Comparative <i>in vitro</i> activity of various antibiotic against planktonic and biofilm and the gene expression profile in <i>Pseudomonas aeruginosa</i>

Mohammad Abu-Sini; Mohammad A. Al-Kafaween; Rania M. Al-Groom; Abu Bakar Mohd Hilmi; Mohammad Abu-Sini; Mohammad A. Al-Kafaween; Rania M. Al-Groom; Abu Bakar Mohd Hilmi

doi:10.3934/microbiol.2023017

AIMS Microbiology

2023, Volume 9, Issue 2: 313-331. doi: 10.3934/microbiol.2023017

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Research article

Comparative in vitro activity of various antibiotic against planktonic and biofilm and the gene expression profile in Pseudomonas aeruginosa

1.
Department of Pharmacy, Faculty of Pharmacy, Al-Zaytoonah University of Jordan, Amman, Jordan
2.
Department of Medical Laboratory Science, Faculty of Allied Medical Sciences, Zarqa University, Zarqa, Jordan
3.
Department of Allied Medical Sciences, Zarqa University College, Balqa Applied University, Al-Salt, Jordan
4.
Department of Biomedicine, Faculty of Health Sciences, Universiti Sultan Zainal Abidin, Terengganu, Malaysia

Received: 08 December 2022 Revised: 10 March 2023 Accepted: 13 March 2023 Published: 30 March 2023

P. aeruginosa is an opportunistic pathogen that is commonly found in nosocomial infections. The purpose of this study was to investigate the effects of seven antibiotics on P. aeruginosa planktonic growth, biofilm formation, and the expression of virulence factors. These antibiotics included Ciprofloxacin (CP), Amikacin (AMK), Vancomycin (VAN), Tetracycline (TET), Gentamicin (GEN), Erythromycin (Ery), and Clindamycin (CLI). Antibiotic susceptibility testing, Minimum Bactericidal Concentration (MBC), Minimum Inhibitory Concentration (MIC), growth curve, time-kill curve, biofilm inhibition and reduction assay, and RT-qPCR were used to assess the effects of these antibiotics on P. aeruginosa planktonic and biofilm. The clear zones of inhibition against P. aeruginosa for the CP, AMK, VAN, TET, GEN, Ery, and CLI were 26 mm, 20 mm, 21 mm, 22 mm, 20 mm, 25 mm and 23 mm, respectively. The MIC values for CP, AMK, VAN, TET, GEN, Ery and CLI against P. aeruginosa ranged from 0.25 to 1 µg/mL while the MBC values ranged from 1 and 0.5 to 2 µg/mL respectively. The growth, total viable counts (TVCs), bacterial adhesion and biofilm formation of P. aeruginosa were reduced after exposure to all the tested antibiotics in a dose-dependent manner. The RT-qPCR analysis showed that all the tested antibiotics share a similar overall pattern of gene expression, with a trend toward reduced expression of the virulence genes of interest (lasR, lasI, fleN, fleQ and fleR, oprB and oprC) in P. aeruginosa. The results indicate that all of the tested antibiotics possess antimicrobial and anti-biofilm activities, and that they may be multiple inhibitors and moderators of P. aeruginosa virulence via a variety of molecular targets. This deduction requires to be investigated in vivo.

Keywords:

Citation: Mohammad Abu-Sini, Mohammad A. Al-Kafaween, Rania M. Al-Groom, Abu Bakar Mohd Hilmi. Comparative in vitro activity of various antibiotic against planktonic and biofilm and the gene expression profile in Pseudomonas aeruginosa[J]. AIMS Microbiology, 2023, 9(2): 313-331. doi: 10.3934/microbiol.2023017

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Abstract

1. Introduction

Nonlinear partial differential equation is a very important branch of the nonlinear science, which has been called the foreword and hot topic of current scientific development. In theoretical science and practical application, nonlinear partial differential is used to describe the problems in the fields of optics, mechanics, communication, control science and biology ^{[1,2,3,4,5,6,7,8,9]}. At present, the main problems in the study of nonlinear partial differential equations are the existence of solutions, the stability of solutions, numerical solutions and exact solutions. With the development of research, especially the study of exact solutions of nonlinear partial differential equations has important theoretical value and application value. In the last half century, many important methods for constructing exact solutions of nonlinear partial differential equations have been proposed, such as the planar dynamic system method ^[10], the Jacobi elliptic function method ^[11], the bilinear transformation method ^[12], the complete discriminant system method for polynomials ^[13], the unified Riccati equation method ^[14], the generalized Kudryashov method ^[15], and so on ^{[16,17,18,19,20,21,22,23,24]}.

