Unveiling solitons and dynamic patterns for a (3+1)-dimensional model describing nonlinear wave motion

Muhammad Bilal Riaz; Syeda Sarwat Kazmi; Adil Jhangeer; Jan Martinovic; Muhammad Bilal Riaz; Syeda Sarwat Kazmi; Adil Jhangeer; Jan Martinovic

doi:10.3934/math.2024992

AIMS Mathematics

2024, Volume 9, Issue 8: 20390-20412. doi: 10.3934/math.2024992

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Research article Special Issues

Unveiling solitons and dynamic patterns for a (3+1)-dimensional model describing nonlinear wave motion

1.
IT4 Innovations, VSB–Technical University of Ostrava, Ostrava, Czech Republic
2.
Department of Computer Science and Mathematics, Lebanese American University, Byblos, Lebanon
3.
Department of Mathematics, Namal University, Talagang Road, Mianwali 42250, Pakistan

Received: 17 March 2024 Revised: 29 April 2024 Accepted: 07 May 2024 Published: 24 June 2024
MSC : 34H10, 35C08

In this study, the underlying traits of the new wave equation in extended (3+1) dimensions, utilized in the field of plasma physics and fluids to comprehend nonlinear wave scenarios in various physical systems, were explored. Furthermore, this investigation enhanced comprehension of the characteristics of nonlinear waves present in seas and oceans. The analytical solutions of models under consideration were retrieved using the sub-equation approach and Sardar sub-equation approach. A diverse range of solitons, including bright, dark, combined dark-bright, and periodic singular solitons, was made available through the proposed methods. These solutions were illustrated through visual depictions utilizing 2D, 3D, and density plots with carefully chosen parameters. Subsequently, an analysis of the dynamical nature of the model was undertaken, encompassing various aspects such as bifurcation, chaos, and sensitivity. Bifurcation analysis was conducted via phase portraits at critical points, revealing the system's transition dynamics. Introducing an external periodic force induced chaotic phenomena in the dynamical system, which were visualized through time plots, two-dimensional plots, three-dimensional plots, and the presentation of Lyapunov exponents. Furthermore, the sensitivity analysis of the investigated model was executed utilizing the Runge-Kutta method. The obtained findings indicated the efficacy of the presented approaches for analyzing phase portraits and solitons over a wider range of nonlinear systems.

Keywords:

Citation: Muhammad Bilal Riaz, Syeda Sarwat Kazmi, Adil Jhangeer, Jan Martinovic. Unveiling solitons and dynamic patterns for a (3+1)-dimensional model describing nonlinear wave motion[J]. AIMS Mathematics, 2024, 9(8): 20390-20412. doi: 10.3934/math.2024992

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Abstract

1. Introduction

The field of mathematical analysis that deals with the study of arbitrary order integrals and derivatives is known as fractional calculus. Because of its numerous applications across a wide range of fields, this field has increased in importance and recognition over the past few years. According to researchers, this field is the most effective at identifying anomalous kinetics and has numerous uses in a variety of fields. Ordinary differential equations with fractional derivatives can be used to simulate a variety of issues, including statistical, mathematical, engineering, chemical, and biological issues. Several distinct forms of fractional integrals and derivative operators (see e.g., ^[1,2,3,4]), including Riemann-Liouville, Caputo, Riesz, Hilfer, Hadamard, Erdélyi-Kober, Saigo, Marichev-Saigo-Maeda and others, have been thoroughly investigated by researchers. From an application perspective, we suggest the readers to see the work related to the fractional differential equations presented by ^[5,6,7,8]. In ^[9], the authors studied symmetric and antisymmetric solitons in the defocused saturable nonlinearity and the PT-symmetric potential of the fractional nonlinear Schrödinger equation. In ^[10], the fractional exponential function approach is used to study a time-fractional Ablowitz-Ladik model, and bright and dark discrete soliton solutions, discrete exponential solutions, and discrete peculiar wave solutions are discovered. In ^[11], the authors presented the rich vector exact solutions for the coupled discrete conformable fractional nonlinear Schrödinger equations by taking into account the conformable fractional derivative.

On the other hand, special functions like Gamma, Beta, Mittag-Leffler, et al. play a vital part in the foundation of fractional calculus. Moreover, the Mittag-Leffler function is regarded as the fundamental function in fractional calculus. The Prabhakar fractional operator containing a three-parameter version of the aforementioned function in the kernel. The M-L function has been extensively studied to construct solutions of fractional PDEs, such as dynamical characteristic of analytical fractional solitons for the space-time fractional Fokas-Lenells equation, soliton dynamics based on exact solutions of conformable fractional discrete complex cubic Ginzburg-Landau equation and Abundant fractional soliton solutions of a space-time fractional perturbed Gerdjikov-Ivanov equation by a fractional mapping method, see ^[12,13,14]. Strong generalizations of the univariate and bivariate Mittag-Leffler functions, which are known to be important in fractional calculus, are the multivariate Mittag-Leffler functions.

The well-known one-parameter Mittag-Leffler (M-L) function is defined by ^[15,16] as follows

$\begin{equation} \varepsilon_{a}(z_1) = \sum\limits_{l = 0}^{\infty}\frac{z_1^l}{\Gamma(al+1)}\,\,\,\,\,\,\,\,\,(a\in\mathbb{C};\Re(a) > 0,z_1\in\mathbb{C}), \end{equation}$

(1.1)

where $\mathbb{C}$ represents the set of complex numbers and $\Re(a)$ denotes the real part of the complex number.

The generalization of (1.1) with two parameters is defined by ^[17,18] as

$\begin{equation} \varepsilon_{a,b}(z_1) = \sum\limits_{l = 0}^{\infty}\frac{z_1^l}{\Gamma(al+b)}\,\,\,\,\,\,\,\,\,(a,b\in\mathbb{C};\Re(a) > 0,\Re(b) > 0), \end{equation}$

(1.2)

Later on, Agarwal ^[19], Humbert ^[20] and Humbert and Agarwal ^[21] studied the properties and applications of M-L functions. In ^[22], the generalization of (1.1) and (1.2) is defined by

$\begin{equation} \varepsilon_{a,b}^{c}(z_1) = \sum\limits_{l = 0}^{\infty}\frac{(c)_l}{\Gamma(a l+b)}\frac{z_1^l}{l!}\,\,\,\,\,\,\,\,\,(a,b,c\in\mathbb{C};\Re(a) > 0,\Re(b) > 0). \end{equation}$

(1.3)

In ^[23], the following generalization of the M-L function is defined by

$\begin{equation} \varepsilon_{a,b}^{c,q}(z_1) = \sum\limits_{l = 0}^{\infty}\frac{(c)_{lq}}{\Gamma(a l+b)}\frac{z_1^l}{l!}\,\,\,\,\,\,\,\,\,(a,b,c\in\mathbb{C};\Re(a) > 0,\Re(b) > 0, q > 0). \end{equation}$

(1.4)

In ^[24], Rahman et al. proposed the following generalized of M-L function by

$\begin{align} \varepsilon_{a,b,p}^{c,q,d}(z_1) = \sum\limits_{l = 0}^{\infty}\frac{B_p(c +lq; d-c)(d)_{lq}}{B(c, d-c)\Gamma(a l+b)}\frac{z_1^l}{l!}, \end{align}$

(1.5)

where $a, b, c, d\in\mathbb{C}; \Re(c) > 0, \Re(a) > 0, \Re(b) > 0, q > 0$ and $B_p(x, y) = \int_0^1 t^{x-1}(1-t)^{y-1}e^{-t-\frac{p}{t}}dt$ is the extension of beta function (see ^[25]).

Gorenflo et al. ^[26] and Haubold et al. ^[27]) studied the various properties of generalized M-L function. In ^[28], a new generalization of M-L function (1.3) is presented by

$\begin{equation} \varepsilon_{a,b,p}^{c}(z_1) = \sum\limits_{l = 0}^{\infty}\frac{(c;p)_l}{\Gamma(a l+b)}\frac{z_1^l}{l!}\,\,\,\,\,\,\,\,\,(p\geq0, a,b,c\in\mathbb{C}; , \Re(a) > 0,\Re(b) > 0,), \end{equation}$

(1.6)

where $(\lambda; p)_l$ is the Pochhammer symbol defined by Srivastava et al. ^[29,30] as

$\begin{equation} (\lambda;p)_{\mu} = \begin{cases} \frac{\Gamma_{p}(\lambda+\mu)}{\Gamma(\lambda)}; &(p > 0, \lambda,\mu\in\mathbb{C} )\\ \\ (\lambda)_{\mu}; & (p = 0,\ {\lambda,\mu}\in \mathbb{C}\setminus\{{0}\}.\ \ \ \ \ \ \ \ \ \ \ \ \end{cases} \end{equation}$

(1.7)

The researchers examined the developments of these extension, (1.6) and (1.7) and studied their related features and applications. In ^[30], Srivastava et al. proposed the following generalized hypergeometric function

$\begin{equation} _sF_t[(\delta_1; p),\cdots, (\delta_s); (\zeta_1),\cdots, (\zeta_t); z_1] = \sum\limits_{l = 0}^{\infty}\frac{(\delta_{1}; p)_l\cdots (\delta_s)_l}{(\zeta_1)_l\cdots (\zeta_t)_l}\ \frac{z_1^l}{l!}\, , \end{equation}$

(1.8)

where $\delta_{j}\in\mathbb{C}$ for j = 1, 2, $\cdots$ , s, $\zeta_{k}\in\mathbb{C}$ for $k = 1, 2, \cdots, t,$ and $\zeta_{k}\neq$ 0, -1, -2, $\cdots$ .

