Sociopolitical conflicts on the establishment of protected natural areas: The case of Portofino National Park (Genoa, North-West Italy)

Lorenzo Brocada; Lorenzo Brocada

doi:10.3934/geosci.2023038

AIMS Geosciences

2023, Volume 9, Issue 4: 713-733. doi: 10.3934/geosci.2023038

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Research article Special Issues

Sociopolitical conflicts on the establishment of protected natural areas: The case of Portofino National Park (Genoa, North-West Italy)

Lorenzo Brocada ^,

Department of Human and Social Science, Doctoral School, University of Sassari, Italy

Received: 12 July 2023 Revised: 13 November 2023 Accepted: 20 November 2023 Published: 28 November 2023

This work traces the main stages of environmental and landscape protection of the Portofino Promontory, located in Riviera Ligure di Levante (N-W Italy), with particular regard on the recent establishment of Portofino National Park. From 2017, when the institution law was enacted, to date, the park has not yet been established due to the socio-political conflicts that have arisen between some stakeholders and institutions of the territory. These conflicts include not only environmentalists against hunters and constructors but also disagreement between municipalities and region (Regione Liguria) and between region and the Ministry of Environment. Today the situation is still stalled, and funds for a park larger than the current one (Portofino Regional Park) have not been allocated. In spite of this, the tug-of-war continues through legal actions. The aim of the article is to analyze the perception of the enlargement of the park by the community and local governance and how this is communicated by the press. The research was conducted through the analysis of the results of a questionnaire aimed at understanding the level of knowledge of the main functions of a national park and the position of the people with respect to it. Second, an analysis of the press was carried out to understand the narratives on this environmental measure. The results of the questionnaire showed a positive consensus toward the park, while press analysis showed little involvement of experts on the subject to foster a political debate without concrete arguments, which damaged the park's image.

Keywords:

Citation: Lorenzo Brocada. Sociopolitical conflicts on the establishment of protected natural areas: The case of Portofino National Park (Genoa, North-West Italy)[J]. AIMS Geosciences, 2023, 9(4): 713-733. doi: 10.3934/geosci.2023038

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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.