There is no unified method to obtain the exact solution of nonlinear partial differential equations. Although predecessors have obtained some analytical solutions with different methods, no scholar has studied the system with complete discrimination system for polynomial method.

The Fokas system is a very important class of nonlinear partial differential equations. In this article, we focus on the Fokas system, which is given as follows ^{[25,26,27,28,29,30,31,32,33,34,35,36,37]}

$\left\{ \begin{array}{l} i{p_t} + {r_1}{p_{xx}} + {r_2}pq = 0, \hfill \\ {r_3}{q_y} - {r_4}{(|p{|^2})_x} = 0, \hfill \end{array} \right.$

(1.1)

where $p = p(x, y, t)$ and $q = q(x, y, t)$ are the complex functions which stand for the nonlinear pulse propagation in monomode optical fibers. The parameters ${r_1}, {r_2}, {r_3}$ and ${r_4}$ are arbitrary non-zero constants, which are coefficients of nonlinear terms in Eq (1.1) and reflect different states of optical solitons.

This paper is arranged as follows. In Section 2, we describe the method of the complete discrimination system for polynomial method. In Section 3, we substitute traveling wave transformation into nonlinear ordinary differential equations and obtain the different new single traveling wave solutions for the Fokas system by complete discrimination system for polynomial method. At the same time, we draw some images of solutions. In Section 4, the main results are summarized.

2. Description of the method

First, we consider the following partial differential equations:

$\left\{ \begin{array}{l}F(u, v, {u}_{x}, {u}_{t}, {v}_{x}, {v}_{t}, {u}_{xx}, {u}_{xt}, {u}_{tt}, \cdots ) = 0\text{, }\\ G(u, v, {u}_{x}, {u}_{t}, {v}_{x}, {v}_{t}, {u}_{xx}, {u}_{xt}, {u}_{tt}, \cdots ) = 0\text{, }\end{array} \right.$

(2.1)

where $F$ and $G$ is polynomial function which is about the partial derivatives of each order of $u(x, t)$ and $v(x, t)$ with respect to $x$ and $t$ .

Step 1: Taking the traveling wave transformation $u(x, t) = u(\xi), v(x, t) = v(\xi), \xi = kx + ct$ into Eq (2.1), then the partial differential equation is converted to an ordinary differential equation

$\left\{ \begin{array}{l} F(u, v, u', v', u'', v'', \cdots ) = 0, \hfill \\ G(u, v, u', v', u'', v'', \cdots ) = 0. \hfill \end{array} \right.$

(2.2)

Step 2: The above nonlinear ordinary differential equations (2.2) are reduced to the following ordinary differential form after a series of transformations:

${(u')^2} = {u^3} + {d_2}{u^2} + {d_1}u + {d_0}.$

(2.3)

The Eq (2.3) can also be written in integral form as:

$\pm (\xi - {\xi _0}) = \int {\frac{{du}}{{\sqrt {{u^3} + {d_2}{u^2} + {d_1}u + {d_0}} }}} .$

(2.4)

Step 3: Let $\phi (u) = {u^3} + {d_2}{u^2} + {d_1}u + {d_0}$ . According to the complete discriminant system method of third-order polynomial

$\left\{\begin{array}{l} \Delta=-27(\frac{2 d_2^3}{27}+d_0-\frac{d_1 d_2}{3})^2-4(d_1-\frac{d_2^2}{3})^3, \\ D_1=d_1-\frac{d_2^2}{3}, \end{array}\right.$

(2.5)

the classification of the solution of the equation can be obtained, and the classification of traveling wave solution of the Fokas system will be given in the following section.