The integral representation of $(\mu; p)_\eta$ is explained by

$\begin{equation} (\mu;p)_{\eta} = \frac{1}{\Gamma(\mu)}\ \int_0^\infty\ s^{\mu+\eta-1}\ e^{-s-\frac{p}{s}} d s, \end{equation}$

(1.9)

where $\Re(\rho) > 0$ and $\Re(\mu+\eta) > 0$ . In particular, the related confluent hypergeometric function $_1F_1$ and the Gauss hypergeometric function $_2F_1$ are given by

$\begin{equation} _2F_1[(\delta_1; p),b;\lambda; z_1] = \sum\limits_{l = 0}^{\infty}\frac{(\delta_{1}; p)_l (b)_l}{(\lambda)_l}\ \frac{z_1^l}{l!}\, , \end{equation}$

(1.10)

and

$\begin{equation} _1F_1[(\delta_1; p);\lambda; z_1] = \Phi[(\delta_1; p);\lambda; z_1] = \sum\limits_{l = 0}^{\infty}\frac{(\delta_{1}; p)_l}{(\lambda)_l}\ \frac{z_1^l}{l!}\, . \end{equation}$

(1.11)

The expansion of the generalised hypergeometric function $_rF_s$ , which was studied by ^[30], has $r$ numerator and $s$ denominator parameters. Researchers recently developed several extensions of special functions, together with their corresponding characteristics and applications. Using extended beta functions as its foundation, Nisar et al. ^[31], Bohner et al. ^[32] and Rahman et al. ^[33] developed an enlargement of fractional derivative operators.

The multivariate M-L function is defined by ^[34] as follows:

$\begin{align} \mathcal{E}_{(a_j), b}^{(c_j)} (z_1, z_2, \dots, z_j)& = \mathcal{E}_{(a_1, a_2,\dots,a_j), b}^{(c_1, c_2 ,\dots, c_j)} (z_1, z_2, \dots z_j) \\ = & \sum\limits_{m_1, m_2, \dots, m_j = 0}^{\infty} \frac{(c_1)_{m_1}(c_2)_{m_2} \dots (c_j)_{m_j} (z_1)^{m_1} \dots (z_j)^{m_j}}{\Gamma(a_{1}m_1+a_2 m_2+\dots a_j m_j + b)m_1! \dots m_j!}, \end{align}$

(1.12)

where $z_i, a_i, b, c_i \in \mathbb{C}$ ; $i = 1, 2, \ldots, j$ , $\Re{(a_i)} > 0$ , $\Re{(b)} > 0$ and $\Re{(c_i)} > 0$ .

In ^{[35,36,37,38,39]}, the authors have studied various properties and applications of different type of generalized M-L functions. For real (complex) valued functions, the Lebesgue measurable space is defined by

$\begin{equation} \textsf{L}(r,s) = \left\{ \begin{array}{c} h: \|h\|_1 = \int_{r}^{s}|h(x)|dx < \infty \\ \end{array} \right\}. \end{equation}$

(1.13)

The left and right sides fractional integral operators of the Riemann-Liouville type are defined by ^[3,4] as follows:

$\begin{equation} (\mathfrak{I}_{r+}^{\lambda}h)(x) = \frac{1}{\Gamma(\lambda)}\int\limits_{r}^{x}\frac{h(\varrho)}{(x-\varrho)^{1-\lambda}}d\varrho,\; \; \; (x > r), \end{equation}$

(1.14)

and

$\begin{equation*} (\mathfrak{I}_{s-}^{\lambda}h)(x) = \frac{1}{\Gamma(\lambda)}\int\limits_{x}^{s}\frac{h(\varrho)}{(\varrho-x)^{1-\lambda}}d\varrho,\; \; \; (x < s), \end{equation*}$

where $h\in \textsf{L}(r, s)$ , $\lambda\in\mathbb{C}$ and $\Re(\lambda) > 0$ .

The left and right sides Riemann-Liouville fractional derivatives for the function $h(x)\in \textsf{L}(r, s)$ , $\lambda\in\mathbb{C}$ , $\Re(\lambda) > 0$ and $n = [\Re(\lambda)]+1$ are defined in ^[3,4] by

$\begin{equation} (\mathfrak{D}_{r+}^{\lambda}h)(x) = (\frac{d}{dx})^n(\mathfrak{I}_{r+}^{n-\lambda}h)(x) \end{equation}$

(1.15)

and

$\begin{equation*} (\mathfrak{D}_{s-}^{\lambda}h)(x) = (-\frac{d}{dx})^n(\mathfrak{I}_{s-}^{n-\lambda}h)(x), \end{equation*}$

respectively. The generalized differential operator $\mathfrak{D}_{r+}^{\lambda, v}$ of order $0 < \lambda < 1$ and type $0 < v < 1$ with respect to $x$ can be found in ^[2,4] as

$\begin{equation} (\mathfrak{D}_{r+}^{\lambda,v}h) = \left( \begin{array}{c} \mathfrak{I}_{r+}^{v(1-\lambda)}\frac{d}{dx}(\mathfrak{I}_{r+}^{(1-v)(1-\lambda)}h) \\ \end{array} \right)(x). \end{equation}$

(1.16)

In particular, if $v = 0,$ then (1.16) will lead to the operator $\mathfrak{D}_{r+}^{\lambda}$ defined in (1.15).

We also take into account the aforementioned well-known results.

Theorem 1.1. In ^[40], the following result for the fractional integral is presented by

$\begin{equation} \mathfrak{I}_{r+}^{\lambda}(\varrho-r)^{\eta-1} = \frac{\Gamma(\eta)}{\Gamma(\lambda+\eta)}(x-r)^{\lambda+\eta-1}, \end{equation}$

(1.17)

where $\lambda$ , $\eta\in\mathbb{C}$ , $\Re(\lambda) > 0$ , $\Re(\eta) > 0$ ,

Theorem 1.2. ^[41] Suppose that the function $h(z)$ has a power series expansion $h(z) = \sum\limits_{k = 0}^{\infty}k_nz^k$ and it is analytic in the disc $|z| < R$ , then we have the following result

$\begin{equation*} \label{a11} \mathfrak{D}_{z}^{\lambda}\{z^{\eta-1}h(z)\} = \frac{\Gamma(\eta)}{\Gamma(\lambda+\eta)}\sum\limits_{n = 0}^{\infty}\frac{a_n(\eta)_n}{(\lambda+\eta)_n}z^n. \end{equation*}$

Lemma 1.1. (Srivastava and Tomovski ^[42]) Suppose that $x > r$ , $\lambda\in(0, 1)$ , $v\in[0, 1]$ , $\Re(\eta) > 0$ and $\Re(\lambda) > 0$ , then we have

$\begin{equation} \mathfrak{D}_{r+}^{\lambda}[(\varrho-r)^{\eta-1}](x) = \frac{\Gamma(\eta)}{\Gamma(\eta-\lambda)}(x-r)^{\eta-\lambda-1}. \end{equation}$

(1.18)

The generalized multivariate M-L function (1.12) is then defined in terms of the modified Pochhammer symbol (1.7) and its different features as well as the accompanying integral operators are examined. This is driven by the aforementioned modifications of special functions.

Motivated by the above results and literature, the paper has the following structure: First, we describe and investigate a novel generalization of the multivariate M-L function using a generalized Pochhammer symbol. Secondly, we offer a few differential and fractional integral formulas for the explored multivariate M-L function. By using the new form of the multivariate M-L function, a new generalization of the fractional integral operator is introduced, and some fundamental characteristics of the operator are discussed.

