to uncover wave solutions for NLPDEs, including the variational iteration method [11], Sardar-subequation method [12,13], Kudryashov’s method [14], extended direct algebraic equation method [15], exp-function algorithm [16], Lie symmetry analysis [17–19], F-expansion scheme [20], generalised exponential rational function [21–23], mapping method [24], and modified rational sine-cosine method [25] and many others [26–31]. These diverse methodologies offer various avenues for exploring the analytical solutions, each contributing uniquely to the understanding and application of nonlinear wave equations.

The Schrödinger’s equation serves as a fundamental framework for describing particle behaviour within a force field or tracking the temporal evolution of physical quantities [32]. Due to its widespread applications, several enhanced formulations of this equation have been documented in the literature, such as an extended version of the nonlinear Schrödinger equation (NLSE) proposed in [33]. Investigations on the perturbed NLSE hierarchy have been conducted in [34]. In [35], the generally projective Riccati equation method is applied to ascertain wave solutions for a specific variant of NLSE. Furthermore, Li and Chen [36] delve into studying the NLSE featuring varying coefficients. This article focuses on the modified NLSE as formulated in [37,38]:

(1) ι q t + a 1 q x x + a 2 ∣ q ∣ 2 q − ι b 1 q x x x − ι b 2 q 2 q x * + ι b 3 ∣ q ∣ 2 q x − b 4 q = 0 ,

where

a 1 = δ 8 k 2 ( − 3 cos 2 ( θ ) + 2 ) , a 2 = − δ k 2 2 , b 1 = δ cos ( θ ) k 16 k 3 ( − 5 cos 2 ( θ ) − 6 ) , b 2 = − δ k cos ( θ ) 4 , b 3 = 3 δ k 2 2 , b 4 = k ∣ q ∣ x 2 ∣ x = 0 .

In the aforementioned equation, δ and k represent frequency and wave number, respectively. The notation ∣ ∣ represents the absolute value of q ( x , t ) , while the symbol * denotes the conjugate of q ( x , t ) . This model finds extensive application in ocean engineering, particularly in tsunami waves and intricate structures that factor in nonlinearities and dispersion terms. In the field of ocean engineering, the prediction and investigation of tsunami waves are of paramount importance due to their potential catastrophic impact on coastal regions. The modified NLSE plays an important role in this context, enabling a highly accurate analysis of these waves. The proposed governing model primarily addresses water waves, especially in situations where long-wave phenomena are observed. These occurrences of long waves are prevalent in both optical wave guides and fluid dynamics, where unexpectedly substantial displacements become noticeable. In the domain of ocean engineering, understanding long-wave occurrences is critical. In optical wave guides, the modified NLSE aids researchers in controlling the behaviour of light pulses, as seen in their applications in fibre optics for communication.

The primary objective of this article is to demonstrate the effectiveness of three methodologies, namely, the SEM [39,40], the 1 φ ( ς ) , φ ′ ( ς ) φ ( ς ) method [41], and METM [42,43], in extracting precise solutions for the modified NLSE. Through the utilization of these three techniques, our aim is to derive a diverse spectrum of wave solutions including solitary waves, dark and bright solitons, singular solitons, multisoliton, and combined solutions. This study seeks to demonstrate and compare the efficacy of these methodologies in obtaining accurate and varied solutions for the modified NLSE. The rest of the article is organised as follows: Section 2 delves into the mathematical analysis of the governed equation. Section 3 comprehensively presents the SEM, accompanied by its practical application on the equation. In Section 4, the methodology involving the 1 φ ( ς ) , φ ′ ( ς ) φ ( ς ) method is detailed, along with its application to the equation. Section 6 elaborates on the METM, providing insights into its methodology and application. Section 7 serves as the results and discussion. Finally, Section 8 encapsulates the conclusions drawn from the article.