3. Traveling wave solution of Eq (1.1)

In the current part, we obtain all exact solutions to Eq (1.1) by complete discrimination system for polynomial method. According to the wave transformation

$p(x, y, t) = \varphi (\eta ){e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}}, q(x, y, t) = \phi (\eta ), \eta = x + y - vt,$

(3.1)

where ${\lambda _1}, {\lambda _2}, {\lambda _3}, {\lambda _4}$ and $v$ are real parameters, and $v$ represents the wave frame speed.

Substituting the above transformation Eq (3.1) into Eq (1.1), we get

$\left\{ \begin{array}{l} ( - v + 2{r_1}{\lambda _1})i\varphi ' - {\lambda _3}\varphi + {r_1}\varphi '' - {r_1}\lambda _1^2\varphi + {r_2}\varphi \phi = 0, \hfill \\ {r_3}\phi ' - 2{r_4}\varphi \varphi ' = 0. \hfill \end{array} \right.$

(3.2)

Integrating the second equation in (3.2) and ignoring the integral constant, we get

$\phi (\eta ) = \frac{{{r_4}{\varphi ^2}(\eta )}}{{{r_3}}}.$

(3.3)

Substituting Eq (3.3) into the first equation in (3.2) and setting $v = 2{r_1}{\lambda _1},$ we get the following:

${r_1}\varphi '' - ({\lambda _3} + {r_1}\lambda _1^2)\varphi + \frac{{{r_2}{r_4}{\varphi ^3}}}{{{r_3}}} = 0.$

(3.4)

Multiplying $\varphi '$ both sides of the Eq (3.4), then integrating once to get

${(\varphi ')^2} = {a_4}{\varphi ^4} + {a_2}{\varphi ^2} + {a_0},$

(3.5)

where ${a_4} = - \frac{{{r_2}{r_4}}}{{2{r_1}{r_3}}}, {a_2} = \frac{{{\lambda _3} + {r_1}\lambda _1^2}}{{{r_1}}},$ ${a_0}$ is the arbitrary constant.

${\rm{Let}} \ \ \varphi = \pm \sqrt {{{(4{a_4})}^{ - \frac{1}{3}}}\omega } , \ {b_1} = 4{a_2}{(4{a_4})^{ - \frac{2}{3}}}, {b_0} = 4{a_0}{(4{a_4})^{ - \frac{1}{3}}}, {\eta _1} = {(4{a_4})^{\frac{1}{3}}}\eta .$

(3.6)

Equation (3.5) can be expressed as the following:

${({\omega '_{{\eta _1}}})^2} = {\omega ^3} + {b_1}{\omega ^2} + {b_0}\omega .$

(3.7)

Then we can get the integral expression of Eq (3.7)

$\pm ({\eta _1} - {\eta _0}) = \int {\frac{{d\omega }}{{\sqrt {\omega ({\omega ^2} + {b_1}\omega + {b_0})} }}} ,$

(3.8)

where ${\eta _0}$ is the constant of integration.

Here, we get the $F(\omega) = {\omega ^2} + {b_1}\omega + {b_0}$ and $\Delta = b_1^2 - 4{b_0}.$ In order to solve Eq (3.7), we discuss the third order polynomial discrimination system in four cases.

Case 1: $\Delta = 0$ and $\omega \gt 0.$

When ${b_1} \lt 0,$ the solution of Eq (3.7) is

${\omega _1} = - \frac{{{b_1}}}{2}{\tanh ^2}(\frac{1}{2}\sqrt { - \frac{{{b_1}}}{2}} ({\eta _1} - {\eta _0})).$

(3.9)

${\omega _2} = - \frac{{{b_1}}}{2}{\coth ^2}(\frac{1}{2}\sqrt { - \frac{{{b_1}}}{2}} ({\eta _1} - {\eta _0})).$

(3.10)

Thus, the classification of all solutions of Eq (3.7) is obtained by the third order polynomial discrimination system. The exact traveling wave solutions of the Eq (1.1) are obtained by combining the above solutions and the conditions (3.6) with Eq (3.1), can be expressed as below:

${p_1}(x, y, t) = \pm \sqrt {\frac{{{r_3}({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_2}{r_4}}}} \tanh (\frac{1}{2}\sqrt { - \frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot ({(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{\frac{1}{3}}}}\eta + {\eta _0})) \cdot {e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}}.$

(3.11)

In Eq (3.11), ${p_1}(x, y, t)$ is a dark soliton solution, it expresses the energy depression on a certain intensity background. Figure 1 depict two-dimensional graph, three-dimensional graph, contour plot and density plot of the solution.