References

[1]	Pinna M (1984) Atti del convegno sul tema: I parchi nazionali e i parchi regionali in Italia. Memorie della Società Geografica Italiana, 33.
[2]	Ugolini GM (2001) I parchi in Liguria: un equilibrio difficile fra territorio vocato e resistenza sociale, L'importanza sociale ed economica di un'efficiente gestione del sistema dei parchi e delle aree protette, Atti della conferenza internazionale, Genova: Brigati, 281–298.
[3]	Giuntarelli P (2008) Parchi, politiche ambientali e globalizzazione, Milano: Franco Angeli.
[4]	Dixon JA, Sherman PB (1991) Economics of Protected Areas. Ambio 20: 68–74.
[5]	Strickland-Munro JK, Allison HE, Moore SA (2010) Using resilience concepts to investigate the impacts of protected area tourism on communities. Ann Tourism Res 37: 499–519. https://doi.org/10.1016/j.annals.2009.11.001 doi: 10.1016/j.annals.2009.11.001
[6]	Cassola P (2005) Turismo sostenibile e aree naturali protette. Concetti, strumenti e azioni, Edizioni ETS.
[7]	Zanolin G (2022) Geografia dei parchi nazionali italiani, Carocci.
[8]	Colvin RM, Witt GB, Lacey J (2015) The social identity approach to understanding socio-political conflict in environmental and natural resources management. Global Environ Change 34: 237–246. https://doi.org/10.1016/j.gloenvcha.2015.07.011 doi: 10.1016/j.gloenvcha.2015.07.011
[9]	Staniscia B, Komatsu G, Staniscia A (2019) Nature Park establishment and environmental conflicts in coastal areas: The case of the Costa Teatina National Park in central Italy, Ocean Coast Manage 182: 104947. https://doi.org/10.1016/j.ocecoaman.2019.104947 doi: 10.1016/j.ocecoaman.2019.104947
[10]	Cadoret A, Cazals C, Diaw M, et al. (2021) Dynamiques Conflictuelles dans les parcs nationaux de la Réunion et des Calanques. Effort environnemental et équité. Les politiques publiques de l'eau et de la biodiversité en France, Bruxelles: Peter Lang, 195–224.
[11]	Deboudt P (2012) La construction du Parc national des Calanques (1971–2012). Le Parc national des Calanques: Construction territoriale, formes de concertation et principes de légitimité. In Action environnementale: que peut-on encore attendre de la concertation, Versailles: Éditions Quae, 25–51.
[12]	Stringa P (1984) Il Golfo Paradiso: da Genova a Portofino: ragioni e strutture di un paesaggio, Stringa.
[13]	Balletti F, Soppa S (2015) The Landscapes of the Portofino Nature Regional Park. Nature Policies and Landscape Policies. Urban and Landscape Perspectives, Cham: Springer. 18: 415–422.
[14]	Turri E (1998) Il paesaggio come teatro, Venezia: Marsilio.
[15]	Piana P (2020) Paper landscapes. Topographical art and environmental change in Liguria, Aracne Editrice.
[16]	Cavanna M (2010) Verso e dentro il Monte: percorsi di accesso e percorsi all'interno, Sentieri sacri sul monte di Portofino, Milano: Silvana Editoriale, 13–24.
[17]	Zanini A (2012) Un secolo di turismo in Liguria. Dinamiche, percorsi, attori, Franco Angeli.
[18]	Mangano S (2007) Turismo e tempo libero nelle aree naturali protette, Roma: Carocci.
[19]	Gastaldi F (2013) Portofino, fra turismo d'élite e spopolamento. Territorio della ricerca su insediamenti e ambiente 6: 105–114.
[20]	Brandolini P, Mandarino A, Paliaga G, et al. (2021) Anthropogenic landforms in an urbanized alluvial-coastal plain (Rapallo city, Italy). J Maps 17: 86–97. https://doi.org/10.1080/17445647.2020.1793818 doi: 10.1080/17445647.2020.1793818
[21]	Scarin ML (1972) Camogli e Recco nel Golfo Paradiso (ricerche di geografia urbana), Città di Castello: Arti Grafiche.
[22]	Leardi E (1991) Il prezzo e il costo del Mar Ligure. In A. Vallega, La Liguria e il mare, Genova: Pubblicazioni dell'Istituto di Scienze Geografiche dell'Università di Genova.