2 Mathematical analysis

Consider the following transformation:

(2) q ( x , t ) = Q ( ς ) e ι φ , ς = κ ( x − w t ) , φ = − η x + v t + θ ,

where κ , w , η , v , and θ represent amplitude, velocity, frequency, wave number, and phase component of the soliton, respectively. By putting the aforementioned equation into (1), the real part of (1) is given as follows:

(3) κ 2 ( − 3 b 1 η + a 1 ) Q ″ + ( a 2 + ( b 2 + b 3 ) η ) Q 3 + ( − v − a 1 η 2 + b 1 η 3 − b 4 ) Q = 0 ,

and the imaginary part is

(4) ( 3 b 1 η 2 − w − 2 a 1 η ) Q ′ − b 1 κ 2 Q ‴ + ( b 2 − b 3 ) Q 2 Q ′ = 0 .

By integrating the (4) of both sides

(5) 3 ( 3 b 1 η 2 − w − 2 a 1 η ) Q − 3 b 1 κ 2 Q ″ + ( b 3 − b 2 ) Q 3 = 0 ,

where the constant of integration is taken as zero. Therefore, from (3) and (5), the following constants are derived.

w = − b 1 v + b 1 b 4 + 2 η ( a 1 − 2 b 1 η ) 2 a 1 − 3 b 1 η , η = a 1 ( b 2 − b 3 ) − 3 a 2 b 1 6 b 1 b 2 .

3 Simplest equation method

Assuming that the solution to equation (3) can be represented in a finite series form,

(6) Q ( ς ) = ∑ i = 0 M g i R ( ς ) i ,

where M can be determined by using balancing rule in which the highest derivative is compared with the nonlinear term and R ( ς ) i satisfies the simple equation (SE) [44]. The important stage in utilising the SEM nvolves selecting the SE, which serves as the foundational framework for generating novel solutions to the nonlinear equations under investigation. In the pursuit of uncovering solitary and multisoliton solutions, the coupled Burgers’ equation, outlined by Wazwaz [45], is specifically selected as it represents the most SE for this purpose.

The Burger equation:

To investigate the multisoliton solutions of the given equation, we selected the coupled Burger’s equations as the SE due to their complete integrability within a (1 + 1)-dimensional framework. These equations can be represented as follows [45]:

(7) ν t − 2 ν ν x − ν x x = 0 .

Let ν ( x , t ) = R ( ς ) .

(8) − κ 2 R ς ς − c R ς − 2 κ R R ς = 0 ,

where c = − κ w . Performing integrations of equation (8) with respect to ς and setting the resulting integral constant to zero results in

(9) R ς = − c κ 2 R − 1 κ R 2 .

The dispersion relation is expressed as follows:

(10) c = − κ 2 .

Hence, (9) is reduced to

(11) R ς = R − 1 κ R 2 .

The generalised multisoliton solution for (11) derived using Hirota’s method is formulated as [45]

(12) R = ∑ j = 1 N κ j e κ j x + c j t 1 + ∑ j = 1 N e κ j x + c j t ,

where κ j and c j are arbitrary constants. Now by putting (11) into (6), we can find the solution of given equation.

3.1 Application of SEM

By using the homogeneous rule on (3), we compare Q ″ and Q 3 ⇒ M + 2 = 3 M , we obtain M = 1 . Let the following solution

(13) Q ( ς ) = g 0 + g 1 R ( ς ) .

By putting (13) along (11), into (3), we obtained following constants:

g 0 = g 0 , g 1 = g 1 , a 2 = − a 1 ( b 2 + b 3 ) 3 b 1 , η = a 1 3 b 1 , v = − 2 a 1 3 − 27 b 1 2 b 4 27 b 1 2 .

By putting these constants in (13), we have the following solutions:

(14) q 1 ( x , t ) = g 0 + g 1 κ 1 e κ 1 x + c 1 t 1 + e κ 1 x + c 1 t e ι φ .

(15) q 2 ( x , t ) = g 0 + g 1 κ 1 e κ 1 x + c 1 t + κ 2 e κ 2 x + c 2 t 1 + e κ 1 x + c 1 t + e κ 2 x + c 2 t e ι φ .

(16) q 3 ( x , t ) = g 0 + g 1 κ 1 e κ 1 x + c 1 t + κ 2 e κ 2 x + c 2 t + κ 3 e κ 3 x + c 3 t 1 + e κ 1 x + c 1 t + e κ 2 x + c 2 t + e κ 3 x + c 3 t e ι φ .

