Unveiling the Transformative Power: Exploring the Nonlocal Potential Approach in the (3 + 1)-Dimensional Yu–Toda–Sasa–Fukuyama Equation

Abstract

This paper focuses on the investigation of the Yu–Toda–Sasa–Fukuyama (YTSF) equation in its three-dimensional form. Based on the well-known Euler operator, a set of seven non-singular local multipliers is explored. Using these seven non-singular multipliers, the corresponding local conservation laws are derived. Additionally, an auxiliary potential-related system of partial differential equations (PDEs) is constructed. This study delves into nonlocal systems, which reveal numerous intriguing exact solutions of the YTSF equation. The nonlinear systems exhibit stable structures such as kink solitons, representing transitions, and breather or multi-solitons, modeling localized energy packets and complex interactions. These are employed in materials science, optics, communications, and plasma. Additionally, patterns such as parabolic backgrounds with ripples inform designs involving structured or varying media such as waveguides.

1. Introduction

Nonlinear evolution equations (NLEEs) are central to mathematical models of many physical phenomena, ranging from fluid mechanics and optical communications to plasma physics and quantum field theory. One of them is the Yu–Toda–Sasa–Fukuyama (YTSF) equation, which is a typical (3 + 1)-dimensional integrable model with a wide variety of nonlinear dynamics. The Yu–Toda–Sasa–Fukuyama (YTSF) equation is a significant integrable nonlinear partial differential equation that emerged as a spin-off of work in soliton theory and integrable systems. It was originally proposed by Yu, Toda, Sasa, and Fukuyama early in the 1980s [1] as a multi-dimensional generalization of the seminal Kadomtsev–Petviashvili (KP) and Korteweg–de Vries (KdV) equations, i.e., to the (3 + 1)-dimensional setting. The YTSF equation can be applied to model an enormous variety of nonlinear wave phenomena and is a simple model for the explanation of interactions in multi-dimensional physical systems. Its integrability, coupled with its susceptibility to existing analytical techniques, renders it highly appropriate for exploring complex wave dynamics in higher dimensions. Consequently, the YTSF equation remains of significant interest in the area of mathematical physics and applied mathematics.

Various techniques have been employed to discover different families of solutions for nonlinear evolution equations and the YTSF equation in particular. A transformed rational function [2] and the exp-function method [3] were used to construct exact solutions. Another technique used an extended homoclinic test approach and its modified form [4,5]. Moreover, Hirota bilinear [6,7,8,9,10], Bäcklund transformation [11,12,13,14,15], Lie infinitesimals, and group similarity were also employed to construct and detect some other families of solutions for the YTSF equation [16,17,18,19,20]. In this study, the nonlocal potential similarity transformation method, a robust technique with proven effectiveness, was utilized [21,22,23,24,25] in conjunction with Lie infinitesimal and symmetry transformation methods to construct novel and exact solutions to the Yu–Toda–Sasa–Fukuyama (YTSF) equation [1,5,7,20]. Nonlocal similarity methods can lead to novel solutions that are not accessible through classical Lie infinitesimal symmetry analysis from earlier work because they consider broader types of transformations and incorporate dependencies that lie outside the local structure of the original differential equations. While Lie symmetry methods rely on infinitesimal transformations that act locally on the independent and dependent variables (and their derivatives), nonlocal symmetries involve auxiliary variables, integral terms, or potentials that are not directly present in the original formulation. This extension beyond local variables enables nonlocal methods to uncover hidden structures and generate solution families that classical symmetries may overlook. For instance, nonlocal symmetries can be associated with conservation laws, potential systems, or inverse transformations, which significantly enrich the solution space. Therefore, solutions obtained via nonlocal similarity reductions often exhibit broader behavior.

The investigation of the interplay and transmission of plasma and electromagnetic waves holds great importance in understanding these phenomena. The Yu–Toda–Sasa–Fukuyama (YTSF) equation has emerged as a widely used theoretical framework for analyzing soliton dynamics and nonlinear wave phenomena in various fields, such as plasma physics, fluid dynamics, and weakly dispersive media. The YTSF equation was initially introduced by Yu et al. [1] as a pioneering contribution and is presented in the following form:

$${\left(w_{xxz}-4w_t+4w{w}_z+2w_x{\partial }_x^{-1}{w}_z\right)}_x+3w_{yy}=0.$$

(1)

where ${\partial }_x^{-1}f=\int fdx,$ setting $w=u_x,$ then integrating to x considering the zero constant of integration.

Theis confers the potential form of the YTSF equation:

$$-4 u_{x t}+u_{xxxz}+4 u_x u_{x z}+2 u_{x x} u_z+3u_{yy}=0.$$

(2)

This study utilizes potential symmetry analysis to obtain novel exact solutions, which are nonlocal in nature, for the nonlinear (3 + 1)-dimensional YTSF equation. The structure of this paper is as follows. Section 2 presents a succinct discussion of the symmetry analysis approach. Section 3 is dedicated to the determination of conservation multipliers and the identification of local conservation laws for the equation through the direct method. Section 4 focuses on the construction of auxiliary potential systems for each conservation law. Subsequently, the Lie point symmetries of these auxiliary systems are computed in order to determine the nonlocal symmetries and nonlocal solutions of the YTSF equation. The conclusions of this paper are summarized in Section 5.

2. Nonlocal Potential Transformation Method Overview

Bluman and Kumei [26] developed a methodology for detecting a nonlocal associated potential system when at least one equation within a system of partial differential equations (PDEs) is formulated in a conserved form. The equation in its conserved form provides a logical starting point for identifying auxiliary potential variables and establishing an auxiliary system of PDEs, commonly referred to as the potential system [27,28]. Nonlocal-associated systems are of significant importance in generating solutions for the given system of PDEs. These PDEs arise through symmetry reductions that can be attributed to nonlocal symmetries [29,30,31], as opposed to invariant solutions obtained from point symmetries. The methods utilized in this study can be concisely summarized as follows:

• Conservation laws are constructed using families of multipliers obtained from determining equations derived via Euler differential operators;
• Subsequently, auxiliary potential systems are formulated to correspond to the identified conservation laws;
• Finally, an examination is conducted on the nonlocal symmetries, followed by the identification and visualization of nonlocal solutions that arise from these symmetries.

This process is illustrated in the flowchart shown in Figure 1.