2. The generalized multivariate of M-L function

We are in a position to present the generalized multivariate M-L function by utilizing the extended Pochhammer symbol in (1.7) as follows:

$\begin{equation} \varepsilon_{(a_j),b;p}^{(c_j)}(z_1,z_2,\cdots,z_j) = \sum\limits_{l_1,\cdots,l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}(c_2)_{l_2}\cdots(c_j)_{l_j}}{\Gamma(a_1 l_1+a_2l_2+\cdots+c_jl_j+b)}\frac{z_1^{l_1} z_2^{l_2}\cdots z_j^{l_j}}{l_1!\cdots l_j!}, \end{equation}$

(2.1)

where $a_i, b, c_i\in\mathbb{C}; \Re(a_i) > 0, \Re(b) > 0, , p\geq0$ for $i = 1, 2, \cdots, j$ . The special case for $a_1 = 1$ and $l_2 = \cdots = l_j = 0$ in (2.1) can be reduced to extended confluent hypergeometric function (1.11) as follows:

$\begin{equation} \varepsilon_{1,b;p}^{c_1}(z_1) = \frac{1}{\Gamma(b)} {_1F_1}[(c_1; p);b; z_1] = \frac{1}{\Gamma(b)} \Phi[(c_1; p);b; z_1]. \end{equation}$

(2.2)

In coming results, we demonstrate some fundamental characteristics and integral representations of the generalized multivariate M-L function.

Theorem 2.1. For the multivariate M-L function defined in (2.1), the following relation holds true:

$\begin{equation} \varepsilon_{(a_j),b;p}^{(c_j)}(z_1,z_2,\cdots,z_j) = b \varepsilon_{(a_j),b+1;p}^{(c_j)}(z_1,z_2,\cdots,z_j) \end{equation}$

(2.3)

$\begin{equation*} +[a_1z_1\frac{d}{dz_1}+\cdots+ a_jz_j\frac{d}{dz_j}] \varepsilon_{(a_j),b+1;p}^{(c_j)}(z_1,\cdots,z_j), \end{equation*}$

where $a_i, b, c_i\in\mathbb{C}; \Re(a_i) > 0, \Re(b) > 0, , p\geq0$ for $i = 1, 2, \cdots, j$ .

Proof. From (2.1), we have

$\begin{align*} &b \varepsilon_{(a_j),b+1,p}^{(c_j)}(z_1,\cdots,z_j)+[a_1z_1\frac{d}{dz_1}+\cdots+ a_jz_j\frac{d}{dz_j}]\varepsilon_{(a_j),b+1;p}^{(c_j)}(z_1,\cdots,z_j)\\ = &b \sum\limits_{l_1,\cdots,l_j = 0}^{\infty}\frac{ (c_1,p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b+1)}\frac{z_1^{l_1}\cdots z_j^{l_j}}{l_1!\cdots l_j!}\\+&[a_1z_1\frac{d}{dz_1}+\cdots+ a_jz_j\frac{d}{dz_j}]\sum\limits_{l_1,\cdots,l_j = 0}^{\infty}\frac{ (c_1,p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b+1)}\frac{z_1^{l_1}\cdots z_j^{l_j}}{l_1!\cdots l_j!}\\ = &b \sum\limits_{l_1,\cdots,l_j = 0}^{\infty}\frac{ (c_1,p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b+1)}\frac{z_1^{l_1}\cdots z_j^{l_j}}{l_1!\cdots l_j!}\\+&[a_1z_1 \frac{d}{dz_1}+\cdots +a_jz_j \frac{d}{dz_j}]\sum\limits_{l_1,\cdots,l_j = 0}^{\infty}\frac{ (c_1,p)_{l_1}(c_2)_{l_2}\cdots(c_j)_{l_j}}{\Gamma(a_1 l_1+\cdots+a_jl_j+b+1)}\frac{z_1^{l_1}\cdots z_jl_j}{l_1!\cdots l_j!}\\ = &b \sum\limits_{l_1,\cdots,l_j = 0}^{\infty}\frac{ (c_1,p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b+1)}\frac{z_1^{l_1}\cdots z_j^{l_j}}{l_1!\cdots l_j!}\\+&\sum\limits_{l_1,\cdots,l_j = 0}^{\infty}\frac{(c_1,p)_{l_1}(c_2)_{l_2}\cdots(c_j)_{l_j} }{\Gamma(a_1 l_1+\cdots+a_jl_j+b+1)}\frac{z_1^{l_1}\cdots z_j^{l_j}}{l_1!\cdots l_j!}(a_1l_1+\cdots+a_jl_j)\\ = & \sum\limits_{l_1,\cdots,l_j = 0}^{\infty}\frac{ (c_1,p)_{l_1}(c_2)_{l_2}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b+1)}\frac{z_1^{l_1}\cdots z_j^{l_j}}{l_1!\cdots l_j!}(a_1l_1+\cdots+a_jl_j+b)\ \ (\mbox{using} \ \Gamma(z_1+1) = z_1 \Gamma(z_1))\\ = &\sum\limits_{l = 0}^{\infty}\frac{(c_1,p)_{l_1}(c_2)_{l_2}\cdots(c_j)_{l_j} }{ \Gamma(a_1l_1+\cdots+a_jl_j+b)}\frac{z_1^{l_1}\cdots z_j^{l_j}}{l_1!\cdots l_j!}\\ = &\varepsilon_{(a_j),b,p}^{(c_j)}(z_1,z_2,\cdots,z_j), \end{align*}$

which is the desired result (2.3).

Theorem 2.2. For the generalized multivariate M-L function defined in (1.12), the following relations hold true:

$\begin{equation} \big(\frac{d}{dz_1}\big)^m\cdots(\frac{d}{dz_j}\big)^m \varepsilon_{(a_j),b;p}^{(c_j)}(z_1,z_2,\cdots,z_j) = (c_1)_m\cdots(c_j)_m \varepsilon_{(a_j),b+(a_j)m;p}^{(c_j)+m}(z_1,\cdots,z_j), \end{equation}$

(2.4)

and

$\begin{equation} \big(\frac{d}{dz_1}\big)^m [z_1^{b-1} \varepsilon_{(a_j),b;p}^{(c_j)}(\varpi_1 z_1^{a_1},\cdots,\varpi_j z_1^{a_j}))] = z_1^{b-m-1} \varepsilon_{(a_j),b-m;p}^{(c_j)}(\varpi_1 z_1^{a_1},\cdots,\varpi_j z_1^{a_j}), \end{equation}$

(2.5)

where $a_i, b, c_i\in\mathbb{C}; \Re(a_i) > 0, \Re(b) > 0, , p\geq0$ for $i = 1, 2, \cdots, j$ , and $\Re(b-m) > 0$ with $m\in\mathbb{N}.$

Proof. Differentiating (1.12) $m$ times with respect to $z_1, z_2, \cdots, z_j$ respectively, we get

$\begin{align*} & (\frac{d}{dz_1}\big)^m\cdots(\frac{d}{dz_j}\big)^m \varepsilon_{(a_j),b;p}^{(c_j)}(z_1,\cdots,z_j) = (\frac{d}{dz_1}\big)^m\cdots(\frac{d}{dz_j}\big)^m \sum\limits_{l_1 = l_2 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}(c_2)_{l_2}\cdots(c_j)_{l_j}}{\Gamma(a_1 l_1+\cdots+a_jl_j+b)}\frac{z_1^{l_1}\cdots z_j^{l_j}}{l_1!\cdots l_j!}\\ = &\sum\limits_{l_1 = \cdots = l_j = m}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b)}\frac{l_1!\cdots l_j!\ z_1^{l_1-m}\cdots z_j^{l_j-m}}{(l_1-m)!\cdots(l_j-m)!\ l1!\cdots l_j!}\\ & = \sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p)_{l_1+m}\cdots(c_j)_{l_j+m}}{\Gamma(a_1(l_1+m)+\cdots a_j(l_j+m)+b)}\frac{z_1^{l_1}\cdots z_j^{l_j}}{l_1!\cdots l_j!}\ \ (\mbox{Replacing}\ l_i \ \mbox{by}\ l_i+m) \\ & = \sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1)_m\cdots(c_j)_m\ (c_1+m;p)_{l_1}\cdots(c_j+m)_{l_j}}{\Gamma(a_1l_1+\cdots a_jl_j+b+(a_1+\cdots+a_j)m)}\frac{z_1^{l_1}\cdots z_j^{l_j}}{l_1!\cdots l_j!}. \end{align*}$

Now using $(\lambda; \sigma)_{\mu+p} = (\lambda)_{\mu} (\lambda+\mu; \sigma)_p$ and $(\lambda)_{\mu+p} = (\lambda)_{\mu} (\lambda+\mu)_p$ , we get