4 Description of 1 φ ( ς ) , φ ′ ( ς ) φ ( ς ) method

Let the solution of (3) is

(17) Q ( ς ) = g 0 + ∑ j = 1 N g j + h j φ ′ ( ς ) j φ ( ς ) ,

where g 0 , g j , and h j ( j = 1 , 2 , … , N ) are constants. N can be obtained by using homogeneous balancing rule and φ ( ς ) represents the following ODE:

(18) φ ′ ( ς ) 2 = φ ( ς ) 2 − υ ,

where

(19) φ ( ς ) = d e ς + υ 4 d e ς .

Now, by inserting (17) along (18) into (19), the system of equations is attained and by solving it, we obtain the values of constants.

4.1 Application of 1 φ ( ς ) , φ ′ ( ς ) φ ( ς ) method

Now by using the homogeneous balance rule, we obtain N = 1 .

(20) Q ( ς ) = g 0 + g 1 + h 1 φ ′ ( ς ) φ ( ς ) .

Now, by using (20) along (18) into (19), the system of equations is attained and by solving it, we obtain the following values of constants.

Set 1:

g 0 = 0 , g 1 = g 1 , h 1 = 0 , a 1 = a 2 g 1 2 + 6 b 1 κ 2 η υ + b 2 η g 1 2 + b 3 η g 1 2 2 κ 2 υ , v = − a 2 η 2 g 1 2 + a 2 κ 2 g 1 2 − 2 b 4 κ 2 − 4 b 1 κ 2 η 3 υ − b 2 κ 3 g 1 2 − b 3 κ 3 g 1 2 + b 2 κ 2 η g 1 2 + b 3 κ 2 η g 1 2 2 κ 2 υ .

By substituting these values into (20), we have

q 1 , * ( x , t ) = g 1 4 d e ς 4 d 2 e 2 ς + υ e ι φ .

By taking υ = ± 4 d 2 , we obtain

(21) q 1 , 1 ( x , t ) = g 1 sech ( ς ) 2 d e ι φ .

(22) q 1 , 2 ( x , t ) = g 1 csch ( ς ) 2 d e ι φ .

Set 2:

g 0 = 0 , g 1 = 0 , h 1 = h 1 , a 1 = − a 2 h 1 2 + 6 b 1 κ 2 η + b 2 ( − η ) h 1 2 − b 3 η h 1 2 2 κ 2 , v = a 2 η 2 h 1 2 + 2 a 2 κ 2 h 1 2 − 2 b 4 η 2 − 4 b 1 κ 2 η 3 + b 2 η 3 h 1 2 + b 3 η 3 h 1 2 + 2 b 2 κ 2 η h 1 2 + 2 b 3 κ 2 η h 1 2 2 κ 2 .

By substituting these values into Eq. (20), we have

q 2 , * ( x , t ) = h 1 ( 4 d 2 e 2 ς − υ ) 4 d 2 e 2 ς + υ e ι φ .

By taking υ = ± 4 d 2 , we obtain

(23) q 1 , 3 ( x , t ) = h 1 tanh ( ς ) e ι φ .

(24) q 1 , 4 ( x , t ) = h 1 coth ( ς ) e ι φ .

5 Modified extended tanh-function method

In this method, the solution of (3) is written as follows:

(25) Q ( ς ) = ω 0 + ∑ j = 1 N ω j Y j ( ς ) + ∑ j = 1 N Z j Y j ( ς ) .

The function Y j ( ς ) satisfies the following equation:

(26) Y ′ ( ς ) = Y 2 ( ς ) + p .

The solution of (26) can be expressed as follows:

(Case 1): If p < 0 , then

Y 1 ( ς ) = − − p tanh ( − − p ς ) , Y 2 ( ς ) = − − p coth ( − − p ς ) .

(Case 2): If p > 0 , then

Y 3 ( ς ) = p tan ( p ς ) , Y 4 ( ς ) = − p cot ( p ς ) .

(Case 3): If p = 0 , then

Y 5 ( ς ) = − 1 ς .