3. The Conservative Forms of the (3 + 1)-Dimensional YTSF Equation

In this section, the conservation laws are presented to identify the conserved forms of the YTSF equation under consideration.

3.1. YTSF Conservation Laws

The conservation laws relate to the preservation of physical quantities, such as mass, energy, electrical charges, momentum, angular momentum, and other motion-related qualities, within the realm of physics and engineering applications. For any PDE, the local conservation law can be considered in the form of a divergence expression:

$${\sum }_{i=1}^n{D}_i{f}^i(x,u_{\left(r\right)})=0.$$

(3)

where $f^i\left(x,u_{\left(r\right)}\right)$ are the fluxes, $u_{\left(r\right)}$ is the dependent variable of the highest-order derivative with n independent variables $x=(x_1,x_2,x_3,\dots ,x_n)$, and D_i can be written as:

$$D_i={\displaystyle \frac{\partial }{\partial x^i}}+u_i^j{\displaystyle \frac{\partial }{\partial u^j}}+u_{i{i}_1}^j{\displaystyle \frac{\partial }{\partial u_{i_1}^j}}+\dots +u_{i{i}_{1\dots .n}}^j{\displaystyle \frac{\partial }{\partial u_{i_{1\dots .n}}^j}}, \mathrm{where} u_i^j={\displaystyle \frac{\partial u^j}{\partial x_i}}.$$

(4)

The conservation forms are found by multiplying the PDE by $\Lambda \left(x;U_{\left(l\right)}\right)$

$$\Lambda \left(x;U_{\left(l\right)}\right)\mathrm{R}\left(x;U_{\left(k\right)}\right)\equiv {\sum }_{i=1}^n{D}_i{f}^i\left(x,u_{\left(r\right)}\right).$$

(5)

The multiplier, $\Lambda \left(x;U_{\left(l\right)}\right)$, is non-singular and U is an arbitrary solution of $\mathrm{R}\left(x;U_{\left(k\right)}\right)$, l represents the dependent variable differential order and $\sum _{i=1}^n{D}_i{f}^i(x,u_{\left(r\right)})$ is a summation of fluxes $f$ⁱ for the $i^{th}$ independent variable x.

Theorem 1.

A set of all non-singular local conservation multipliers for the PDE $R\left(x;U_{\left(k\right)}\right)$ yields a local conservation law if:

$$E_{U^j}\left(\Lambda \left(x;U_{\left(l\right)}\right)\mathrm{R}\left(x;U_{\left(k\right)}\right)\right)\equiv 0, j=1,\dots ,m.$$

(6)

where $E_{U^j}$ is the Euler operator defined by

$$E_{U^j}={\displaystyle \frac{\partial }{\partial U^j}}-D_i{\displaystyle \frac{\partial }{\partial U^j}}+\cdots +{\left(-\right)}^s{D}_{i_1}\dots D_{i_s}{\displaystyle \frac{\partial }{\partial U_{i_1\dots i_s}^j}}$$

(7)

Solving the determining system, (6), for an arbitrary function $U\left(x\right)$ leads to a group of multipliers. Hence, the corresponding fluxes $f^i(x,u_r)$ of these multipliers are derived.

3.2. Derivation of the (3 + 1) YTSF Conservation Multipliers

The direct method is a systematic method used to construct local conservation laws, as proposed [32,33,34]. According to this method, one seeks a set of local multipliers of a given PDE system, which depend on independent and dependent variables and derivatives of dependent variables up to some fixed order.

As indicated in Section 2, the local multipliers ${\Lambda }_j$ are determined by employing a direct method that relies on the Euler operation, as described below:

$$\begin{gathered} E_u\left({\Lambda }_j\left(t, x, y, z, u, u_x, u_y,u_z, u_t, u_{x x},u_{x y},u_{x z},u_{x t},u_{y y},u_{y z},u_{y t},u_{zz},u_{zt}, \\ {u}_{t t}\right) \left(-4 u_{x t}+u_{xxxz}+4 u_x u_{x z}+2 u_{x x} u_z+3u_{yy}\right)\right)=0. \end{gathered}$$

(8)

By utilizing Maple v2021, the symbolic computation software, Equation (8) can be solved, resulting in a collection of determining equations. The solution to these equations results in a family of second-order conservation multipliers:

$$\begin{array}{c}{\Lambda }_j\left(t, x, y, z, u, u_x, u_y,u_z, u_t, u_{x x},u_{x y}, u_{x z},u_{x t},u_{y y},u_{y z},u_{y t},u_{zz},u_{zt},u_{t t}\right)\hfill \\ ={\displaystyle \frac{\left(\left(6 {F_1}_t\left(t\right) y+6 F_2\left(t\right)\right) u_x\right)}{6}}+{\displaystyle \frac{\left(\left(9 F_1\left(t\right)+6\right) u_y\right)}{6}}+2 F_{1_{t,t} }\left(t\right) y z+{\displaystyle \frac{2 {F3}_{t, t} y^2}{3}}+2 {F_2}_t\left(t\right) z\hfill \\ +{F_3}_t\left(t\right)x+u_z F_3\left(t\right)+F_4\left(t\right) y+F_5(t)\hfill \end{array}$$

(9)

where $F_1, F_2,$…, $F_5$ are arbitrary functions. Consequently, the following local multipliers are obtained:

$${\Lambda }_1=1, {\Lambda }_2=t, {\Lambda }_3=y, {\Lambda }_4=u_t,{\Lambda }_5=u_x, {\Lambda }_6=u_y, { \Lambda }_7=u_z .$$

(10)

In the next step, the conservation laws corresponding to these seven multipliers are constructed.

3.3. Derivation of Conservation Laws

The process of obtaining local conservation law fluxes involves the inversion of the divergence differential operator. Figure 2 illustrates a flowchart of this process.

• For the first multiplier ${\Lambda }_1=1$, using the direct method, the (3 + 1)-dimensional YTSF equation can be written in a conserved form:

$${-\left({4u}_x\right)}_t+{\left(2 u_x u_z\right)}_x+{\left(3 u_y \right)}_y+{\left(u_x^2+u_{xxx}\right)}_z=0$$

(11)

• For multiplier ${\Lambda }_2=t$, Equation (1) could be written in the following form:

$$\begin{gathered} t \left(-4 u_{x t}+u_{xxxz}+4 u_x u_{x z}+2 u_{x x} u_z+3u_{yy}\right)=D_t{f}^1+D_x{f}^2+D_y{f}^3+D_z{f}^4. \\ ={-\left({4 t u}_x\right)}_t+{\left(2 {t u}_x u_z+4u+t u_{xxz}\right)}_x+{\left(3t{ u}_y\right)}_y+{\left(t{ u}_x^2\right)}_z \end{gathered}$$

(12)

In the same way, the corresponding conservation laws of multipliers ${\Lambda }_j, j=1...7$ can be obtained. The results are tabulated hereafter in

Table 1. The multipliers and the conservation laws.

