# Invariant Subspaces of the Two-Dimensional Nonlinear Evolution Equations

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Direct Extensions of Invariant Subspaces

#### 2.1. Direct Extensions in ${\mathbb{R}}^{2}$

**Theorem**

**1.**

**Remark**

**1.**

#### 2.2. Invariant Subspaces of a Quadratic Operator in ${\mathbb{R}}^{2}$

#### 2.2.1. The Space ${W}_{n+n-1}^{xy}$

**Proposition**

**1.**

- (1)
- $A[u]=\gamma [u{\Delta}_{2}u-|\nabla u{|}^{2}]$, with:$${L}_{x}^{3}[v]=\frac{{d}^{3}v}{d{x}^{3}}-{b}_{1}\frac{dv}{dx}=0,\phantom{\rule{1.em}{0ex}}{L}_{y}^{3}[w]=\frac{{d}^{3}w}{d{y}^{3}}+{b}_{1}\frac{dw}{dy}=0;$$
- (2)
- $A[u]=\alpha [{({\Delta}_{2}u)}^{2}-{b}_{2}^{2}|\nabla u{|}^{2}]$, with:$${L}_{x}^{3}[v]=\frac{{d}^{3}v}{d{x}^{3}}=0,\phantom{\rule{1.em}{0ex}}{L}_{y}^{3}[w]=\frac{{d}^{3}w}{d{y}^{3}}+{b}_{2}\frac{{d}^{2}w}{d{y}^{2}}=0;$$
- (3)
- $A[u]=\alpha [{({\Delta}_{2}u)}^{2}-\frac{8}{9}{b}_{2}^{2}u{\Delta}_{2}u+\frac{16}{81}{b}_{2}^{4}{u}^{2}]$, with:$${L}_{x}^{3}[v]=\frac{{d}^{3}v}{d{x}^{3}}-\frac{4}{9}{b}_{2}^{2}\frac{dv}{dx}=0,\phantom{\rule{1.em}{0ex}}{L}_{y}^{3}[w]=\frac{{d}^{3}w}{d{y}^{3}}+{b}_{2}\frac{{d}^{2}w}{d{y}^{2}}+\frac{2}{9}{b}_{2}^{2}\frac{dw}{dy}=0;$$
- (4)
- $A[u]=\gamma [({a}_{1}+{b}_{1}){({\Delta}_{2}u)}^{2}+4{a}_{1}{b}_{1}u{\Delta}_{2}u+{({a}_{1}-{b}_{1})}^{2}|\nabla u{|}^{2}+{a}_{1}{b}_{1}({a}_{1}+{b}_{1}){u}^{2}]$, with:$${L}_{x}^{3}[v]=\frac{{d}^{3}v}{d{x}^{3}}+{a}_{1}\frac{dv}{dx}=0,\phantom{\rule{1.em}{0ex}}{L}_{y}^{3}[w]=\frac{{d}^{3}w}{d{y}^{3}}+{b}_{1}\frac{dw}{dy}=0;$$
- (5)
- $A[u]=\alpha [{({\Delta}_{2}u)}^{2}+{b}_{1}|\nabla u{|}^{2}]$, with:$${L}_{x}^{3}[v]=\frac{{d}^{3}v}{d{x}^{3}}=0,\phantom{\rule{1.em}{0ex}}{L}_{y}^{3}[w]=\frac{{d}^{3}w}{d{y}^{3}}+{b}_{1}\frac{dw}{dy}=0;$$
- (6)
- $A[u]=\alpha {({\Delta}_{2}u)}^{2}+\gamma u{\Delta}_{2}u+(\gamma {b}_{1}-\alpha {b}_{1}^{2}){u}^{2}$, with:$${L}_{x}^{3}[v]=\frac{{d}^{3}v}{d{x}^{3}}+{b}_{1}\frac{dv}{dx}=0,\phantom{\rule{0.277778em}{0ex}}{L}_{y}^{3}[w]=\frac{{d}^{3}w}{d{y}^{3}}+{b}_{1}\frac{dw}{dy}=0;$$
- (7)
- $A[u]=\alpha {({\Delta}_{2}u)}^{2}+\gamma u{\Delta}_{2}u+\delta {|\nabla u|}^{2}$, with:$${L}_{x}^{3}[v]=\frac{{d}^{3}v}{d{x}^{3}}=0,\phantom{\rule{1.em}{0ex}}{L}_{y}^{3}[w]=\frac{{d}^{3}w}{d{y}^{3}}=0;$$