Now, by using (25) and (26) into (3), the system of equations in obtained and solving this system, a set of constants is obtained. By using these values of constants along above cases, we obtain the solution of (1).

5.1 Application of modified extended tanh-function method

By using the homogeneous balancing rule, the N = 1 is acquired. Hence, from (25), suppose the following solutions:

(27) Q ( ς ) = ω 0 + ω 1 Y ( ς ) + Z 1 Y ( ς ) .

By putting (27) along (26) into (3), we obtained the following constants.

Set: 1

ω 0 = 0 , ω 1 = ω 1 , b 1 = a 1 3 η , a 2 = b 2 ( − κ ) − b 3 κ , Z 1 = 0 , b 4 = 1 3 ( − 2 a 1 η 2 − 3 v ) .

(Case 1): If p < 0 , then

q 2 , 1 ( x , t ) = ( ω 1 ( − − p tanh ( − − p ς ) ) ) e ι φ , q 2 , 2 ( x , t ) = ( ω 1 ( − − p coth ( − − p ς ) ) ) e ι φ .

(Case 2): If p > 0 , then

q 2 , 3 ( x , t ) = ( ω 1 ( p tan ( p ς ) ) ) e ι φ , q 2 , 4 ( x , t ) = ( ω 1 ( p cot ( p ς ) ) ) e ι φ .

(Case 3): If p = 0 , then

q 2 , 5 ( x , t ) = − ω 1 ς e ι φ .

Set: 2

ω 0 = 0 , Z 1 = ω 1 p , a 2 = − 2 a 1 κ 2 + b 2 ω 1 2 ( − η ) − b 3 ω 1 2 η + 6 b 1 κ 2 η ω 1 2 , b 4 = − 4 a 1 κ 2 p − a 1 η 2 + 12 b 1 κ 2 p η + b 1 η 3 − v .

(Case 1): If p < 0 , then

q 2 , 6 ( x , t ) = ω 1 p coth ( ς − p ) − Z 1 tanh ( ς − p ) − p e ι φ , q 2 , 7 ( x , t ) = ω 1 p tanh ( ς − p ) − Z 1 coth ( ς − p ) − p e ι φ .

(Case 2): If p > 0 , then

q 2 , 8 ( x , t ) = ω 1 p tan ( ς p ) + Z 1 cot ( ς p ) p e ι φ , q 2 , 9 ( x , t ) = ω 1 p cot ( ς p ) + Z 1 tan ( ς p ) p e ι φ .

Note: It is important to highlight that q 2 , 6 ( x , t ) and q 2 , 7 ( x , t ) exhibit similarities to q 2 , 8 ( x , t ) and q 2 , 9 ( x , t ) . This is due to the mathematical identity tanh 2 ( − p ) = − tan 2 ( p ) , which holds true irrespective of the value of p . This equivalence underlines the relation between hyperbolic and trigonometric functions in the given context.

6 Graphical representation

This study employed three distinct techniques to investigate the solutions of the modified NLSE. The obtained solutions showcase the diverse and rich behaviour of waves described by the modified NLSE. Multisoliton solutions derived from SEM indicate complex wave structures involving multiple solitons, while the solutions obtained through the 1 φ ( ς ) , φ ′ ( ς ) φ ( ς ) method encompass dark, bright, and singular waveforms, each depicting distinct characteristics of wave concentration and propagation. Moreover, the application of the METM unveiled dark, periodic, and singular solutions alongside combined soliton patterns, further expanding the spectrum of possible wave behaviours described by the NLSE. The comparison between these methods is given in Table 1. Graphical representations in both 3D and 2D forms, illustrating absolute, real, and imaginary values, give a visual understanding of behaviour of waves in different dimensions. These visualisations serve as important tools in understanding the dynamics of waves and help in validating the accuracy of the attained solutions. These graphical representations depict different wave patterns influenced by constants and have applications in marine engineering for designing resilient structures and controlling wave propagation.