Multiplier	Conservation Law
${\mathbf{\Lambda }}_1=1$	${-\left({4u}_x\right)}_t+{\left(2 u_x u_z\right)}_x+{\left(3 u_y \right)}_y+{\left(u_x^2+u_{xxx}\right)}_z=0.$
${\mathbf{\Lambda }}_2=\mathit{t}$	${-\left({4 t u}_x\right)}_t+{\left(2 {t u}_x u_z+4u+t u_{xxz}\right)}_x+{\left(3t{ u}_y\right)}_y+{\left(t{ u}_x^2\right)}_z=0.$
${\mathbf{\Lambda }}_3=\mathit{y}$	${-\left({4 y u}_x\right)}_t+{\left(2 {y u}_x u_z+4u+y u_{xxz}\right)}_x+{\left(3y{ u}_y-3 u\right)}_y+{\left(y{ u}_x^2\right)}_z=0.$
${\mathbf{\Lambda }}_4={\mathit{u}}_{\mathit{t}}$	$$\begin{gathered} { \left(-{\displaystyle \frac{3}{2}} {{(u}_y)}^2-{{(u}_{x })}^2{u}_z+{\displaystyle \frac{1}{2}} u_{x x}{ u}_{x z} \right)}_t+{\left(-{\displaystyle \frac{1}{2}} u_{x t}{ u}_{x z}-2 {{(u}_t)}^2+u_t{ u}_{x x z}+2 u_t{u}_x u_y \right)}_x+{\left(3 u_y u_t \right)}_y \\ -{\left(- u_t {{(u}_{ x})}^2+{\displaystyle \frac{1}{2}} u_{x x}{ u}_{x t} \right)}_z=0. \end{gathered}$$
${\mathbf{\Lambda }}_5={\mathit{u}}_{\mathit{x}}$	${-\left(2 {{(u}_x)}^2\right)}_t+{\left(-{\displaystyle \frac{3}{2}} {{(u}_y)}^2+{{(u}_x)}^2 u_z+u_x{ u}_{x x z} \right)}_x+{\left(3 u_y u_x \right)}_y-{\left( {\displaystyle \frac{1}{2}} {{(u}_{x x})}^2-{{(u}_x)}^3\right)}_z=0.$
${\mathbf{\Lambda }}_6={\mathit{u}}_{\mathit{y}}$	$$\begin{gathered} {-\left(2 u_x u_y \right)}_t+{\left( 2 u_x u_y u_z-2 u_t u_y+u_y{ u}_{x x z}- {\displaystyle \frac{1}{2}} { u}_{x z} { u}_{x y} \right)}_x \\ +{\left(-{{(u}_x)}^2{u}_z+{\displaystyle \frac{3}{2}} {{(u}_y)}^2+2 u_t u_x+{\displaystyle \frac{1}{2}} u_{x x} u_{x z} \right)}_y-{\left( {\displaystyle \frac{1}{2}} u_{x x} u_{x y}-{{(u}_x)}^2{u}_y\right)}_z=0. \end{gathered}$$
${\mathbf{\Lambda }}_7={\mathit{u}}_{\mathit{z}}$	$$\begin{gathered} {-\left(2 u_x u_z \right)}_t+{\left(-2 u_t u_z+u_z{ u}_{x x z}- {\displaystyle \frac{1}{2}} {\left({ u}_{x z}\right)}^2+2 {{(u}_z)}^2{u}_x \right)}_x+{\left( 2 u_y u_z \right)}_y-{\left(-2 u_{ x} u_t+{\displaystyle \frac{3}{2}} {{(u}_y)}^2\right)}_z \\ =0. \end{gathered}$$

.

4. Potential Systems

Consider a scalar (3 + 1)-dimensional PDE of order $k$ with the independent variables $(t, x, y, z)$ and a single dependent variable $u$, which can be expressed in a conservative form:

$${\sum }_{i=1}^4{D}_i{f}^i\left(t, x, y, z,u_{\left(k-1\right)}\right)=0.$$

(13)

where $f^i\left(t, x, y, z,u_{\left(k-1\right)}\right)$ are the fluxes. This conserved form allows for introducing three new variables $v=\left(v^{\left(1\right)},v^{\left(2\right)},v^{\left(3\right)}\right)$. These variables are used to build up the auxiliary potential system:

$$\psi \left(t,x,y,z;u,v^{\left(1\right)}, v^{\left(2\right)}, v^{\left(3\right)}\right)=\left\{\begin{array}{c}{f}^1=v_x^{\left(1\right)},\\ f^2=-\left(v_y^{\left(2\right)}+v_t^{\left(1\right)}\right),\\ f^3=v_z^{\left(3\right)}+v_x^{\left(2\right)},\\ f^4={-v}_y^{\left(3\right)}.\end{array}\right.$$

(14)

4.1. The Auxiliary Potential Systems of the (3 + 1)-Dimensional YTSF Equation

Applying Equations (13) and (14) to the conserved form of the YTSF equation, Equation (11), results in:

$$D_t\left(-{4u}_x\right)+D_x\left(2 u_x u_z\right)+D_y\left(3 u_y\right)+D_z\left(u_x^2+u_{xxx}\right)=D_t\left(f^1\right)+D_x\left(f^2\right)+D_y\left(f^3\right)+D_z\left(f^4\right)$$

(15)

This conserved form allows the potential variables $v=\left(v^{\left(1\right)},v^{\left(2\right)},v^{\left(3\right)}\right)$ to be introduced, and the auxiliary potential system corresponding to the (3 + 1)-dimensional YTSF equation can be described as follows:

$$\psi \left(t,x,y,z;u,v^{\left(1\right)}, v^{\left(2\right)}, v^{\left(3\right)}\right)=\left\{\begin{array}{c}-{4u}_x=v_x^{\left(1\right)},\\ 2 u_x u_z=-\left(v_y^{\left(2\right)}+v_t^{\left(1\right)}\right),\\ 3 u_y=v_z^{\left(3\right)}+v_x^{\left(2\right)},\\ u_x^2+u_{xxx}={-v}_y^{\left(3\right)}.\end{array}\right.$$

(16)

For each conservation law presented in Table 1, the corresponding potential system is derived and systematically tabulated in

Table 2. Auxiliary potential system.