$\begin{align*} & (\frac{d}{dz_1}\big)^m\cdots(\frac{d}{dz_j}\big)^m \varepsilon_{(a_j),b;p}^{(c_j)}(z_1,\cdots,c_j)\\ = &(c_1)_m \cdots(c_j)_m \sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1+m;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots a_jl_j+b+(a_1+\cdots +a_j)m)}\frac{z_1^{l}\cdots z_j^{l_j}}{l_1!\cdots l_j!}\\ & = (c_1)_m\cdots (c_j)_{m} \ \varepsilon_{(a_j),b+(a_j)m;p}^{(c_j)+m}(z_1,z_2,\cdots, z_j), \end{align*}$

which is the desired result (2.4). Similarly, to prove (2.5), we have

$\begin{align*} &\big(\frac{d}{dz_1}\big)^m [z_1^{b-1} \varepsilon_{(a_j),b;p}^{(c_j)}(\varpi_1 z_1^{a_1},\cdots \varpi_j z_j^{a_j})]\\ = &\big(\frac{d}{dz_1}\big)^m z_1^{b-1}\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b)}\frac{(\varpi_1 z_1^{a_1})^{l_1}\cdots(\varpi_jz_1^{a_j})^{l_j}}{l_1!\cdots l_j!}\\ = &\big(\frac{d}{dz_1}\big)^m \sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+ a_jl_j+b)} \frac{z_1^{b-1+a_1l_1+\cdots+a_jl_j}}{l_1!\cdots l_j!} \varpi_1^{l_1}\cdots\varpi_j^{l_j}\\ = &\sum\limits_{l_1 = \cdots l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b)} \frac{\varpi_1^{l_1}\cdots\varpi_j^{l_j}}{l_1!\cdots l_j!} \frac{(a_1l_1+\cdots+a_jl_j+b-1)!}{(a_1l_1+\cdots+a_jl_j+b-m-1)!}\ z_1^{a_1l_1+\cdots+a_jl_j+b-m-1}. \end{align*}$

Differentiating $m$ times and using the relation $l(l-1)! = l!$ , we get

$\begin{align*} &\big(\frac{d}{dz_1}\big)^m [z_1^{b-1} \varepsilon_{(a_j),b;p}^{(c_j)}(\varpi_1 z_1^{a_1},\cdots \varpi_j z_j^{a_j})]\\ = &\sum\limits_{l_1 = \cdots l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b)} \frac{\Gamma(a_1l_1+\cdots+a_jl_j+b)}{\Gamma(a_1l_1+\cdots+a_jl_j+b-m)}\, \varpi_1^{l_1}\cdots\varpi_j^{l_j} \, \frac{z_1^{a_1l_1+\cdots+a_jl_j+b-1-m}}{l_1!\cdots l_j!} \\ = & z_1^{b-m-1}\sum\limits_{l_1 = \cdots l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b-m)}\frac{(\varpi_1 z_1^{a_1})^{l_1}\cdots(\varpi_1 z_1^{a_j})^{l_j}}{l_1!\cdots l_j!}\\ = &z_1^{b-m-1}\varepsilon_{(a_j),b-m;p}^{(c_j)}(\varpi_1 z_1^{a_1},\cdots,\varpi_j z_1^{a_j}). \end{align*}$

The proof is completed.

Corollary 2.1. The generalized multivariate M-L function has the following integral representations:

$\begin{equation*} \int_{0}^{z_1}t^{b-1}\varepsilon_{(a_j),b;p}^{(c_j)}(\varpi_1 t^{a_1},\cdots,\varpi_j t^{a_j}) d t = z_1^{b}\varepsilon_{(a_j),b+1;p}^{(c_j)}(\varpi_1 z_1^{a_1},\cdots,\varpi_jz_1^{a_j}), \end{equation*}$

where $a_i, b, c_i, \varpi_i\in\mathbb{C}; \Re(a_i) > 0, \Re(b) > 0, p\geq0$ for $i = 1, 2, \cdots, j$ .

3. Fractional integral and differential formulas of generalized M-L function

In this section, we present some fractional integration and differentiation formulas of generalized M-L function given in (2.1).

Theorem 3.1. Suppose $x > r\, (r\in\mathbb{R_+} = [0, \infty))$ , $a_i$ , $b$ , $c_i$ , $\varpi\in\mathbb{C}$ , $\Re(a_i) > 0$ and $\Re(c_i) > 0$ , $\Re(b) > 0$ and $\Re(\lambda) > 0$ , then the following relations hold true:

$\begin{align} &\mathfrak{I}_{r+}^{\lambda}\left[(\varrho-r)^{b-1}\varepsilon_{(a_j),b;p}^{(c_j)}(\varpi_1(\varrho-r)^{a_1},\cdots,\varpi_j(\varrho-r)^{a_j})\right](x) \\ = &(x-r)^{\lambda+b-1} \varepsilon_{(a_i), b+\lambda;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j}), \end{align}$

(3.1)

$\begin{align} &\mathfrak{D}_{r+}^{\lambda}\left[(\varrho-r)^{b-1}\varepsilon_{(a_i),b;p}^{( c_i)}(\varpi_1(\varrho-r)^{a_1},\cdots,\varpi_j(\varrho-r)^{a_j})\right](x) \\ = &(x-r)^{b-\lambda-1} \varepsilon_{(a_i),b-\lambda;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j}) \end{align}$

(3.2)

and

$\begin{align} &\mathfrak{D}_{r+}^{\lambda,v}\left[(\varrho-r)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j})\right](x) \\ = &(x-r)^{b-\lambda-1} \varepsilon_{(a_i),b-\lambda;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j}). \end{align}$

(3.3)

Proof. Consider

$\begin{align*} &\mathfrak{I}_{r+}^{\lambda}\left[(\varrho-r)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j})\right](x)\\ = &\frac{1}{\Gamma(\lambda)}\int_{r}^{x}\frac{(x-r)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(\varrho-r)^{a_1},\cdots,\varpi_1(\varrho-r)^{a_j})} {(x-\varrho)^{1-\lambda}}d\varrho\\ = &\frac{1}{\Gamma(\lambda)}\sum\limits_{n = 0}^{\infty}\frac{(c_1;p,v)_{l_1}\cdots(c_j)_{l_n}\varpi^{l_1}\cdots\varpi^{l_j}}{\Gamma(a_1 l_1+\cdots+a_jl_j+b)l_1!\cdots l_j!}\int_{r}^{x}(\varrho-r)^{b+a_1 l_1+\cdots+a_jl_j-1}(x-\varrho)^{\lambda-1}d\varrho\\ = &\sum\limits_{n = 0}^{\infty}\frac{(c_1;p,v)_{l_1}\cdots(c_j)_{l_n}\varpi^{l_1}\cdots\varpi^{l_j}}{\Gamma(a_1 l_1+\cdots+a_jl_j+b)l_1!\cdots l_j!} \left( \mathfrak{I}_{r+}^{\lambda}[(\varrho-r)^{b+a_1 l_1+\cdots+a_jl_j-1}] \right). \end{align*}$

By the use of (1.17), we have

$\begin{align*} &\mathfrak{I}_{r+}^{\lambda}\left[(\varrho-r)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j})\right](x)\\ = &\sum\limits_{n = 0}^{\infty}\frac{(c_1;p,v)_{l_1}\cdots(c_j)_{l_n}\varpi^{l_1}\cdots\varpi^{l_j}}{\Gamma(a_1 l_1+\cdots+a_jl_j+b)l_1!\cdots l_j!} (x-r)^{b+\lambda+a_1l_1+\cdots+a_jl_j-1}. \frac{\Gamma(a_1l_1+\cdots+a_jl_j+b)}{\Gamma(a_1l_1+\cdots+a_jl_j+b+\lambda)}\\ = &(x-r)^{b+\lambda-1}\sum\limits_{n = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b+\lambda)} \frac{[\varpi_1^{l_1}(x-r)^{a_1l_1}\cdots\varpi_j^{l_j}(x-r)^{a_jl_j}]}{l_1!\cdots l_j!}\\ = &(x-r)^{b+\lambda-1}\varepsilon_{(a_i),b+\lambda;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j}), \end{align*}$

which gives the proof of (3.1).

Next, we have

$\begin{align*} &\mathfrak{D}_{r+}^{\lambda}\left[(\varrho-r)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(\varrho-r)^{a_1},\cdots,\varpi_j(\varrho-r)^{a_j})\right]\\ = &(\frac{d}{dx})^n \left\{ \mathfrak{I}_{r+}^{n-\lambda}[(\varrho-r)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(\varrho-r)^{a_1},\cdots,\varpi_j(\varrho-r)^{a_j})] \right\}, \end{align*}$

which on using (3.1) takes the following form:

$\begin{align*} &\mathfrak{D}_{r+}^{\lambda}\left[(\varrho-r)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(\varrho-r)^{a_1},\cdots,\varpi_j(\varrho-r)^{a_j})\right] \\ = &(\frac{d}{dx})^n \left\{ (x-r)^{b-\lambda+n-1}\varepsilon_{(a_i),b-\lambda+n;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j}) \right\}. \end{align*}$

Applying (2.5), we get

$\begin{align*} &\mathfrak{D}_{r+}^{\lambda}\left[(\varrho-r)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j})\right](x) \\ = &\left\{ (x-r)^{\eta-\lambda-1}\varepsilon_{(a_i),b-\lambda;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots, \varpi_j(x-r)^{a_j}) \right\}, \end{align*}$

which gives the proof of (3.2).