**Proposition**

**2.**

- (1)
- $A[u]=\gamma [({a}_{1}^{2}+{b}_{1}^{2}){({\Delta}_{2}u)}^{2}-4{a}_{1}^{2}{b}_{1}^{2}u{\Delta}_{2}u-{({a}_{1}^{2}-{b}_{1}^{2})}^{2}|\nabla u{|}^{2}+{a}_{1}^{2}{b}_{1}^{2}({a}_{1}^{2}+{b}_{1}^{2}){u}^{2}]$, with:$${L}_{x}^{2}[v]=\frac{{d}^{2}v}{d{x}^{2}}+{a}_{1}\frac{dv}{dx}=0,\phantom{\rule{1.em}{0ex}}{L}_{y}^{2}[w]=\frac{{d}^{2}w}{d{y}^{2}}+{b}_{1}\frac{dw}{dy}=0;$$
- (2)
- $A[u]=\alpha [{({\Delta}_{2}u)}^{2}-{b}_{1}^{2}|\nabla u{|}^{2}]$, with:$${L}_{x}^{2}[v]=\frac{{d}^{2}v}{d{x}^{2}}=0,\phantom{\rule{1.em}{0ex}}{L}_{y}^{2}[w]=\frac{{d}^{2}w}{d{y}^{2}}+{b}_{1}\frac{dw}{dy}=0;$$
- (3)
- $A[u]=\alpha {({\Delta}_{2}u)}^{2}+\gamma u{\Delta}_{2}u-(\alpha {b}_{1}^{2}+\gamma ){b}_{1}^{2}{u}^{2}$, with:$${L}_{x}^{2}[v]=\frac{{d}^{2}v}{d{x}^{2}}+{b}_{1}\frac{dv}{dx}=0,\phantom{\rule{1.em}{0ex}}{L}_{y}^{2}[w]=\frac{{d}^{2}w}{d{y}^{2}}+{b}_{1}\frac{dw}{dy}=0;$$
- (4)
- $A[u]=\alpha {({\Delta}_{2}u)}^{2}+\gamma u{\Delta}_{2}u-(\alpha {b}_{1}^{2}+\gamma ){b}_{1}^{2}{u}^{2}$, with:$${L}_{x}^{2}[v]=\frac{{d}^{2}v}{d{x}^{2}}-{b}_{1}\frac{dv}{dx}=0,\phantom{\rule{1.em}{0ex}}{L}_{y}^{2}[w]=\frac{{d}^{2}w}{d{y}^{2}}+{b}_{1}\frac{dw}{dy}=0;$$
- (5)
- $A[u]=\alpha {({\Delta}_{2}u)}^{2}+\gamma u{\Delta}_{2}u+\delta {|\nabla u|}^{2}$, with:$${L}_{x}^{2}[v]=\frac{{d}^{2}v}{d{x}^{2}}=0,\phantom{\rule{1.em}{0ex}}{L}_{y}^{2}[w]=\frac{{d}^{2}w}{d{y}^{2}}=0;$$

#### 2.2.2. The Space ${W}_{nn}^{xy}$

**Proposition**

**3.**

- (1)
- $A[u]=\alpha {({\Delta}_{2}u)}^{2}+\gamma u{\Delta}_{2}u-({a}_{0}+{b}_{0})[\alpha ({a}_{0}+{b}_{0})-\gamma ]{u}^{2}$, with:$${L}_{x}^{2}[v]=\frac{{d}^{2}v}{d{x}^{2}}+{a}_{0}v=0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{L}_{y}^{2}[w]=\frac{{d}^{2}v}{d{y}^{2}}+{b}_{0}v=0;$$
- (2)
- $A[u]=\alpha [{({\Delta}_{2}u)}^{2}-{b}_{1}^{2}(2u{\Delta}_{2}u+{|\nabla u|}^{2})+2{b}_{1}^{4}{u}^{2}]$, with:$${L}_{x}^{2}[v]=\frac{{d}^{2}v}{d{x}^{2}}+{b}_{1}\frac{dv}{dx}=0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{L}_{y}^{2}[w]=\frac{{d}^{2}v}{d{y}^{2}}+{b}_{1}\frac{dw}{dy}=0;$$
- (3)
- $A[u]=\alpha [{({\Delta}_{2}u)}^{2}-{b}_{1}^{2}(2u{\Delta}_{2}u+{|\nabla u|}^{2})+2{b}_{1}^{4}{u}^{2}]$, with:$${L}_{x}^{2}[v]=\frac{{d}^{2}v}{d{x}^{2}}-{b}_{1}\frac{dv}{dx}=0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{L}_{y}^{2}[w]=\frac{{d}^{2}v}{d{y}^{2}}+{b}_{1}\frac{dw}{dy}=0.$$