${q_1}(x, y, t) = \frac{{{\lambda _3} + {r_1}\lambda _1^2}}{{{r_2}}}{\tanh ^2}(\frac{1}{2}\sqrt { - \frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot ({(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{\frac{1}{3}}}}\eta + {\eta _0}))$

(3.12)

$\begin{array}{l} {p_2}(x, y, t) = \pm \sqrt {\frac{{{r_3}({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_2}{r_4}}}} \coth (\frac{1}{2}\sqrt { - \frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot \\ ({(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{\frac{1}{3}}}}\eta + {\eta _0})) \cdot {e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}}, \end{array}$

(3.13)

Figure 1. Module length graphs of Eq (3.12) when

${r_1} = - 2, {r_2} = 1, {r_3} = - 1, {r_4} = 1, {\lambda _1} = - 1, {\lambda _3} = 3, {\eta _0} = 0$ .

DownLoad: Full-Size Img PowerPoint

where ${p_1}(x, y, t), {q_1}(x, y, t), {p_2}(x, y, t), {q_2}(x, y, t)$ are hyperbolic function solutions. Specially, ${p_2}(x, y, t)$ is a bright soliton solution.

${q_2}(x, y, t) = \frac{{{\lambda _3} + {r_1}\lambda _1^2}}{{{r_2}}}{\coth ^2}(\frac{1}{2}\sqrt { - \frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot ({(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{\frac{1}{3}}}}\eta + {\eta _0})).$

(3.14)

When ${b_1} \gt 0,$ the solution of Eq (3.7) is

${\omega _3} = \frac{{{b_1}}}{2}{\tan ^2}(\frac{1}{2}\sqrt {\frac{{{b_1}}}{2}} ({\eta _1} - {\eta _0})).$

(3.15)

The exact traveling wave solutions of the Eq (1.1) can be expressed as below:

$\begin{array}{l} {p_3}(x, y, t) = \pm \sqrt { - \frac{{{r_3}({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_2}{r_4}}}} \tan (\frac{1}{2}\sqrt {\frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot \\ ({(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{\frac{1}{3}}}}\eta + {\eta _0})) \cdot {e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}}. \end{array}$

(3.16)

${q_3}(x, y, t) = - \frac{{{\lambda _3} + {r_1}\lambda _1^2}}{{{r_2}}}{\tan ^2}(\frac{1}{2}\sqrt {\frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot ({(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{\frac{1}{3}}}}\eta + {\eta _0})).$

(3.17)

In Eq (3.16) and Eq (3.17), ${p_3}(x, y, t)$ and ${q_3}(x, y, t)$ are trigonometric function solutions. ${q_3}(x, y, t)$ is a periodic wave solution, and it Shows the periodicity of ${q_3}(x, y, t)$ in Figure 2(a), (b).

Figure 2. Module length graphs of Eq (3.17) when

${r_1} = 2, {r_2} = 1, {r_3} = 1, {r_4} = 1, {\lambda _1} = 1, {\lambda _3} = - 1, {\eta _0} = 0$ .

DownLoad: Full-Size Img PowerPoint

When ${b_1} = 0,$ the solution of Eq (3.7) is

${\omega _4} = \frac{4}{{{{({\eta _1} - {\eta _0})}^2}}}.$

(3.18)

The exact traveling wave solutions of the Eq (1.1) can be expressed as below:

${p_4}(x, y, t) = \pm \sqrt { - {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{ - \frac{1}{3}}}} \frac{2}{{{{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0}}}{e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}},$

(3.19)

${q_4}(x, y, t) = - \frac{{{r_4}}}{{{r_3}}}{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{ - \frac{1}{3}}}\frac{4}{{{{({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0})}^2}}},$

(3.20)

where ${p_4}(x, y, t)$ is exponential function solution, and ${q_4}(x, y, t)$ is rational function solution.