Multiplier	Auxiliary Potential System
${\mathbf{\Lambda }}_1=1$	${\psi }^1\left(t,x,y,z;u,v^{\left(1\right)}, v^{\left(2\right)}, v^{\left(3\right)}\right)=\left\{\begin{array}{c}-{4u}_x=v_x^{\left(1\right)},\\ 2 u_x u_z=-\left(v_y^{\left(2\right)}+v_t^{\left(1\right)}\right),\\ 3 u_y=v_z^{\left(3\right)}+v_x^{\left(2\right)},\\ u_x^2+u_{xxx}={-v}_y^{\left(3\right)}.\end{array}\right.$
${\mathbf{\Lambda }}_2=\mathit{t}$	${\psi }^2\left(t,x,y,z;u,v^{\left(4\right)}, v^{\left(5\right)}, v^{\left(6\right)}\right)=\left\{\begin{array}{c}-{4 t u}_x=v_x^{\left(4\right)},\\ 2 {t u}_x u_z+4u+t u_{xxz}=-\left(v_y^{\left(5\right)}+v_t^{\left(4\right)}\right),\\ 3t{ u}_y=v_z^{\left(6\right)}+v_x^{\left(5\right)},\\ t{ u}_x^2={-v}_y^{\left(6\right)}.\end{array}\right.$
${\mathbf{\Lambda }}_3=\mathit{y}$	${\psi }^3\left(t,x,y,z;u,v^{\left(7\right)}, v^{\left(8\right)}, v^{\left(9\right)}\right)=\left\{\begin{array}{c}-{4 y u}_x=v_x^{\left(7\right)},\\ 2 {y u}_x u_z+4u+y u_{xxz}=-\left(v_y^{\left(8\right)}+v_t^{\left(7\right)}\right),\\ 3y{ u}_y-3 u=v_z^{\left(9\right)}+v_x^{\left(8\right)},\\ y{ u}_x^2=-v_y^{\left(9\right)}.\end{array}\right.$
${\mathbf{\Lambda }}_4={\mathit{u}}_{\mathit{t}}$	${\psi }^4\left(t,x,y,z;u,v^{\left(10\right)}, v^{\left(11\right)}, v^{\left(12\right)}\right)=\left\{\begin{array}{c}-{\displaystyle \frac{3}{2}} {{(u}_y)}^2-{{(u}_{x })}^2{u}_z+{\displaystyle \frac{1}{2}} u_{x x}{ u}_{x z}=v_x^{\left(10\right)},\\ -{\displaystyle \frac{1}{2}} u_{x t}{ u}_{x z}-2 {{(u}_t)}^2+u_t{ u}_{x x z}+2 u_t{u}_x u_y=-\left(v_y^{\left(11\right)}+v_t^{\left(10\right)}\right),\\ 3 u_y u_t=v_z^{\left(12\right)}+v_x^{\left(11\right)},\\ - u_t {{(u}_{ x})}^2+{\displaystyle \frac{1}{2}} u_{x x}{ u}_{x t}=v_y^{\left(12\right)}.\end{array}\right.$
${\mathbf{\Lambda }}_5={\mathit{u}}_{\mathit{x}}$	${\psi }^5\left(t,x,y,z;u,v^{\left(13\right)}, v^{\left(14\right)}, v^{\left(15\right)}\right)=\left\{\begin{array}{c}-2 {{(u}_x)}^2=v_x^{\left(13\right)},\\ -{\displaystyle \frac{3}{2}} {{(u}_y)}^2+{{(u}_x)}^2 u_z+u_x{ u}_{x x z}=-\left(v_y^{\left(14\right)}+v_t^{\left(13\right)}\right),\\ 3 u_y u_x=v_z^{\left(15\right)}+v_x^{\left(14\right)},\\ {\displaystyle \frac{1}{2}} {{(u}_{x x})}^2-{{(u}_x)}^3=v_y^{\left(15\right)}.\end{array}\right.$
${\mathbf{\Lambda }}_6={\mathit{u}}_{\mathit{y}}$	${\psi }^6\left(t,x,y,z;u,v^{\left(16\right)}, v^{\left(17\right)}, v^{\left(18\right)}\right)=\left\{\begin{array}{c}-2 u_x u_y=v_x^{\left(16\right)},\\ 2 u_x u_y u_z-2 u_t u_y+u_y{ u}_{x x z}-{\displaystyle \frac{1}{2}} { u}_{x z} { u}_{x y}=-\left(v_y^{\left(17\right)}+v_t^{\left(16\right)}\right),\\ -{{(u}_x)}^2{u}_z+{\displaystyle \frac{3}{2}} {{(u}_y)}^2+2 u_t u_x+{\displaystyle \frac{1}{2}} u_{x x} u_{x z}=v_z^{\left(18\right)}+v_x^{\left(17\right)},\\ {\displaystyle \frac{1}{2}} u_{x x} u_{x y}-{{(u}_x)}^2{u}_y=v_y^{\left(18\right)}.\end{array}\right.$
${\mathbf{\Lambda }}_7={\mathit{u}}_{\mathit{z}}$	${\psi }^7\left(t,x,y,z;u,v^{\left(19\right)}, v^{\left(20\right)}, v^{\left(21\right)}\right)=\left\{\begin{array}{c}-2 u_x u_z=v_x^{\left(19\right)},\\ -2 u_t u_z+u_z{ u}_{x x z}- {\displaystyle \frac{1}{2}} {\left({ u}_{x z}\right)}^2+2 {{(u}_z)}^2{u}_x=-\left(v_y^{\left(20\right)}+v_t^{\left(19\right)}\right),\\ 2 u_y u_z=v_z^{\left(21\right)}+v_x^{\left(20\right)},\\ -2 u_{ x} u_t+{\displaystyle \frac{3}{2}} {{(u}_y)}^2=v_y^{\left(21\right)}.\end{array}\right.$

where, $\left(v^{\left(1\right)},v^{\left(2\right)},...,v^{\left(21\right)}\right)$ represent the variables of potentials.

.