**Example**

**1.**

- (1)
- ${W}_{2+2-1}^{xy}$, determined by the system:$${L}_{x}^{2}[v]=\frac{{d}^{2}v}{d{x}^{2}}+{a}_{1}\frac{dv}{dx}=0,\phantom{\rule{1.em}{0ex}}{L}_{y}^{2}[w]=\frac{{d}^{2}w}{d{y}^{2}}+{b}_{1}\frac{dw}{dy}=0,\phantom{\rule{1.em}{0ex}}\mathrm{with}\phantom{\rule{4pt}{0ex}}{a}_{1}{b}_{1}({a}_{1}^{2}-{b}_{1}^{2})=0;$$
- (2)
- ${W}_{3+3-1}^{xy}$, determined by any of the following systems:$$\begin{array}{c}{L}_{x}^{3}[v]=\frac{{d}^{3}v}{d{x}^{3}}+{a}_{1}\frac{dv}{dx}=0,\phantom{\rule{1.em}{0ex}}{L}_{y}^{3}[w]=\frac{{d}^{3}w}{d{y}^{3}}+{a}_{1}\frac{dw}{dy}=0;\hfill \\ {L}_{x}^{3}[v]=\frac{{d}^{3}v}{d{x}^{3}}-{b}_{2}^{2}\frac{dv}{dx}=0,\phantom{\rule{1.em}{0ex}}{L}_{y}^{3}[w]=\frac{{d}^{3}w}{d{y}^{3}}+{b}_{2}\frac{{d}^{2}w}{d{y}^{2}}=0;\hfill \\ {L}_{x}^{3}[v]=\frac{{d}^{3}v}{d{x}^{3}}\pm {a}_{2}\frac{{d}^{2}v}{d{x}^{2}}=0,\phantom{\rule{1.em}{0ex}}{L}_{y}^{3}[w]=\frac{{d}^{3}w}{d{y}^{3}}+{a}_{2}\frac{{d}^{2}w}{d{y}^{2}}=0;\hfill \end{array}$$
- (3)
- ${W}_{4+4-1}^{xy}$, determined by the system:$${L}_{x}^{4}[v]=\frac{{d}^{4}v}{d{x}^{4}}+{a}_{2}\frac{{d}^{2}v}{d{x}^{2}}=0,\phantom{\rule{1.em}{0ex}}{L}_{y}^{4}[w]=\frac{{d}^{4}w}{d{y}^{4}}+{a}_{2}\frac{{d}^{2}w}{d{y}^{2}}=0;$$
- (4)
- ${W}_{22}^{xy}$, determined by any of the following systems:$$\begin{array}{c}{L}_{x}^{2}[v]=\frac{{d}^{2}v}{d{x}^{2}}=0,\phantom{\rule{1.em}{0ex}}{L}_{y}^{2}[w]=\frac{{d}^{2}w}{d{y}^{2}}+{b}_{1}\frac{dw}{dy}=0;\hfill \\ {L}_{x}^{2}[v]=\frac{{d}^{2}v}{d{x}^{2}}+{a}_{0}v=0,\phantom{\rule{1.em}{0ex}}{L}_{y}^{2}[w]=\frac{{d}^{2}w}{d{y}^{2}}+{b}_{0}w=0.\hfill \end{array}$$