To obtain (3.3), we have

$\begin{align*} &\left( \mathfrak{D}_{r+}^{\lambda,v}\left[(\varrho-r)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(\varrho-r)^{a_1},\cdots,\varpi_j(\varrho-r)^{a_j})\right] \right)(x)\\ = &\left( \mathfrak{D}_{r+}^{\lambda,v}\left[\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p,v)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j +b)}\frac{\varpi^{l_1}\cdots\varpi^{l_j}}{l_1!\cdots l_j!}(\varrho-r)^{a_1 l_1+\cdots+a_jl_j+b-1}\right] \right)(x)\\ = &\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p,v)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j +b)}\frac{\varpi^{l_1}\cdots\varpi^{l_j}}{l_1!\cdots l_j!}\\ \times&\left( \mathfrak{D}_{r+}^{\lambda,v}[(\varrho-r)^{a_1 l_1+\cdots+a_jl_j+b-1}] \right)(x). \end{align*}$

By applying (1.18), we get

$\begin{align*} &\left( \mathfrak{D}_{r+}^{\lambda,v}[(\varrho-r)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(\varrho-r)^{a_1},\cdots,\varpi_j(\varrho-r)^{a_j})] \right)(x)\\ = &\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p,v)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j +b)}\frac{\varpi^{l_1}\cdots\varpi^{l_j}}{l_1!\cdots l_j!}\\ \times&\frac{\Gamma(a_1l_1+\cdots+ a_jl_j+b)}{\Gamma(a_1l_1+\cdots+a_jl_j +b-\lambda)}(x-r)^{a_1l_1+ \cdots+a_jl_j+b-\lambda-1}\\ = &(x-r)^{b-\lambda-1}\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p,v)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j +b-\lambda)}\frac{\varpi^{l_1}(x-r)^{a_1}\cdots\varpi^{l_j}(x-r)^{a_j}}{l_1!\cdots l_j!}\\ = &(x-r)^{b-\lambda-1}\varepsilon_{(a_i),b-\lambda;p}^{(c_i)}(\varpi^{l_1}(x-r)^{a_1},\cdots,\varpi^{l_j}(x-r)^{a_j}), \end{align*}$

which completes the required proof.

Remark 3.1. Applying Theorem 3.1 for $p = 0$ , then we obtain the result presented in ^[34].

4. Generalization of fractional integral and its properties

In this section, we define a fractional integral involving newly defined multivariate M-L function and discuss its properties.

Definition 4.1. Let $b, a_i, c_i, \varpi_i\in\mathbb{C}$ , $\Re(c_i) > 0$ , $\Re(a_i) > 0$ and $\Re(b) > 0$ and $h\in L(r, s)$ . Then the generalized left and right sided fractional integrals are defined by

$\begin{align} &\left(\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}h\right)(x) = \int_{r}^{x}(x-\varrho)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)} (\varpi_1(x-\varrho)^{a_1},\cdots,\varpi_j(x-\varrho)^{a_j})h(\varrho)d\varrho, (x > r) \end{align}$

(4.1)

and

$\begin{align} &\left(\mathfrak{R}_{s-;(a_i),b;p}^{(\varpi_i);(c_i)}h\right)(x) = \int_{x}^{s}(\varrho-x)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)} (\varpi_1(\varrho-x)^{a_1},\cdots,\varpi_j(\varrho-x)^{a_j})h(\varrho)d\varrho, (x < s), \end{align}$

(4.2)

respectively.

Remark 4.1. If we consider $p = 0$ , then the operators defined in (4.1) and (4.2) will take the form defined earlier by ^[34]. Similarly, if we consider $p = 0$ and $j = 1$ , then the operators defined in (4.1) and (4.2) will take the form defined by ^[22]. If we take $j = 1$ , then the work done in this paper will lead to the work presented by ^[28]. Also, if we consider one of $\varpi_i = 0$ , for $i = 1, 2, \cdots, j$ , then the operators defined in (4.1) and (4.2) will take the form of the classical operators.

Next, we prove the following properties of integral operator defined in (4.1).

Theorem 4.1. Suppose that $b, a_i, \lambda, c_i, \varpi_i\in\mathbb{C}$ , $\Re(a_i) > 0$ , $\Re(b) > 0$ , $\Re(\lambda) > 0$ , $p\geq0$ and $\Re(c_i) > 0$ for $i = 1, 2, \cdots, j$ , then the following result holds true:

$\begin{align*} \left(\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}[(\varrho-r)^{\lambda-1}] \right) (x) = (x-r)^{\lambda+b-1}\Gamma(\lambda)\varepsilon_{(a_i),b+\lambda}^{(c_i);p}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j}). \end{align*}$

Proof. By the use of definition (4.1), we have

$\begin{equation*} \left( \mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}h \right)(x) = \int_{r}^{x}(x-\varrho)^{b-1}\varepsilon_{(a_i),b}^{(c_i)}(\varpi_1(x-\varrho)^{a_1},\cdots,\varpi_j(x-\varrho)^{a_j})h(\varrho)d\varrho. \end{equation*}$

Therefore, we get

$\begin{align*} &\left( \mathfrak{R}_{r+;(a_i), b;p}^{(\varpi_i);(c_i)}[(\varrho-r)^{\lambda-1}] \right) (x) = \int_{r}^{x}(x-\varrho)^{b-1}(\varrho-r)^{\lambda-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-\varrho)^{a_1},\cdots,\varpi_j(x-\varrho)^{a_j})d\varrho\\ = &\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b))}\frac{\varpi_1^{l_1}\cdots\varpi_j^{l_j}} {l_1!\cdots l_j!} \left( \int_{r}^{x}(\varrho-r)^{\lambda-1}(x-\varrho)^{\lambda+a_1l_1+ \cdots+a_jl_j-1}d\varrho \right)\\ = &\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b))}\frac{\varpi_1^{l_1}\cdots\varpi_j^{l_j}} {l_1!\cdots l_j!} \mathfrak{I}_{r+}^{a_1l_1+\cdots+a_jl_j+b}[(\varrho-r)^{\lambda-1}]\\ & = (x-r)^{b+\lambda-1} \sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b))} \frac{[\varpi_1(x-r)^{a_1l_1}\cdots\varpi_j(x-r)^{a_jl_j}]}{l_1!\cdots l_j!}\\ \times& \frac{\Gamma(\lambda)\Gamma(a_1l_1+\cdots+a_jl_j +b)}{\Gamma(a_1l_1+\cdots+a_jl_j+b+\lambda)}\\ = &(x-r)^{b+\lambda-1}\Gamma(\lambda)\varepsilon_{(a_i),b+\lambda;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j}), \end{align*}$

which gives the desired proof.

Theorem 4.2. Suppose that $c_i, a_i, b, \varpi_i \in\mathbb{C}$ , $\Re(a_i) > 0$ , $\Re(b) > 0,$ $p\geq0$ for $i = 1, 2, \cdots, j$ , then the following result holds true:

$\begin{equation*} \|\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}\Phi\|_1\leq K\|\Phi\|_1. \end{equation*}$

Where

$\begin{align*} &K: = (s-r)^{\mathfrak{Re}(b)}\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{|(c_1;p)_{l_1}\cdots(c_j)_{l_j}|}{\Gamma(a_1l_1+\cdots+a_jl_j +b)(\Re(b)+\Re(a_1)l_1+\cdots+\Re(a_j)l_j)}\\ \times&\frac{|\varpi_1^{l_1}(s-r)^{a_1l_1}\cdots\varpi_j^{l_j}(s-r)^{a_jl_j}|}{l_1!\cdots l_j!}. \end{align*}$

Proof. By the use of (1.13) and (4.1) and by interchanging integration and summation order, we have