[23]	Brandolini P, Faccini F, Paliaga G, et al. (2017) Urban geomorphology in coastal environment: Man-made morphological changes in a seaside tourist resort (Rapallo, Eastern Liguria, Italy). Quaestiones Geographicae 36: 97–110. https://doi.org/10.1515/quageo-2017-0027 doi: 10.1515/quageo-2017-0027
[24]	Dell'Agnese E, Bagnoli L (2004) Modi e mode del turismo in Liguria. Da Giovanni Ruffini a Rick Steves, Milano: Cuem.
[25]	Istat, Censimento della popolazione e delle abitazioni, 2021. Available from: http://dati-censimentipermanenti.istat.it/.
[26]	Brandolini P, Faccini F, Piccazzo M (2006) Geomorphological hazard and tourist vulnerability along Portofino Park trails (Italy). Nat Hazards Earth Syst Sci 6: 563–571. https://doi.org/10.5194/nhess-6-563-2006 doi: 10.5194/nhess-6-563-2006
[27]	Celata F, Romano A (2022) Overtourism and online short-term rental platforms in Italian cities. J Sustain Tour 30: 1020–1039. https://doi.org/10.1080/09669582.2020.1788568 doi: 10.1080/09669582.2020.1788568
[28]	Jokela S, Minoia P (2021) Tourism Platforms, Situating Sustainability: A Handbook of Contexts and Concepts, Helsinki: Helsinki University Press, 223–237.
[29]	Candia S, Pirlone F, Spadaro I (2018) Sustainable development and the plan for tourism in Mediterranean coastal areas: Case study of the region of Liguria, Italy. WIT Trans Ecol Environ 217: 523–534.
[30]	Galeotti R (2023) La Cooperativa dei Pescatori in liquidazione. Camogli ammaina un pezzo della sua storia. Available from: https://www.ilsecoloxix.it/levante/2023/02/08/news/la_cooperativa_dei_pescatori_in_liquidazione_camogli_ammaina_un_pezzo_della_sua_storia-12629541/.
[31]	Armiero M, Graf von Hardenberg W (2013) Green rhetoric in blackshirts: Italian Fascism and the environment. Environ Hist 19: 283–311. https://doi.org/10.3197/096734013X13690716950064 doi: 10.3197/096734013X13690716950064
[32]	Spotorno M (2005) Le Parc naturel régional de Portofino en Ligurie. Méditerranée 105: 47–52. https://doi.org/10.4000/mediterranee.342 doi: 10.4000/mediterranee.342
[33]	Girani A (2013) Parco di Portofino: una storia lunga 80 anni. Portofino per terra e per mare 3: 6–12. Available from: http://www.parcoportofino.com/parcodiportofino/resources/cms/documents/articolo_una_storia_lunga_80_anni.pdf
[34]	Graziani CA (2019) Appunti per una riflessione critica sui parchi naturali. Ambiente e territorio. I parchi tra crisi e rilancio, Assago: Edizioni ETS, 17–34.
[35]	Piccioni L (2023) Parchi naturali. Storia delle aree protette in Italia, Bologna: Il Mulino.
[36]	Scanu G, Madau C (2001) Prospettive di tutela dell'ambiente in Sardegna nel quadro delle nuove politiche di valorizzazione e gestione delle risorse naturali, Il caso del Monte Arci. Atti della Conferenza Internazionale, a cura di Brandis P, Università di Sassari, Istituto e Laboratorio di geografia, Genova, Brigati, 241–280.
[37]	Generalitat de Catalunya, Parc Natural de Cap de Creus, Història de protecció. Available from: https://parcsnaturals.gencat.cat/ca/xarxa-de-parcs/cap-creus/el-parc/historia-de-proteccio/.
[38]	Piana P (2019) La Valle dei Mulini dell'Acquaviva nel Parco di Portofino. Evoluzione e prospettive di sviluppo di un paesaggio produttivo della Liguria di Levante. Annali di ricerche e studi di geografia 75–76: 39–54.
[39]	Brocada L, Girani A (2022) Itinerari di turismo lento e processi partecipativi per la valorizzazione del territorio nel Golfo Paradiso (Genova): tra conflittualità e collaborazione. Itinerari per la rigenerazione territoriale. Promozione e valorizzazione dei territori: sviluppi reticolari e sostenibili, Milano: Franco Angeli, 362–371.
[40]	Turconi L, Faccini F, Marchese A, et al. (2020) Implementation of nature-based solutions for hydro-meteorological risk reduction in small mediterranean catchments: The case of Portofino Natural Regional Park, Italy. Sustainability 12: 1240. https://doi.org/10.3390/su12031240 doi: 10.3390/su12031240