4.2. Investigation of the Lie Infinitesimals of the Auxiliary Potential Systems

This subsection focuses on the solutions of auxiliary potential systems through the utilization of both nonlocal and point symmetries. The subsequent discussion focuses on the invariant solutions associated with these potential systems.

4.2.1. Point Symmetries of the (3 + 1)-Dimensional YTSF Equation

The point symmetries of the (3 + 1)-dimensional YTSF Equation (1) and its corresponding auxiliary systems, ${\psi }^i, (i=1, 2,\dots ,7)$, were determined using Maple software. The results obtained are summarized in

Table 3. The point symmetries of the YTSF equation and its auxiliary systems.

Equation	Point Symmetries
YTSF Equation (1)	${\xi }_x={\displaystyle \frac{2 {F_1}_t\left(t\right) y}{3}}+F_3\left(t\right)+{\displaystyle \frac{\left(6 C_1-3 C_2\right)x}{3}},$ ${\xi }_y=C_1 y+F_1\left(t\right),$ ${\xi }_z=\left(-4 C_1+3 C_2\right) z+F_2\left(t\right),$ ${\xi }_t=C_2 t+C_3,$ ${\eta }_u=-{\displaystyle \frac{4zy{F_1}_{t, t}\left(t\right)}{3}}-{\displaystyle \frac{2y^2{F_1}_{t, t}\left(t\right)}{3}}-x{F_2}_t\left(t\right)-2z{F_3}_t\left(t\right)+y{F}_4\left(t\right)+F_5\left(t\right)+{\displaystyle \frac{\left(-6 C_1+3 C_2\right) u}{3}}.$
${\mathit{\psi }}^1$	${\xi }_x=\left(-C_1+2 C_3\right)x+F_1\left(t\right),$ ${ \xi }_y=C_3 y+C_4,$ ${ \xi }_z=\left(3 C_1-4 C_3\right) z+C_5,$ ${\xi }_t=C_1 t+C_2,$ ${\eta }_u=\left(C_1-2 C_3\right)u-2 {z{F}_1}_t\left(t\right)+y{F}_2\left(t\right)+F_3\left(t\right),$ ${\eta }_{v1}=\left(C_1-2 C_3\right) v_1+{F_4}_y\left(y, z, t\right),$ ${\eta }_{v2}=-C_3 v_2-{F_4}_t\left(y, z, t\right)+{F_5}_z\left(x, z, t\right),$ ${\eta }_{v3}=\left(4C_1-7C_3\right) v_3-{F_5}_x\left(x, z, t\right)+3z{F}_2\left(t\right)+F_6\left(x, t\right).$
${\mathit{\psi }}^2$	${\xi }_x=F_1\left(t\right)+{\displaystyle \frac{\left(C_1-C_2\right) x}{2}},$ ${ \xi }_y={\displaystyle \frac{\left(3 C_1-C_2\right) y}{4}}+C_4 ,$ ${ \xi }_z=C_2 z+C_3,$ ${\xi }_t=C_1 t,$ ${\eta }_u=-2{z{F}_1}_t(t)+y{F}_2(t)+F_3(t)+{\displaystyle \frac{\left(-C_1+C_2\right)u}{2}},$ ${\eta }_{v1}={F_5}_y\left(y, z, t\right)+{\displaystyle \frac{\left(\left( C_1+C_2\right) v_1\right)}{2}},$ ${\eta }_{v2}=-{F_4}_z\left(x, z, t\right)-{F_5}_t\left(y, z, t\right)+8yz{F_1}_t\left(t\right)+F_6\left(z, t\right)+{\displaystyle \frac{\left(\left(12 t x-8 y^2\right) F_2\left(t\right)\right)}{4}}-4 y{F}_3\left(t\right)+{\displaystyle \frac{\left(\left(C_1+C_2 \right)v_2 \right)}{4}},$ ${\eta }_{v3}={F_4}_x\left(x, z, t\right)+{\displaystyle \frac{\left(\left(-C_1+7 C_2\right) v_3 \right)}{4}}.$
${\mathit{\psi }}^3$	${\xi }_x=F_1\left(t\right)+{\displaystyle \frac{x}{2}}\left( C_1- C_3\right),$ ${\xi }_y={\displaystyle \frac{1}{4}}y \left(-C_3+3C_1\right),$ ${\xi }_z=C_3 z+C_4,$ ${\xi }_t=C_1 t+C_2,$ ${\eta }_u=-2{z{F}_1}_t\left(t\right)+y{F}_2\left(t\right)+F_3\left(t\right)+{\displaystyle \frac{1}{2}}\left(-C_1+C_3\right)u,$ ${\eta }_{v1}={F_4}_y\left(y, z, t\right)+{\displaystyle \frac{1}{4}}\left(C_1+C_3\right) v_1,$ ${\eta }_{v2}=6xz{F_1}_t\left(t\right) -3x{F}_3\left(t\right)+F_6(z, t)-{F_5}_z(x, z, t)-{ F_4}_t(y, z, t),$ ${\eta }_{v3}={F_5}_x\left(x, z, t\right)+{\displaystyle \frac{1}{2}}\left(-C_1+3 C_3\right) v_3.$
${\mathit{\psi }}^4$	${\xi }_x={\displaystyle \frac{1}{5}}{C}_1 x+C_5,$ ${\xi }_y={\displaystyle \frac{3}{5}}{C}_1 y+C_3,$ ${\xi }_z={\displaystyle \frac{3}{5}} C_1 z+C_4,$ ${\xi }_t=C_1 t+C_2,$ ${\eta }_u=-{\displaystyle \frac{1}{5}}{C}_1 u+C_6,$ ${\eta }_{v1}=-{\displaystyle \frac{7}{5}}{C}_1 v_1+{F_2}_y\left(y, z, t\right),$ ${\eta }_{v2}=-{\displaystyle \frac{9}{5}}{C}_1 v_2-{F_1}_z\left(x, z, t\right)-{F_2}_t\left(y, z, t\right)+F_3\left(z, t\right),$ ${\eta }_{v3}=-{\displaystyle \frac{7}{5}}{C}_1 v_3+{F_1}_x(x, z, t).$
${\mathit{\psi }}^5$	${\xi }_x=F_1\left(t\right)+{\displaystyle \frac{\left(\left(C_3-C_1\right) x\right)}{2}},$ ${\xi }_y={\displaystyle \frac{\left(-C_1+3 C_3\right) y}{4}}+C_5,$ ${\xi }_z=C_1 z+C_2,$ ${\xi }_t=C_3 t+ C_4,$ ${\eta }_u=-2{ F_1}_t\left(t\right) z+F_2\left(t\right) y+F_3\left(t\right)+{\displaystyle \frac{\left(-C_3+C_1\right)u }{2}},$ ${\eta }_{v1}={ F_4}_y\left(y, z, t\right)+{\displaystyle \frac{\left(\left(-3C_3+3C_1\right) v_1 \right)}{2}},$ ${\eta }_{v2}=-{ F_4}_t\left(y, z, t\right)-{F_5}_z\left(x, z, t\right)+F_6\left(z, t\right)+3F_2\left(t\right) u+{\displaystyle \frac{ \left(-7C_3+5C_1\right)v_2}{4}},$ ${\eta }_{v3}={F_5}_x\left(x, z, t\right)+{\displaystyle \frac{\left(\left(11C_1-9 C_3\right)v_3 \right)}{4}}.$
${\mathit{\psi }}^6$	${\xi }_x=C_3 x+F_1\left(t\right),$ ${\xi }_y={\displaystyle \frac{\left(\left(C_3+C_1\right) y \right)}{2}}+C_5,$ ${\xi }_z=\left(C_1-2C_3\right) z+C_4,$ ${\xi }_t=C_1 t+C_2,$ ${\eta }_u=-C_{3}u-2z{F_1}_t\left(t\right)+F_2\left(t\right),$ ${\eta }_{v1}={ F_4}_y\left(y, z, t\right)+{\displaystyle \frac{\left(\left(-5 C_3-C_1\right)v_1\right)}{2}},$ ${\eta }_{v2}=-{F_3}_z\left(x, z, t\right)-{ F_4}_t\left(y, z, t\right)-4zu{F_1}_{t,t}\left(t\right)+2u{F_2}_t\left(t\right)+F_5\left(z, t\right)-\left(C_1+2 C_3\right)v_2,$ ${\eta }_{v3}=-5C_3{v}_3+{F_3}_x(x, z, t).$
${\mathit{\psi }}^7$	${\xi }_x=\left(-C_1+2 C_3\right) x+C_6,$ ${\xi }_y=C_{3}y+C_4,$ ${\xi }_z=\left(-4 C_3+3C_1\right)z+C_5,$ ${\xi }_t=C_{1}t+C_2,$ ${\eta }_u=\left(C_1-2C_3\right)u+C_7,$ ${\eta }_{v1}=-C_{1}v1+{F_1}_y\left(y, z, t\right),$ ${\eta }_{v2}=\left(-2C_1+C_3\right)v_2-{F_2}_z\left(x, z, t\right)-{F_1}_t\left(y, z, t\right)+F_3\left(z, t\right),$ ${\eta }_{v3}=\left(-5C_3+2 C_1\right)v_3+{F_2}_x(x, z, t).$