$\begin{align*} \|\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i) }\Phi\|_1& = \int\limits_{r}^{s}|\int_{r}^{x}(x-\varrho)^{b-1} \varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-\varrho)^{a_1},\cdots,\varpi_j(x-\varrho)^{a_j}) \Phi(\varrho)d\varrho|dx\\ \leq&\int_{r}^{s}\left[ \int_{\varrho}^{x} (x-\varrho)^{\Re(b)-1}|\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-\varrho)^{a_1},\cdots,\varpi_j(x-\varrho)^{a_j})|dx \right]|\Phi(\varrho)|d\varrho\\ = &\int_{r}^{s}\left[ \int_{0}^{x-\varrho} u^{\Re(b)-1}|\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1 u^{a_1},\cdots,\varpi_j u^{a_j})|du \right]|\Phi(\varrho)|d\varrho, \end{align*}$

by setting $u = x-\varrho.$ After simplification, we obtain

$\begin{align*} &\|\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}\Phi\|_1\\ \leq&\int_{r}^{s} \Bigr[ \sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{|(c_1;p)_{l_1}\cdots(c_j)_{l_j}|}{\Gamma(a_1l_1+\cdots+a_jl_j+b)}\frac{|\varpi_1^{a_1}\cdots\varpi_j^{l_j}|} {l_1!\cdots l_j!}\\ \times& \left( \frac{(u)^{\Re(b)+\Re(a_1)l_1+\cdots+\Re(a_j)l_j}}{(\Re(b)+\Re(a_1)l_1+\cdots+\Re(a_j)l_j)} \right)_{0}^{s-r} \Bigr] |\Phi(\varrho)|d\varrho. \end{align*}$

It follows that

$\begin{align*} &\|\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}\Phi\|_1\leq \Bigr\{ (s-r)^{\Re(b)} \sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{|(c_1;p)_{l_1}\cdots(c_j)_{l_j}|}{\Gamma(a_1l_1+\cdots+a_jl_j +b)(\Re(b)+\Re(a_1)l_1+\cdots+\Re(a_j)l_j)}\\ \times&\frac{|\varpi_1^{l_1}(s-r)^{a_1l_1}\cdots\varpi_j^{l_j}(s-r)^{a_jl_j}|}{l_1!\cdots l_j!} \Bigr\} \int\limits_{r}^{s}|\Phi(\varrho)|d\varrho\\ = &K||\Phi||_1, \end{align*}$

where

$\begin{align*} &K = (s-r)^{\mathfrak{Re}(b)}\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{|(c_1;p)_{l_1}\cdots(c_j)_{l_j}|}{\Gamma(a_1l_1+\cdots+a_jl_j +b)(\Re(b)+\Re(a_1)l_1+\cdots+\Re(a_j)l_j)}\\ \times&\frac{|\varpi_1^{l_1}(s-r)^{a_1l_1}\cdots\varpi_j^{l_j}(s-r)^{a_jl_j}|}{l_1!\cdots l_j!}, \end{align*}$

which gives the desired result.

Corollary 4.1. If we take $a_i, b, c_i, \varpi_i \in \mathbb{C}$ , $\Re(a_i) > 0$ , $\Re(b) > 0$ , $\Re(c_i) > 0$ with $i = 1, 2, \cdots, j$ , then the following result holds true:

$\begin{align*} \left(\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}1\right)(x) = (x-r)^{b}\varepsilon_{(a_i),b+1;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j}). \end{align*}$

Proof. Consider

$\begin{align*} &\left(\mathfrak{R}_{r+;(a_i),b}^{(\varpi_i);(c_i)}1\right)(x) = \int_{r}^x(x-\varrho)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-\varrho)^{a_1},\cdots,\varpi_j(x-r)^{a_j})d\varrho\\ = &\int_{r}^x(x-\varrho)^{b-1}\left(\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}\varpi_1^{l_1}(x-\varrho)^{a_1l_1}\cdots\varpi_j^{l_j}(x-\varrho)^{a_jl_j}} {\Gamma(a_1l_1+\cdots+a_jl_j+b)l_1!\cdots l_j!}\right)d\varrho. \end{align*}$

It follows that

$\begin{align*} \left(\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}1\right)(x) & = \sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}\varpi_1^{l_1}\cdots\varpi_j^{l_j}} {\Gamma(a_1l_1+\cdots+a_jl_j+b)l_1!\cdots l_j!} \int_{r}^x(x-\varrho)^{b+a_1l_1+\cdots+a_jl_j-1}d\varrho\\ = &(x-r)^{b}\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}\varpi_1^{l_1}(x-r)^{a_1l_1}\cdots\varpi_j^{l_j}(x-r)^{a_jl_j}} {\Gamma(a_1l_1+\cdots+a_jl_j+b)(a_1l_1+\cdots+a_jl_j+b)l_1!\cdots l_j!}\\ = &(x-r)^{b}\,\varepsilon_{(a_i),b+1;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j}), \end{align*}$

which gives the desired proof.

Theorem 4.3. The generalized fractional operator can be represented in term of Riemann–Liouville fractional integrals for $c_i$ , $a_i$ , $b$ , $\varpi_i\in\mathbb{C}$ with $\Re(a_i) > 0$ , $\Re(b) > 0$ , $\Re(c_i) > 0$ for $i = 1, 2, \cdots, j$ , $p\geq0$ and $x > r$ as follows:

$\begin{align*} \left(\mathfrak{R}_{r+;(a_i),b}^{(\varpi_i);(c_i)}h\right)(x) = \sum\limits_{l_1 = \cdots = l_j = 0}^\infty \frac{\Gamma(c_1+l_1;p)(c_2)_{l_2}\cdots(c_j)_{l_j}\varpi_1^{a_1}\cdots\varpi_j^{a_j}} {\Gamma(c_1)l_1!\cdots l_j!}\mathfrak{I}_{r+}^{a_1l_1+\cdots+a_jl_j +b}h(x). \end{align*}$

Proof. By utilizing (2.1) in (4.1) and then interchanging the order of summation and integration, we have

$\begin{align*} \left(\mathfrak{R}_{r+;(a_i),b}^{(\varpi_i);(c_i)}h\right)(x)& = \int_{r}^{x}(x-\varrho)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)} (\varpi_1(x-\varrho)^{a_1},\cdots,\varpi_j(x-\varrho)^{a_j})h(\varrho)d\varrho\\ = &\int_{r}^{x}(x-\varrho)^{b-1}\sum\limits_{l_1 = \cdots = l_j = 0}^\infty\frac{\Gamma(c_1+l_1;p)(c_2)_{l_2}\cdots(c_j)_{l_j}\varpi_1^{l_1} (x-\varrho)^{a_1l_1}\cdots\varpi_j^{l_j} (x-\varrho)^{a_jl_j}} {\Gamma(c_1)\Gamma(a_1l_1+\cdots+a_jl_j+b)l_1!\cdots l_j!}h(\varrho)d\varrho\\ = &\sum\limits_{l_1 = \cdots = l_j = 0}^\infty\frac{\Gamma(c_1+l_1;p)(c_2)_{l_2}\cdots(c_j)_{l_j}\varpi_1^{a_1l_1}\cdots\varpi_j^{a_jl_j} }{\Gamma(c_1)l_1!\cdots l_j!} \frac{1}{\Gamma(a_1l_1+\cdots+a_jl_j +b)}\\ \times&\int_{r}^{x}(x-\varrho)^{a_1l_1+\cdots+a_jl_j+b-1}h(\varrho)d\varrho\\ = &\sum\limits_{l_1 = \cdots = l_j = 0}^\infty\frac{\Gamma(c_1+l_1;p)(c_2)_{l_2}\cdots(c_j)_{l_j}\varpi_1^{a_1l_1}\cdots\varpi_j^{a_jl_j} }{\Gamma(c_1)l_1!\cdots l_j!}\mathfrak{I}_{r+}^{a_1l_1 +\cdots+a_jl_j+b}h(x), \end{align*}$

which gives the desired proof.

Theorem 4.4. For $\lambda$ , $c_i$ , $a_i$ , $b$ , $\varpi_i\in\mathbb{C}$ with $\Re(a_i) > 0$ , $\Re(b) > 0$ , $\Re(c_i) > 0$ , $\Re(\lambda) > 0$ , for $i = 1, 2, \cdots, j$ , $p\geq0$ and $x > r$ , then the following result holds true:

$\begin{equation} (\mathfrak{I}_{r+}^{\lambda}[\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}h])(x) = (\mathfrak{R}_{r+;(a_i),b+\lambda}^{(\varpi_i);(c_i)}h)(x) = (\mathfrak{R}_{r+;(a_i),b}^{(\varpi_i);(c_i)}[\mathfrak{I}_{r+}^{\lambda}h])(x), \end{equation}$

(4.3)

where $h\in L(r, s)$ .