[41]	Coratza P, Vandellli V, Fiorentini L, et al. (2019) Bridging terrestrial and marine geoheritage: Assessing geosites in Portofino Natural Park (Italy). Water 11: 2112. https://doi.org/10.3390/w11102112 doi: 10.3390/w11102112
[42]	Marchioro C (2018) Dinamiche socio-economiche nelle aree interne della Liguria. Atti della 22 Conferenza ASITA. Bolzano, Italia.
[43]	ISPRA (2018) Istruttoria per l'istituzione del Parco Nazionale di Portofino, Roma.
[44]	ANSA, Sovraffollamento a Portofino, istituite zone rosse. 2023. Available from: https://www.ansa.it/liguria/notizie/2023/04/09/sovraffollamento-a-portofino-istituite-zone-rosse_17a912c4-45e0-46ee-900f-590d2ff780b8.html.
[45]	ANSA, Per San Fruttuoso ipotesi numero chiuso. 2015. Available from: https://www.ansa.it/liguria/notizie/2015/07/11/per-san-fruttuoso-ipotesi-numero-chiuso_adf4eefb-183b-43eb-802d-da7c72b03677.html.
[46]	Vallega A (1999) Fundamentals of integrated Coastal Management, Amsterdam: Kluwer.
[47]	Cadoret A (2009) Conflict Dynamics in Coastal Zones: A Perspective Using the Example of Languedoc-Rousillon (France). J Coast Conserv 13: 151–163. https://doi.org/10.1007/s11852-009-0048-9 doi: 10.1007/s11852-009-0048-9
[48]	ANSA, Parco Portofino: troppi cinghiali, via restrizioni in area contigua. 2023. Available from: https://www.ansa.it/liguria/notizie/2023/01/02/parco-di-portofino-troppi-cinghiali-sospesa-area-contigua_45358df0-b7c9-431f-a466-13784546d22b.html.
[49]	ISPRA, Sintesi delle misure di controllo e prevenzione della psa. 2022. Available from: https://www.isprambiente.gov.it/files2022/notizie/misure-psa-divulgativo-ispra_def-con-loghi.pdf.
[50]	Camerada MV (2015) L'analisi dei Parchi Naturali Italiani attraverso l'applicazione del modello I.S.A. Verso un nuovo paradigma geopolitico, Raccolta di scritti in onore di Gianfranco Lizza. Tomo I, Roma: Aracne Editrice, 311–336.
[51]	Tillett G, French BJ (2006) Resolving Conflict: A Practical Approach, Oxford University Press.
[52]	Gambino R (1991) I parchi naturali. Problemi ed esperienze di pianificazione nel contesto ambientale, Roma: NIS.
[53]	Brandis P, Scanu G (1995) I parchi e le aree protette, La Sardegna nel mondo mediterraneo, IV convegno internazionale di studi, Bologna: Patron Editore.
[54]	Brandis P (2001) L'importanza sociale ed economica di un'efficiente gestione del sistema dei parchi e delle aree protette, Atti della conferenza internazionale. Genova: Brigati.
[55]	Scarlata R (2015) Aree naturali protette, turismo e sviluppo locale sostenibile. Geotema 49: 5–104.
[56]	Strickland-Munro JK, Allison HE, Moore SA (2010) Using resilience concepts to investigate the impacts of protected area tourism on communities. Ann Tourism Res 37: 499–519. https://doi.org/10.1016/j.annals.2009.11.001 doi: 10.1016/j.annals.2009.11.001
[57]	Martínez Quintana V (2017) El turismo de naturaleza: un producto turístico sostenible. Arbor 193: a396. https://doi.org/10.3989/arbor.2017.785n3002 doi: 10.3989/arbor.2017.785n3002
[58]	Gross M, Pearson J, Arbieu U, et al. (2023) Tourists' valuation of nature in protected areas: A systematic review. Ambio 52: 1065–1084. https://doi.org/10.1007/s13280-023-01845-0 doi: 10.1007/s13280-023-01845-0
[59]	European Commission (2020) EU Biodiversity Strategy for 2030. Bringing nature back into our lives, Bruxelles.
[60]	Brocada L, Piana P (2022) Per un'ecologia politica dei borderscapes: il caso del confine tra Polonia e Bielorussia nella foresta di Białowieża. Documenti Geografici 2: 17–30. https://doi.org/10.19246/DOCUGEO2281-7549/202202_02 doi: 10.19246/DOCUGEO2281-7549/202202_02

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