Case 2: $\Delta = 0$ and ${b_0} = 0.$

When $\omega \gt - {b_1}$ and ${b_1} \lt 0,$ the solution of Eq (3.7) is

${\omega _5} = \frac{{{b_1}}}{2}{\tanh ^2}(\frac{1}{2}\sqrt {\frac{{{b_1}}}{2}} ({\eta _1} - {\eta _0})) - {b_1}.$

(3.21)

${\omega _6} = \frac{{{b_1}}}{2}{\coth ^2}(\frac{1}{2}\sqrt {\frac{{{b_1}}}{2}} ({\eta _1} - {\eta _0})) - {b_1}.$

(3.22)

The exact traveling wave solutions of the Eq (1.1) can be expressed as below:

$\begin{array}{l} {p_5}(x, y, t) = \\ \pm \sqrt { - \frac{{{r_3}({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_2}{r_4}}}({{\tanh }^2}(\frac{1}{2}\sqrt {\frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot ({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0})) - 2)} \cdot {e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}}, \end{array}$

(3.23)

${q_5}(x, y, t) = - \frac{{{\lambda _3} + {r_1}\lambda _1^2}}{{{r_2}}}{\tanh ^2}(\frac{1}{2}\sqrt {\frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot ({(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{\frac{1}{3}}}}\eta + {\eta _0})) + \frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_2}}},$

(3.24)

$\begin{array}{l} {p_6}(x, y, t) = \\ \pm \sqrt { - \frac{{{r_3}({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_2}{r_4}}}({{\coth }^2}(\frac{1}{2}\sqrt {\frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot ({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0})) - 2)} \cdot {e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}}, \end{array}$

(3.25)

${q_6}(x, y, t) = - \frac{{{\lambda _3} + {r_1}\lambda _1^2}}{{{r_2}}}{\coth ^2}(\frac{1}{2}\sqrt {\frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot ({(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{\frac{1}{3}}}}\eta + {\eta _0})) + \frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_2}}},$

(3.26)

where ${p_5}(x, y, t), {q_5}(x, y, t), {p_6}(x, y, t)$ and ${q_6}(x, y, t)$ are hyperbolic function solutions.

When $\omega \gt - {b_1}$ and ${b_1} \gt 0,$ the solution of Eq (3.7) is

${\omega _7} = - \frac{{{b_1}}}{2}{\tan ^2}(\frac{1}{2}\sqrt { - \frac{{{b_1}}}{2}} ({\eta _1} - {\eta _0})) - {b_1}.$

(3.27)

The exact traveling wave solutions of the Eq (1.1) can be expressed as below:

$\begin{array}{l} {p_7}(x, y, t) = \\ \pm \sqrt {\frac{{{r_3}({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_2}{r_4}}}({{\tan }^2}(\frac{1}{2}\sqrt { - \frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot ({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0})) + 2)} \cdot {e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}}, \end{array}$

(3.28)

${q_7}(x, y, t) = \frac{{{\lambda _3} + {r_1}\lambda _1^2}}{{{r_2}}}{\tan ^2}(\frac{1}{2}\sqrt { - \frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot ({(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{\frac{1}{3}}}}\eta + {\eta _0})) + \frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_2}}},$

(3.29)

where ${p_7}(x, y, t)$ and ${q_7}(x, y, t)$ are trigonometric function solutions.

Case 3: $\Delta \gt 0$ and ${b_0} \ne 0.$ Let $u \lt v \lt s,$ there $u, v$ and $s$ are constants satisfying one of them is zero and two others are the root of $F(\omega) = 0.$

When $u \lt \omega \lt v,$ we can get the solution of Eq (3.7) is

${\omega _8} = u + (v - u)s{n^2}(\frac{{\sqrt {s - u} }}{2}({\eta _1} - {\eta _0}), c),$

(3.30)

where ${c^2} = \frac{{v - u}}{{s - u}}.$

The exact traveling wave solutions of the Eq (1.1) can be expressed as below:

${p_8}(x, y, t) = \pm \sqrt { - {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{1}{3}}}}[u + (v - u) \cdot s{n^2}(\frac{{\sqrt {s - u} }}{2}({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0}), c)]} \cdot {e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}}.$