.

4.2.2. Invariant Solutions of the Potential System ${\mathsf{\psi }}^1$

To obtain some exact solutions for YTSF equation, the auxiliary system of ${\psi }^1$ described in Table 3 is solved. For this case, one of the obtained solutions is discussed in detail, while the remaining solutions are presented concisely.

According to appropriate choice of the arbitrary constants in Table 3, the vector, $S={\displaystyle \frac{\partial }{\partial x}}+{\displaystyle \frac{\partial }{\partial t}}$, can be obtained and employed to reduce the number of independent variables $(t,x,y,z)$ to three independent variables ($p, q, r$). The auxiliary system ${\psi }^1$ is then reduced to:

$$-2 {G_1}_s{G_1}_r=-{G_3}_p-{G_2}_s, 3 {G_1}_p={G_4}_r-{G_3}_s, 4 {G_1}_s=- {G_2}_s,{ {G_1}_s}^2- {G_1}_{sss}=-{G_4}_p$$

(17)

where:

$$\begin{gathered} p=y, r=z, s=-x+t, G_1\left(p, r, s\right)=u\left(t,x, y, z\right), G_2\left(p, r, s\right)=v_1\left(t,x, y, z\right), \\ {G}_3(p, r, s)=v_2(t,x, y, z), G_4(p, r, s)=v_3(t,x, y, z) \end{gathered}$$

(18)

The reduced system (17) has a Lie space of infinite dimensions. One of the similarity vectors is the vector $X={\displaystyle \frac{p}{2}} {\displaystyle \frac{\partial }{\partial p}}+r {\displaystyle \frac{\partial }{\partial r}}+2r {\displaystyle \frac{\partial }{\partial G_1}}-{\displaystyle \frac{G_3}{2}} {\displaystyle \frac{\partial }{\partial G_3}}+{\displaystyle \frac{G_4}{2}} {\displaystyle \frac{\partial }{\partial G_4}},$ which could reduce the number of independent variables by one using the following equations:

$$\begin{gathered} -{\displaystyle \frac{2 {F_1}_{t_2} {F_1}_{t_1} t_1-2 p_1}{p1}}={\displaystyle \frac{2 {F_3}_{t_1} t_1+t_1{F}_3-{F_2}_{t_2} p_1}{p1}},-6{F_1}_{ t1}\sqrt{{\displaystyle \frac{p_1}{t_1}}} {\displaystyle \frac{t_1^2}{p_1}} ={\displaystyle \frac{{ F_4}_{t_1}-{ F_3}_{t_2}}{\sqrt{\frac{p_1}{t_1}}}}, \\ 4 {F_1}_{t_2}=-{F_2}_{t_2}, {F_1}_{t_2}^2-{F_1}_{t_2{t}_2{t}_2 }=2t_1{F_4}_{ t_1}-F_4 \end{gathered}$$

(19)

where:

$$\begin{gathered} -{\displaystyle \frac{2 {F_1}_{t_2} {F_1}_{t_1} t_1-2 p_1}{p1}}={\displaystyle \frac{2 {F_3}_{t_1} t_1+t_1{F}_3-{F_2}_{t_2} p_1}{p1}},-6{F_1}_{ t1}\sqrt{{\displaystyle \frac{p_1}{t_1}}} {\displaystyle \frac{t_1^2}{p_1}} ={\displaystyle \frac{{ F_4}_{t_1}-{ F_3}_{t_2}}{\sqrt{\frac{p_1}{t_1}}}}, \\ 4 {F_1}_{t_2}=-{F_2}_{t_2}, {F_1}_{t_2}^2-{F_1}_{t_2{t}_2{t}_2 }=2t_1{F_4}_{ t_1}-F_4 \end{gathered}$$