Table 1

Comparison betwen three methods

Method	Advantages	Limitation
SEM	This method is utilised to extract multisoliton solutions without using the Hirota Bilinear form.	As, this method employs a specialised form of the (1+1)D Burger’s equation, it is capable of solving only (1+1)D NLPDEs to extract multiple solitons.
1 φ ( ς ) , φ ′ ( ς ) φ ( ς ) Method	This method yields solutions in hyperbolic form with parameter values that result in a large number of solutions, including bright, dark, and singular solitons.	This method has limitation in extracting periodic-singular solutions.
METM	METM constructs hyperbolic, trigonometric and rational solutions in the form of dark, singular, periodic singular, and combined dark-singular soliton solutions.	METM fails to construct bright soliton.

Figure 1(a) and (b) display the absolute value graph of q 1 ( x , t ) , showing kink soliton pattern. This soliton structure has transition and shows a significant change in the wave amplitude. In Figure 1(c) and (d), the real value graphs of q 1 ( x , t ) illustrate periodic behaviour. These graphs exhibit repetitive patterns over a given period. Figure 1(e) and (f) portray the imaginary value graphs of q 1 ( x , t ) , also showcasing periodic characteristics similar to the real value graphs. These graphs correspond to the provided constants: a 1 = 0.5 , b 1 = 2 , b 4 = 2 , b 2 = 0.2 , b 3 = 1 , κ 1 = − 1 , g 0 = 1.5 , v = 1 , and ϑ = 1 .

$Figure 1 1-soliton solution of ∣ q 1 ( x , t ) | {q}_{1}\left(x,t) with a 1 = 0.5 {a}_{1}=0.5 , b 1 = 2 {b}_{1}=2 , b 4 = 2 {b}_{4}=2 , b 2 = 0.2 {b}_{2}=0.2 , b 3 = 1 {b}_{3}=1 , κ 1 = ‒ 1 {\kappa }_{1}=‒1 , g 0 = 1.5 {g}_{0}=1.5 , v = 1 v=1 , and ϑ = 1 {\vartheta }=1 .$

Figure 1

1-soliton solution of ∣ q 1 ( x , t ) with a 1 = 0.5 , b 1 = 2 , b 4 = 2 , b 2 = 0.2 , b 3 = 1 , κ 1 = ‒ 1 , g 0 = 1.5 , v = 1 , and ϑ = 1 .

Figure 2(a) and (b) represent the absolute value graph of q 2 ( x , t ) , illustrating the presence of two distinct soliton graphs. These soliton structures indicate the presence of multiple waves within the solution. In Figure 2(c) and (d), the real value graphs of q 2 ( x , t ) display periodic behaviour while Figure 2(e) and (f), portraying the imaginary value graphs, exhibit similar periodic characteristics as observed in the real value graphs. These figures correspond to the defined constants: a 1 = 0.5 , b 1 = 2 , b 4 = 2 , b 2 = 0.2 , b 3 = 1 , κ 1 = − 1 , κ 2 = 0.5 , g 0 = 1.5 , v = 1 , and ϑ = 1 .

$Figure 2 2-soliton solution of q 2 ( x , t ) {q}_{2}\left(x,t) with a 1 = 0.5 {a}_{1}=0.5 , b 1 = 2 {b}_{1}=2 , b 4 = 2 {b}_{4}=2 , b 2 = 0.2 {b}_{2}=0.2 , b 3 = 1 {b}_{3}=1 , κ 1 = ‒ 1 {\kappa }_{1}=‒1 , κ 2 = 0.5 {\kappa }_{2}=0.5 , g 0 = 1.5 {g}_{0}=1.5 , v = 1 v=1 , and ϑ = 1 {\vartheta }=1 .$

Figure 2

2-soliton solution of q 2 ( x , t ) with a 1 = 0.5 , b 1 = 2 , b 4 = 2 , b 2 = 0.2 , b 3 = 1 , κ 1 = ‒ 1 , κ 2 = 0.5 , g 0 = 1.5 , v = 1 , and ϑ = 1 .