Proof. By employing (1.14) and (4.1), we have

$\begin{align*} &(\mathfrak{I}_{r+}^{\lambda}[\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}h])(x)\\ = &\frac{1}{\Gamma(\lambda)}\int_{r}^{x}\frac{[(\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}h)(\varrho)]}{(x-\varrho) ^{1-\lambda}}d\varrho\\ = &\frac{1}{\Gamma(\lambda)}\int_{r}^{x}(x-\varrho)^{\lambda-1} \left[ \int_{r}^{\varrho} (\varrho-u)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(\varrho-u)^{a_1},\cdots,\varpi_j(\varrho-u)^{a_j})h(u)du \right]d\varrho. \end{align*}$

It follows that

$\begin{align*} &(\mathfrak{I}_{r+}^{\lambda}[\mathfrak{R}_{r+;(a_i),b}^{(\varpi_i);(c_i)}h])(x) \\ = &\int_{r}^{x} \left[ \frac{1}{\Gamma(\lambda)}\int_{u}^{x} (x-\varrho)^{\lambda-1}(\varrho-u)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i) }(\varpi_1(\varrho-u)^{a_1},\cdots,\varpi_j(\varrho-u)^{a_j})d\varrho \right]h(u)du. \end{align*}$

By considering $\varrho-u = \theta$ , we get

$\begin{align*} \begin{aligned} &(\mathfrak{I}_{r+}^{\lambda}[\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}h])(x)\\ = &\int_{r}^{x} \left[ \frac{1}{\Gamma(\lambda)}\int_{0}^{x-u} (x-u-\theta)^{\lambda-1}\theta^{b-1}\varepsilon_{(a_i),b;p}^{(c_i) } (\varpi_1\theta^{a_1},\cdots,\varpi_j\theta^{a_j})d\theta \right]h(u)du\\ = &\int_{r}^{x} \left[ \frac{1}{\Gamma(\lambda)}\int_{0}^{x-u} \frac{\theta^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1\theta^{a_1},\cdots,\varpi_j\theta^{a_j})}{(x-u-\theta)^{1-\lambda}}d\theta \right]h(u)du.\end{aligned} \end{align*}$

Hence, from (1.14) and applying (3.1), we obtain

$\begin{align*} &(\mathfrak{I}_{r+}^{\lambda}[\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}h])(x)\\ = &\int_{r}^{x}\left[ \theta^{\lambda+b-1}\varepsilon_{(a_i),b+\lambda;p}^{(c_i)}(\varpi_1\theta^{a_1},\cdots,\varpi_j\theta^{a_j}) \right]h(u)du\\ = &\int_{r}^{x} (x-u)^{\lambda+b-1}\varepsilon_{(a_i),b+\lambda}^{(c_i)}(\varpi_1(x-u)^{a_1},\cdots,\varpi_j(x-u)^{a_j})h(u)du. \end{align*}$

Thus, we have

$\begin{equation} (\mathfrak{I}_{r+}^{\lambda}[\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}h])(x) = \left( \mathfrak{R}_{r+;(a_i),b+\lambda}^{(\varpi_i);(c_i)}h \right)(x). \end{equation}$

(4.4)

Next, consider the right hand side of (4.3) and employing (4.1) to derive the second part, we have

$\begin{align*} &(\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}[\mathfrak{I}_{r+}^{\lambda}h])(x)\\ = &\int_{r}^{x}(x-\varrho)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-\varrho)^{a_1},\cdots,\varpi_j(x-\varrho)^{a_j}) [\mathfrak{I}_{r+}^{\lambda}h] (\varrho)d\varrho\\ = &\int_{r}^{x}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-\varrho)^{a_1},\cdots,\varpi_j(x-\varrho)^{a_j})\left( \frac{1}{\Gamma(\lambda)}\int\limits_{r}^{\varrho}\frac{h(u)}{(\varrho-u)^{1-\lambda}}du \right)d\varrho. \end{align*}$

It follows that

$\begin{align*} &(\mathfrak{R}_{r+;(a_i),b}^{(\varpi_i);(c_i)}[\mathfrak{I}_{r+}^{\lambda}h])(x)\\ = &\int\limits_{r}^{x}\frac{1}{\Gamma(\lambda)} \left[ \int_{u}^{x}(x-\varrho)^{b-1}(\varrho-u)^{\lambda-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-\varrho)^{a_1},\cdots,\varpi_j(x-\varrho)^{a_j})d\varrho \right] h(u)du. \end{align*}$

By setting $x-\varrho = \theta$ , we get

$\begin{align*} &(\mathfrak{R}_{r+;(a_i),b}^{(\varpi_i);(c_i)}[\mathfrak{I}_{r+}^{\lambda}h])(x)\\ = &\int_{r}^{x}\frac{1}{\Gamma(\lambda)} \left[ \int_{x-u}^{0}\theta^{b-1}(x-\theta-u)^{\lambda-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1\theta^{a_1},\cdots,\varpi_j\theta^{a_j}) (-d\theta) \right] h(u)du\\ = &\int\limits_{r}^{x}\frac{1}{\Gamma(\lambda)} \left[ \int_{0}^{x-u}\theta^{b-1}(x-\theta-u)^{\lambda-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1\theta^{a_1},\cdots,\varpi_j\theta^{a_j})d\theta \right] h(u)du. \end{align*}$

Further, by using (1.14) and applying (3.1), we obtain

$\begin{equation} (\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}[\mathfrak{I}_{r+}^{\lambda}h])(x) = \left( \mathfrak{R}_{r+;(a_i),b+\lambda}^{(\varpi_i);(c_i)}h \right)(x). \end{equation}$

(4.5)

Thus, (4.4) and (4.5) gives the desired proof.

5. Conclusions

Nowadays, the theories are developed very rapidly. The scientists are introducing more advanced and generalized forms of the classical ones. In this present study, we introduced a generalized form of the multivariate M-L function (2.1) by employing the generalized Pochhammer symbol and its properties. By using this more extended form of M-L, we introduced a fractional integral operator and studied some of the basic properties of this operator. The special cases of the main results are if we take $p = 0$ , then the operators defined in (4.1) and (4.2) will reduce to the work done by ^[34]. Similarly, if we take $j = 1$ and $p = 0$ , then the operators defined in (4.1) and (4.2) will lead to the work done by ^[22]. If we take $j = 1$ , then the work done in this paper will lead to the work presented by ^[28]. Moreover, if we consider one of $\varpi_i = 0$ , for $i = 1, 2, \cdots, j$ , then the operators defined in (4.1) and (4.2) will reduce to the classical R-L operators. We believe that our proposed operator will be more applicable in the fields of fractional integral inequalities and operator theory.

Acknowledgments

The author T. Abdeljawad would like to thank Prince Sultan University for supporting through TAS research lab. Manar A. Alqudah: Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2023R14), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Conflict of interest

The authors declare no conflict of interest.