(3.31)

${q_8}(x, y, t) = - \frac{{{r_4}}}{{{r_3}}}{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{ - \frac{1}{3}}}}[u + (v - u) \cdot s{n^2}(\frac{{\sqrt {s - u} }}{2}({(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{\frac{1}{3}}}}\eta + {\eta _0}), c)].$

(3.32)

When $\omega \gt s,$ the solution of Eq (3.7) is

${\omega _9} = \frac{{ - v \cdot s{n^2}(\sqrt {s - u} ({\eta _1} - {\eta _0})/2, c) + s}}{{c{n^2}(\sqrt {s - u} ({\eta _1} - {\eta _0})/2, c)}}.$

(3.33)

The exact traveling wave solutions of the Eq (1.1) can be expressed as below:

${p_9}(x, y, t) = \pm \sqrt { - {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{1}{3}}}}\frac{{ - v \cdot s{n^2}(\frac{{\sqrt {s - u} }}{2}({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0}), c)] + s}}{{c{n^2}(\frac{{\sqrt {s - u} }}{2}({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0}), c)}}} {e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}}.$

(3.34)

${q_9}(x, y, t) = - \frac{{{r_4}}}{{{r_3}}}{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{ - \frac{1}{3}}}}\frac{{ - v \cdot s{n^2}(\frac{{\sqrt {s - u} }}{2}({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0}), c)] + s}}{{c{n^2}(\frac{{\sqrt {s - u} }}{2}({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0}), c)}}.$

(3.35)

Case 4: $\Delta \lt 0.$

When $\omega \gt 0,$ similarly we get

${\omega _{10}} = \frac{{2\sqrt {{b_0}} }}{{1 + cn(b_0^{\frac{1}{4}}({\eta _1} - {\eta _0}), c)}} - \sqrt {{b_0}} ,$

(3.36)

where ${c^2} = (1 - \frac{{{b_1}\sqrt {{b_0}} }}{2})/2.$

The exact traveling wave solutions of the Eq (1.1) can be expressed as below:

${p_{10}}(x, y, t) = \pm \sqrt {2\sqrt {{a_0}} {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{1}{2}}}}[\frac{{ - 2}}{{1 + cn({{(4{a_0}{{( - \frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{1}{3}}}})}^{\frac{1}{4}}}({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0}), c)}} + 1]} \cdot {e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}} ,$

(3.37)

${q_{10}}(x, y, t) = - \frac{{{r_4}}}{{{r_3}}}2\sqrt {{a_0}} {(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{ - \frac{1}{2}}}}[\frac{{ - 2}}{{1 + cn({{(4{a_0}{{( - \frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{1}{3}}}})}^{\frac{1}{4}}}({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0}), c)}} + 1],$

(3.38)

where ${p_8}(x, y, t), {q_8}(x, y, t), {p_9}(x, y, t), {q_9}(x, y, t), {p_{10}}(x, y, t)$ and ${q_{10}}(x, y, t)$ are Jacobian elliptic function solutions.

4. Conclusions

In this paper, the complete discrimination system of polynomial method has been applied to construct the single traveling wave solutions of the Fokas system. The Jacobian elliptic function solutions, the trigonometric function solutions, the hyperbolic function solutions and the rational function solutions are obtained. The obtained solutions are very rich, which can help physicists understand the propagation of traveling wave in monomode optical fibers. Furthermore, we have also depicted two-dimensional graphs, three-dimensional graphs, contour plots and density plots of the solutions of Fokas system, which explains the state of solitons from different angles.

Acknowledgments

This work was supported by Scientific Research Funds of Chengdu University (Grant No.2081920034).

Conflict of interest

The authors declare no conflict of interest.

Acknowledgments

This work was supported by Al-Zaytoonah University of Jordan (Grant Number: 2019-2018/18/09) and Universiti Sultan Zainal Abidin (UniSZA) (UniSZA/2018/DPU/13:R0034-R013). The authors thank all the staff members of the Faculty of Pharmacy at Al-Zaytoonah University of Jordan and the Faculty of Health Sciences at UniSZA for their support and commitment.

Conflict of interest

The authors declare that there is no conflict of interest.

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