Figure 3(a) and (b) display the graph of q 3 ( x , t ) showing the presence of three soliton patterns. These soliton graphs illustrate the coexistence of multiple waves within the solution. In Figure 3(c) and (d), the real value graphs of q 3 ( x , t ) demonstrate periodic wave behaviour, depicting repetitive patterns indicative over specific intervals. Similarly, Figure 3(e) and (f), representing the imaginary value graphs of q 3 ( x , t ) , portray periodic characteristics. These visualisations correspond to the specified constants: a 1 = 0.5 , b 1 = 2 , b 4 = 2 , b 2 = 0.2 , b 3 = 1 , κ 1 = − 1 , κ 2 = 0.5 , κ 3 = 0.5 , g 0 = 1.5 , v = 1 , and ϑ = 1 .

$Figure 3 3-solitons solutions of q 3 ( x , t ) {q}_{3}\left(x,t) with a 1 = 0.5 {a}_{1}=0.5 , b 1 = 2 {b}_{1}=2 , b 4 = 2 {b}_{4}=2 , b 2 = 0.2 {b}_{2}=0.2 , b 3 = 1 {b}_{3}=1 , κ 1 = ‒ 1 {\kappa }_{1}=‒1 , κ 2 = 0.5 {\kappa }_{2}=0.5 , κ 3 = 0.5 , g 0 = 1.5 {\kappa }_{3}=0.5,{g}_{0}=1.5 , v = 1 v=1 , and ϑ = 1 {\vartheta }=1 .$

Figure 3

3-solitons solutions of q 3 ( x , t ) with a 1 = 0.5 , b 1 = 2 , b 4 = 2 , b 2 = 0.2 , b 3 = 1 , κ 1 = ‒ 1 , κ 2 = 0.5 , κ 3 = 0.5 , g 0 = 1.5 , v = 1 , and ϑ = 1 .

Figure 4(a) and (b) display the bright soliton pattern observed in the absolute value graph for q 1 , 1 ( x , t ) , while Figure 4(c) and (d) shows the real value graph, portraying a combined dark-bright pattern. Similarly, Figure 4(e) and (f) demonstrate the imaginary value graph also show dark-bright soliton. The constants utilised for generating these graphs include η = 1 , κ = 2 , p = 1 , v = 0.9 , w = − 0.5 , g 1 = 1 , d = 1 , and ϑ = 1 .

$Figure 4 Bright soliton for ∣ q 1 , 1 ( x , t ) {q}_{1,1}\left(x,t) ∣ with η = 1 \eta =1 , κ = 2 \kappa =2 , p = 1 p=1 , v = 0.9 v=0.9 , w = ‒ 0.5 w=‒0.5 , g 1 = 1 {g}_{1}=1 , d = 1 d=1 , and ϑ = 1 {\vartheta }=1 .$

Figure 4

Bright soliton for ∣ q 1 , 1 ( x , t ) ∣ with η = 1 , κ = 2 , p = 1 , v = 0.9 , w = ‒ 0.5 , g 1 = 1 , d = 1 , and ϑ = 1 .

Figure 5(a) and (b) presents the absolute value graph of q 1 , 3 ( x , t ) , exhibiting a dark soliton pattern. These graphs illustrate the concentrated and stable nature of the dark soliton within the solution. In Figure 5(c) and (d), the real value graphs of q 1 , 3 ( x , t ) show periodic behaviour. Similarly, Figure 5(e) and (f) portray the imaginary value graphs of q 1 , 3 ( x , t ) , also displaying periodic characteristics akin to the real value graphs. The specified parameters utilised to generate these graphs are η = 1 , κ = 2 , p = 1 , v = 0.9 , w = − 0.5 , h 1 = 1 , and ϑ = 1 .

$Figure 5 Dark soliton of q 1 , 3 ( x , t ) {q}_{1,3}\left(x,t) with η = 1 \eta =1 , κ = 2 \kappa =2 , p = 1 p=1 , v = 0.9 v=0.9 , w = ‒ 0.5 w=‒0.5 , h 1 = 1 {h}_{1}=1 , and ϑ = 1 {\vartheta }=1 .$

Figure 5

Dark soliton of q 1 , 3 ( x , t ) with η = 1 , κ = 2 , p = 1 , v = 0.9 , w = ‒ 0.5 , h 1 = 1 , and ϑ = 1 .