References

[1]	S. Kumar, A. Kumar, B. Mohan, Evolutionary dynamics of solitary wave profiles and abundant analytical solutions to a (3+1)-dimensional burgers system in ocean physics and hydrodynamics, J. Ocean Eng. Sci., 8 (2021), 1–14. https://doi.org/10.1016/j.joes.2021.11.002 doi: 10.1016/j.joes.2021.11.002
[2]	N. Zobeiry, K. D. Humfeld, A physics-informed machine learning approach for solving heat transfer equation in advanced manufacturing and engineering applications, Eng. Appl. Artif. Intel., 101 (2021), 104232. https://doi.org/10.1016/j.engappai.2021.104232 doi: 10.1016/j.engappai.2021.104232
[3]	S. Kumar, S. Rani, N. Mann, Diverse analytical wave solutions and dynamical behaviors of the new (2+1)-dimensional Sakovich equation emerging in fluid dynamics, Eur. Phys. J. Plus, 137 (2022), 1226. https://doi.org/10.1140/epjp/s13360-022-03397-w doi: 10.1140/epjp/s13360-022-03397-w
[4]	V. Jadaun, N. R. Singh, S. Singh, R. Shankar, Impact of solitons on the progression of initial lesion in aortic dissection, Int. J. Biomath., 15 (2022), 2150096. https://doi.org/10.1142/S1793524521500960 doi: 10.1142/S1793524521500960
[5]	V. Jadaun, A. Srivastav, A special phenomenon of wave interactions: an application of nonlinear evolution equation in (3+1)-dimension, Commun. Nonlinear Sci. Numer. Simul., 130 (2024), 107733. https://doi.org/10.1016/j.cnsns.2023.107733 doi: 10.1016/j.cnsns.2023.107733
[6]	N. Raza, S. S. Kazmi, Qualitative analysis and stationary optical patterns of nonlinear Schrödinger equation including nonlinear chromatic dispersion, Opt. Quant. Electron., 55 (2023), 718. https://doi.org/10.1007/s11082-023-04978-4 doi: 10.1007/s11082-023-04978-4
[7]	S. Duran, B. Karabulut, Nematicons in liquid crystals with Kerr Law by sub-equation method, Alex. Eng. J., 61 (2022), 1695–1700. https://doi.org/10.1016/j.aej.2021.06.077 doi: 10.1016/j.aej.2021.06.077
[8]	A. M. Wazwaz, Painlevé integrability and lump solutions for two extended (3+1)-and (2+1)-dimensional Kadomtsev-Petviashvili equations, Nonlinear Dyn., 111 (2023), 3623–3632. https://doi.org/10.1007/s11071-022-08074-2 doi: 10.1007/s11071-022-08074-2
[9]	R. R. Yuan, Y. Shi, S. L. Zhao, J. X. Zhao, The combined KdV-mKdV equation: bilinear approach and rational solutions with free multi-parameters, Results Phys., 55 (2023), 107188. https://doi.org/10.1016/j.rinp.2023.107188 doi: 10.1016/j.rinp.2023.107188
[10]	S. S. Kazmi, A. Jhangeer, N. Raza, H. I. Alrebdi, A. H. Abdel-Aty, H. Eleuch, The analysis of bifurcation, quasi-periodic and solitons patterns to the new form of the generalized $q$ -deformed Sinh-Gordon equation, Symmetry, 15 (2023), 1324. https://doi.org/10.3390/sym15071324 doi: 10.3390/sym15071324
[11]	G. Akram, I. Zainab, M. Sadaf, A. Bucur, Solitons, one line rogue wave and breather wave solutions of a new extended KP-equation, Results Phys., 55 (2023), 107147. https://doi.org/10.1016/j.rinp.2023.107147 doi: 10.1016/j.rinp.2023.107147
[12]	M. A. Ullah, K. Rehan, Z. Perveen, M. Sadaf, G. Akram, Soliton dynamics of the KdV–mKdV equation using three distinct exact methods in nonlinear phenomena, Nonlinear Eng., 13 (2024), 20220318. https://doi.org/10.1515/nleng-2022-0318 doi: 10.1515/nleng-2022-0318
[13]	V. Jadaun, Soliton solutions of a (3+1)-dimensional nonlinear evolution equation for modeling the dynamics of ocean waves, Phys. Scripta, 96 (2021), 095204. https://doi.org/10.1088/1402-4896/ac0031 doi: 10.1088/1402-4896/ac0031
[14]	O. González-Gaxiola, A. Biswas, L. Moraru, A. A. Alghamdi, Solitons in neurosciences by the Laplace-Adomian decomposition scheme, Mathematics, 11 (2023), 1080. https://doi.org/10.3390/math11051080 doi: 10.3390/math11051080
[15]	B. Li, J. Zhao, W. Liu, Analysis of interaction between two solitons based on computerized symbolic computation, Optik, 206 (2020), 164210. https://doi.org/10.1016/j.ijleo.2020.164210 doi: 10.1016/j.ijleo.2020.164210
[16]	M. Sadaf, S. Arshed, G. Akram, E. Husaain, Dynamical behavior of nonlinear cubic-quartic Fokas-Lenells equation with third and fourth order dispersion in optical pulse propagation, Opt. Quant. Electron., 55 (2023), 1207. https://doi.org/10.1007/s11082-023-05389-1 doi: 10.1007/s11082-023-05389-1
[17]	G. Akram, M. Sadaf, S. Arshed, M. Farrukh, Optical soliton solutions of Manakov model arising in the description of wave propagation through optical fibers, Opt. Quant. Electron., 56 (2024), 906. https://doi.org/10.1007/s11082-024-06735-7 doi: 10.1007/s11082-024-06735-7
[18]	M. Vivas-Cortez, N. Raza, S. S. Kazmi, Y. Chahlaoui, G. A. Basendwah, A novel investigation of dynamical behavior to describe nonlinear wave motion in (3+1)-dimensions, Results Phys., 55 (2023), 107131. https://doi.org/10.1016/j.rinp.2023.107131 doi: 10.1016/j.rinp.2023.107131
[19]	S. Kumar, V. Jadaun, Symmetry analysis and some new exact solutions of Born-Infeld equation, Int. J. Geom. Methods Mod. Phys., 15 (2018), 1850183. https://doi.org/10.1142/S0219887818501839 doi: 10.1142/S0219887818501839
[20]	Y. Li, S. F. Tian, J. J. Yang, Riemann-Hilbert problem and interactions of solitons in the $n$ -component nonlinear Schrödinger equations, Stud. Appl. Math., 148 (2022), 577–605. https://doi.org/10.1111/sapm.12450 doi: 10.1111/sapm.12450
[21]	Z. Q. Li, S. F. Tian, J. J. Yang, E. Fan, Soliton resolution for the complex short pulse equation with weighted Sobolev initial data in space-time solitonic regions, J. Differ. Equations, 329 (2022), 31–88. https://doi.org/10.1016/j.jde.2022.05.003 doi: 10.1016/j.jde.2022.05.003
[22]	Z. Q. Li, S. F. Tian, J. J. Yang, Soliton resolution for the Wadati-Konno-Ichikawa equation with weighted Sobolev initial data, Ann. Henri Poincaré, 23 (2022), 2611–2655. https://doi.org/10.1007/s00023-021-01143-z doi: 10.1007/s00023-021-01143-z
[23]	Z. Q. Li, S. F. Tian, J. J. Yang, On the soliton resolution and the asymptotic stability of $N$ -soliton solution for the Wadati-Konno-Ichikawa equation with finite density initial data in space-time solitonic regions, Adv. Math., 409 (2022), 108639. https://doi.org/10.1016/j.aim.2022.108639 doi: 10.1016/j.aim.2022.108639
[24]	L. Akinyemi, Shallow ocean soliton and localized waves in extended (2+1)-dimensional nonlinear evolution equations, Phys. Lett. A, 463 (2023), 128668. https://doi.org/10.1016/j.physleta.2023.128668 doi: 10.1016/j.physleta.2023.128668
[25]	A. M. Wazwaz, W. Alhejaili, S. A. El-Tantawy, Analytical study on two new (3+1)-dimensional Painlevé integrable equations: Kink, lump, and multiple soliton solutions in fluid mediums, Phys. Fluids, 35 (2023), 093119. https://doi.org/10.1063/5.0169763 doi: 10.1063/5.0169763
[26]	N. Ullah, M. I. Asjad, A. Hussanan, A. Akgül, W. R. Alharbi, H. Algarni, et al., Novel waves structures for two nonlinear partial differential equations arising in the nonlinear optics via Sardar-subequation method, Ale. Eng. J., 71 (2023), 105–113. https://doi.org/10.1016/j.aej.2023.03.023 doi: 10.1016/j.aej.2023.03.023
[27]	D. Lathrop, Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Phys. Today, 68 (2015), 54–55. https://doi.org/10.1063/PT.3.2751 doi: 10.1063/PT.3.2751
[28]	M. B. Riaz, A. Jhangeer, J. Martinovic, S. S. Kazmi, Dynamics and soliton propagation in a modified Oskolkov equation: phase plot insights, Symmetry, 15 (2023), 2171. https://doi.org/10.3390/sym15122171 doi: 10.3390/sym15122171
[29]	M. H. Rafiq, N. Raza, A. Jhangeer, Dynamic study of bifurcation, chaotic behavior and multi-soliton profiles for the system of shallow water wave equations with their stability, Chaos Soliton. Fract., 171 (2023), 113436. https://doi.org/10.1016/j.chaos.2023.113436 doi: 10.1016/j.chaos.2023.113436
[30]	K. Hosseini, E. Hinçal, M. Ilie, Bifurcation analysis, chaotic behaviors, sensitivity analysis, and soliton solutions of a generalized Schrödinger equation, Nonlinear Dyn., 111 (2023), 17455–17462. https://doi.org/10.1007/s11071-023-08759-2 doi: 10.1007/s11071-023-08759-2
[31]	A. M. Talafha, A. Jhangeer, S. S. Kazmi, Dynamical analysis of (4+1)-dimensional Davey Srewartson Kadomtsev Petviashvili equation by employing Lie symmetry approach, Ain Shams Eng. J., 14 (2023), 102537. https://doi.org/10.1016/j.asej.2023.102537 doi: 10.1016/j.asej.2023.102537
[32]	L. Bai, J. Qi, Y. Sun, Further physical study about solution structures for nonlinear $q$ -deformed Sinh-Gordon equation along with bifurcation and chaotic behaviors, Nonlinear Dyn., 111 (2023), 20165–20199. https://doi.org/10.1007/s11071-023-08882-0 doi: 10.1007/s11071-023-08882-0
[33]	L. Yang, M. ur Rahman, M. A. Khan, Complex dynamics, sensitivity analysis and soliton solutions in the (2+1)-dimensional nonlinear Zoomeron model, Results Phys., 56 (2024), 107261. https://doi.org/10.1016/j.rinp.2023.107261 doi: 10.1016/j.rinp.2023.107261

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