(20)

The reduced system (19) has a Lie space of infinite dimensions. One of the similarity vectors is the vector $V={\displaystyle \frac{\partial }{\partial t_2}}+\left(4F_1+F_2\right) {\displaystyle \frac{\partial }{\partial F_2}}$, which could be used to create a new equation with a lower number of independent variables:

$${\displaystyle \frac{2 {H_1}_{\eta }{\eta }^2+\eta H_1-H_4 e^{p_1}{p}_1}{p_1}}=0, H_4 e^{p_1}=0, 2 {H_2}_{\eta }\eta -H_2=0, -6{H_3}_{\eta }\sqrt{{\displaystyle \frac{p_1}{\eta }}}{\displaystyle \frac{{\eta }^2}{p_1}}={\displaystyle \frac{{H_2}_{\eta }}{\sqrt{\frac{p_1}{\eta }}}}$$

(21)

where:

$$\begin{gathered} p_1=t_2, \eta =t_1, H_1\left(\eta \right)=F_3\left(t_1, t_2\right), H_2\left(\eta \right)=F_4\left(t_1, t_2\right), H_3\left(\eta \right)=F_1\left(t_1, t_2\right), \\ {H}_4\left(\eta \right)={\displaystyle \frac{4F_1\left(t_1, t_2\right)+F_2\left(t_1, t_2 \right)}{e^{t_2}}} \end{gathered}$$

(22)

Solving (21) leads to:

$$H_1\left(\eta \right)={\displaystyle \frac{C_1}{\sqrt{\eta }}}, H_2\left(\eta \right)=C_3\sqrt{\eta }, H_3\left(\eta \right)={\displaystyle \frac{C_3}{6\sqrt{\eta }}}+C_2, H_4(\eta )=0.$$

(23)

Using back-substitution, the solution of the YTSF equation (Equation (1)) could be obtained in the form:

$$u_1\left(t, x, y, z\right)={\displaystyle \frac{C_1 y}{6 \sqrt{z}}}-2z+C_2$$

(24)

Following the same procedures, the following solutions are obtained for the YTSF equation:

$$u_2=C_1,$$

(25)

$$u_3=F_1\left(t\right),$$

(26)

$$u_4=F_1\left(z\right),$$

(27)

$$u_5=F_1\left(x\right)$$

(28)

$$u_6={\displaystyle \frac{C_1}{\sqrt{\frac{t}{z}}}}$$

(29)

$$u_7={\displaystyle \frac{\left(\left(7C_3 y^{\frac{4}{3}} z^{\frac{1}{3}}{\left(\frac{t}{y^{\frac{4}{3}}{z}^{\frac{1}{3}}}\right)}^{\frac{1}{4}}+12C_{2}t\right)z^{\frac{1}{3}} \right)}{12 y^{\frac{2}{3}} \sqrt{\frac{t}{y^{\frac{4}{3}}{z}^{\frac{1}{3}}} } t}}$$

(30)

$$u_8={\displaystyle \frac{ C_4\sqrt{3}+C_3\sqrt{\frac{3 t x-y^2}{y^2}}}{\sqrt{\frac{\left(3tx-y^2\right)}{y^2}} x }}$$

(31)

$$u_9=-2 C_2 tanh\left({\displaystyle \frac{\left(\left(4 t-4 x+4 y\right)C_2^2+4C_1{C}_2+7 z\right)}{4 C_2}}\right)+C_3$$

(32)

$$u_{10}=-2C_3 tanh\left(-{\displaystyle \frac{\left(2C_{3}y\sqrt{ 3 C_2 C_3-3} \right)}{3}}+{\displaystyle \frac{{3C}_3\left(t-x\right)}{3}}+C_{2}z+C_1\right)+C_4$$

(33)

$$u_{11}=2C_2 tanh\left( C_2 x-{\displaystyle \frac{2\sqrt{3}{C}_2 y \sqrt{-C_2 C_3} }{3}}+C_{3}z+C_1\right)+C_4,$$

(34)

$$u_{12}=2C_2 tanh\left( C_2 x-{\displaystyle \frac{2\sqrt{3}{C}_2 y \sqrt{-C_2 C_3} }{3}}+C_{3}z+C_1\right)+t+C_4$$

(35)

$$u_{13}=y{F}_1\left(x\right)+F_2\left(x\right)$$

(36)

$$u_{14}=F_1\left(z, t\right)$$

(37)

$$u_{15}=e^{-{\displaystyle \frac{4 \sqrt{ c2} y c1-4 c{1}^2 x-3 c2 t}{4 c1}}} C1 C4 (e^{2 \sqrt{ c2} y} C2+C3)$$

(38)

$$u_{16}=F_2\left(t^2+4 x\right)+F_1\left(t\right)+{\displaystyle \frac{\left(t^3+6 t x+4 y^2\right) c_2}{8}}+tz+C_{1}y+C_2$$

(39)

$$u_{17}=y{F}_1\left(x\right)+{\displaystyle \frac{t^2}{2}}+F_2(x)$$

(40)

$$u_{18}=2C_2 tanh\left( C_2 x-{\displaystyle \frac{2\sqrt{3}{C}_2 y \sqrt{-C_2 C_3} }{3}}+C_{3}z+C_1\right)+{\displaystyle \frac{t^2}{2}}+C_4$$

(41)

$$u_{19}=-{\displaystyle \frac{3 z}{2}}+F_2(y-x)+F_1(t)$$

(42)

$$u_{20}=-2 C_2 tanh(C_2^2{C}_3 t+{\displaystyle \frac{\left(-4 x+4 y-3 t\right) C_2}{4}}+C_{3}z+C_1)+C_4$$

(43)

$$u_{21}=y+F_1(z, t)$$

(44)

$$u_{22}=t y+F_1(z, t)$$

(45)

$$u_{23}=-x+F_1\left(z, t\right)$$

(46)

$$\begin{gathered} u_{24}=z+C_5{e}^{-x{\displaystyle \frac{\sqrt{ 4 c_3^2-6c_2}}{2}}-\sqrt{c_2} y+c_3\left(x+t\right)}\left( C_1{C}_3{e}^{2\sqrt{c_2} y+x \sqrt{4 c_3^2-6c_2}} \\ +C_1{C}_4{e}^{x \sqrt{4 c_3^2-6c_2}}+C2 \left( e^{2 \sqrt{c_2}y} C_3+C_4\right)\right), \end{gathered}$$