Figure 6(a) and (b) represent the graph of q 2 , 2 ( x , t ) exhibiting a singular soliton pattern within the solution. These graphs demonstrate the concentrated nature of the singular soliton. In Figure 6(c) and (d), the real value graphs of q 2 , 2 ( x , t ) display the characteristics of the singular soliton. These graphs illustrate the distinct behaviour and stability of the real component of the solution. Similarly, Figure 6(e) and (f) shows the imaginary value graphs of q 2 , 2 ( x , t ) , also highlighting the singular soliton characteristics within the imaginary component of the solution. The specified parameters used to generate these graphs include η = 1 , κ = 2 , p = 1 , v = 0.9 , w = − 0.5 , ω 1 = 1 , and ϑ = 1 .

$Figure 6 Singular soliton of q 2 , 2 ( x , t ) {q}_{2,2}\left(x,t) with η = 1 \eta =1 , κ = 2 \kappa =2 , p = 1 p=1 , v = 0.9 v=0.9 , w = ‒ 0.5 w=‒0.5 , ω 1 = 1 {\omega }_{1}=1 , and ϑ = 1 {\vartheta }=1 .$

Figure 6

Singular soliton of q 2 , 2 ( x , t ) with η = 1 , κ = 2 , p = 1 , v = 0.9 , w = ‒ 0.5 , ω 1 = 1 , and ϑ = 1 .

Figure 7(a) and (b) shows the graph of q 2 , 7 ( x , t ) at its absolute value, displaying a dark singular pattern within the solution. These graphs illustrate the concentrated and distinct nature of the dark-singular graph. In Figure 7(c) and (d), the real value graphs of q 2 , 7 ( x , t ) demonstrate the characteristics of the singular graph, highlighting its behaviour and stability within the real component of the solution. Similarly, Figure 7(e) and (f) portray the imaginary value graphs of q 2 , 7 ( x , t ) , also displaying the characteristics of the singular graph within the imaginary component of the solution. The used parameters are η = 1 , κ = 2 , p = − 1 , v = 0.9 , w = − 0.5 , ω 1 = 1 , and ϑ = 1 .

$Figure 7 Combined dark-singular soliton of q 2 , 7 ( x , t ) {q}_{2,7}\left(x,t) with η = 1 \eta =1 , κ = 2 \kappa =2 , p = ‒ 1 p=‒1 , v = 0.9 v=0.9 , w = ‒ 0.5 w=‒0.5 , ω 1 = 1 {\omega }_{1}=1 , and ϑ = 1 {\vartheta }=1 .$

Figure 7

Combined dark-singular soliton of q 2 , 7 ( x , t ) with η = 1 , κ = 2 , p = ‒ 1 , v = 0.9 , w = ‒ 0.5 , ω 1 = 1 , and ϑ = 1 .

7 Conclusion

The denouement of this inquiry, centred upon the meticulous derivation of solutions for the modified NLSE, has bestowed illuminating insights into the nuanced domain of water wave propagation within the ambit of ocean engineering. Employing sophisticated methodologies such as the SEM, the 1 φ ( ς ) , φ ′ ( ς ) φ ( ς ) method, and METM, we have adeptly elicited distinctive wave solutions, encompassing manifestations of brightness, periodicity, dark solitons, singularity, as well as amalgamated and multiple wave configurations. Graphical depictions delineating absolute, real, and imaginary constituents have proffered invaluable visual elucidations, engendering a profound comprehension of the intricacies inherent in wave dynamics. These discernments augur transformative implications for the paradigm of ocean engineering technologies, particularly in the governance of wave propagation and the refinement of resilient marine structures.

Acknowledgement

The authors wish to express gratitude for the support provided by the National Science and Technology Council in Taiwan under grant numbers 112-2115-M-006-002 and 112-2321-B-006-020.

Funding information: The authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: The datasets generated and/or analyzed during the current study are accessible within the manuscript.

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Received: 2024-04-09

Revised: 2024-06-05

Accepted: 2024-06-21

Published Online: 2024-09-27

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/nleng-2024-0025

Keywords for this article

nonlinear Schrödinger equation; multisoliton; ocean engineering; simplest equation method; [1/φ(ς), φ′(ς)/φ(ς)] method; modified extended tanh-function method; nonlinear equations

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