(47)

$$u_{25}=F_2\left(t^2+4 x\right)+F_1\left(t\right)+{\displaystyle \frac{\left(t^3+6 t x+4 y^2\right) c_2}{8}}+t z+C_1 y+C_2$$

(48)

$$u_{26}=C_1 C_4 e^{\left(-{\displaystyle \frac{4c_1\sqrt{c_2}y-4 c_1^2 x-3 c_2 t}{4 c_1}}\right)}({ C}_2{e}^{\left(2 \sqrt{c_2} y\right)}+C_3)$$

(49)

$$u_{27}=F_1\left(x\right) y+t y+F_2\left(x\right)$$

(50)

$$u_{28}=2 tanh\left( C_2 x-{\displaystyle \frac{2C_2\sqrt{3}y\sqrt{-C_2 C_3}}{3}}+C_{3}z+C_1\right)C_2+t y+C_4,$$

(51)

$$u_{29}={\displaystyle \frac{\sqrt{z}}{t^{\frac{5}{2}}}}\left({\displaystyle \frac{7}{12}}{C}_1{t}^{{\displaystyle \frac{5}{4}}}{z}^{{\displaystyle \frac{1}{4}}}y-{\displaystyle \frac{4}{3}}\sqrt{tz}{y}^2+C_2{t}^2-2t^{{\displaystyle \frac{3}{2}}}x\sqrt{z}\right)$$

(52)

$$u_{30}=C_1\left(y{F}_1\left(t\right)+F_2\left(t\right)\right)+C_2{C}_3{C}_4{e}^{-\left({\displaystyle \frac{4C_1\sqrt{C_2}y-4C_1^{2}x-3C_{2}t}{4C_1}}\right)}+C_2{C}_3{C}_5{e}^{\left({\displaystyle \frac{4C_1\sqrt{C_2}y+4C_1^{2}x+3C_{2}t}{4C_1}}\right)}$$

(53)

$$\begin{gathered} u_{31}=z+C_5{e}^{-{\displaystyle \frac{x}{2}}\sqrt{4C_3^2-6C_2}-\sqrt{C_2}y+C_3\left(x+t\right)}\left(C_1{C}_3{e}^{x\sqrt{4C_3^2-6C_2}+2\sqrt{C_2}y}+C_1{C}_4{e}^{x\sqrt{4C_3^2-6C_2}} \\ +C_2{C}_3{e}^{2\sqrt{C_2}y}+C_2{C}_4\right) \end{gathered}$$

(54)

$$u_{32}=C_4+{\displaystyle \frac{t^{2}y}{2}}+2C_2\tanh \left(C_{2}x-{\displaystyle \frac{2}{\sqrt{3}}}{C}_2\sqrt{-C_2{C}_3}y+C_{3}z+C_1\right)$$

(55)

All of these closed-form solutions have been checked using the Maple package. Some of these solutions are comparable to some solutions in the literature [1,5,7,20], while others are totally new solutions.

Some interesting waveforms are depicted in Figure 3, Figure 4, Figure 5 and Figure 6. Figure 3 is a kink soliton solution, possibly derived from the YTSF equation. It is a stable localized transition region between two different constant states or vacuum levels in a system. Physically, such solutions can be used to describe phenomena such as domain walls between two different magnetic or electric regions, phase boundaries in materials, or the propagation of sharp fronts in nonlinear media. In engineering, information on these kink solutions is valuable due to their use in nonlinear optics for signal processing, data transmission by optical fibers, and materials science defect or interface modeling. Figure 4 presents a more complex wave solution that is different from the elementary kink. It shows a localized central excitation with tails of decaying oscillations, a representation of a breather soliton or a localized wave packet. This could physically represent a pulsating localized excitation, a temporary concentration of energy within a nonlinear system, or the interaction dynamics of multiple wave components. These intricate wave structures are explored in engineering contexts, such as nonlinear fiber optics for high-level signal modulation, rogue wave analysis in fluid dynamics, or the simulation of energy localization phenomena in plasma physics. Figure 5 displays some localized wave patterns interacting with each other, perhaps breathers or solitons, forming with space and time. Physically, this could be the collision, interaction, or fusion/fission process of nonlinear waves, displaying events such as energy transfer or the creation of complex, transient patterns in the system. Engineering applications of the research on such complex dynamics include modeling multi-pulse interactions in optical communication systems to wave turbulence in plasma or fluids. Figure 6 depicts a solution characterized by a broad parabolic profile modulated by fine periodic ripples. It describes a situation in which a quadratic-varying background potential or field in one direction ($y$) exists with stable, periodic wave patterns in the orthogonal direction ($x$). At a physical level, this might describe situations such as wave propagation or pattern formation in a confining potential, or periodic instabilities arising on a slowly varying background field. Engineering applications range from the design of periodically appearing graded-index waveguides to pattern analysis in space-inhomogeneous condensed matter systems or wave propagation in bounded fluids or plasmas.

5. Conclusions

Some families of analytical solutions of the (3 + 1)-dimensional YTSF equation are obtained by employing the potential similarity transformation approach, which enables the exploration of several families of exact and traveling wave solutions. Different exact solutions have been derived using the multipliers of the nonlocal potential similarities. The solutions include kink, soliton, exponential, and periodic waves. Some of the obtained waveforms are comparable to those previously reported in the literature [1,5,7,20]. The kink soliton solution is a stable transition between two different states in a nonlinear system. These solutions are useful in the modeling of interfaces in materials, signal transmission in nonlinear optics, and defects in engineering systems. The breather soliton, with a localized core and exponentially decaying, oscillatory tails, mimics pulsating wave packets or energy localization in nonlinear media. Such structures are of significant engineering concern in advanced optical modulation, rogue wave prediction, and plasma energy localization. Interacting localized multi-soliton waves, such as colliding breathers or solitons, display nonlinear effects such as wave fusion, energy exchange, and transient complexity. Applications include modeling multi-soliton dynamics in communication systems, as well as in fluid or plasma turbulence. The parabolic background with periodic ripples suggests a wave field under the influence of a varying potential. This pattern mimics wave behavior in confined media and enables engineering designs involving graded-index materials and structured waveguides.