**Example**

**2.**

- (1)
- ${W}_{2+2-1}^{xy}$, determined by the system:$${L}_{x}^{2}[v]=\frac{{d}^{2}v}{d{x}^{2}}+{a}_{1}\frac{dv}{dx}=0,\phantom{\rule{1.em}{0ex}}{L}_{y}^{2}[w]=\frac{{d}^{2}w}{d{y}^{2}}+{b}_{1}\frac{dw}{dy}=0,\phantom{\rule{1.em}{0ex}}with\phantom{\rule{4pt}{0ex}}{a}_{1}{b}_{1}=0;$$
- (2)
- ${W}_{3+3-1}^{xy}$, determined by any of the following systems:$$\begin{array}{cc}& {L}_{x}^{3}[v]=\frac{{d}^{3}v}{d{x}^{3}}=0,\phantom{\rule{1.em}{0ex}}{L}_{y}^{3}[w]=\frac{{d}^{3}w}{d{y}^{3}}+{b}_{2}\frac{{d}^{2}w}{d{y}^{2}}+{b}_{1}\frac{dw}{dy}=0;\hfill \end{array}$$
- (3)
- ${W}_{22}^{xy}$, determined by any of the following systems:$$\begin{array}{cc}& {L}_{x}^{2}[v]=\frac{{d}^{2}v}{d{x}^{2}}=0,\phantom{\rule{1.em}{0ex}}{L}_{y}^{2}[w]=\frac{{d}^{2}w}{d{y}^{2}}+{b}_{1}\frac{dw}{dy}+{b}_{0}w=0,\phantom{\rule{1.em}{0ex}}with\phantom{\rule{4pt}{0ex}}{b}_{0}{b}_{1}=0;\hfill \\ & {L}_{x}^{2}[v]=\frac{{d}^{2}v}{d{x}^{2}}+{a}_{0}v=0,\phantom{\rule{1.em}{0ex}}{L}_{y}^{2}[w]=\frac{{d}^{2}w}{d{y}^{2}}+{b}_{0}w=0;\hfill \end{array}$$
- (4)
- ${W}_{33}^{xy}=\mathcal{L}\{1,x,{x}^{2},y,{y}^{2},xy,{x}^{2}y,x{y}^{2},{x}^{2}{y}^{2}\}$, determined by the system:$${L}_{x}^{3}[v]=\frac{{d}^{3}v}{d{x}^{3}}=0,\phantom{\rule{1.em}{0ex}}{L}_{y}^{3}[w]=\frac{{d}^{3}w}{d{y}^{3}}=0.$$

## 3. Invariant Subspaces under the General Change of Variables

**Proposition**

**4.**

**Example**

**3.**

## 4. Invariant Subspace in $\mathbb{R}$ and Lie’s Classical Symmetries

**Example**

**4.**

- (1)
- ${\tilde{X}}_{1}$. For ${\tilde{X}}_{1}$, its invariants are $\tilde{U}=U$ and $z=x+y$. The corresponding invariant solutions of (21) are $U=\tilde{U}(z,t)$, where $\tilde{U}(z,t)$ satisfies:$${\tilde{U}}_{t}=2(\tilde{U}{\tilde{U}}_{zz}-{\tilde{U}}_{z}^{2})\equiv {\tilde{J}}^{1}[\tilde{U}].$$
- (2)
- ${\tilde{X}}_{2}$. For ${\tilde{X}}_{2}$, its invariants are $\tilde{U}=U$ and $z={x}^{2}+{y}^{2}$. The corresponding invariant solutions of (21) are $v=\tilde{U}(z,t)$, where $\tilde{U}(z,t)$ satisfies:$${\tilde{U}}_{t}=4z\tilde{U}{\tilde{U}}_{zz}-4z{\tilde{U}}_{z}^{2}+4\tilde{U}{\tilde{U}}_{z}\equiv {\tilde{J}}^{2}[\tilde{U}].$$
- (3)
- ${\tilde{X}}_{3}$. For ${\tilde{X}}_{3}$, its invariants are $\tilde{U}=U{x}^{-2}$ and $z=y/x$. The corresponding invariant solutions of (21) are $U={x}^{2}\tilde{U}(z,t)$, where $\tilde{U}(z,t)$ satisfies:$${\tilde{U}}_{t}=(1+{z}^{2})\tilde{U}{\tilde{U}}_{zz}-(1+{z}^{2}){\tilde{U}}_{z}^{2}+2z\tilde{U}{\tilde{U}}_{z}-2{\tilde{U}}^{2}\equiv {\tilde{J}}^{3}[\tilde{U}].$$
- (4)
- ${\tilde{X}}_{4}$. For ${\tilde{X}}_{4}$, its invariants are $\tilde{U}=v{x}^{-2}$ and $z=x+{y}^{2}/x$. The corresponding invariant solutions of (21) are $U={x}^{2}\tilde{U}(z,t)$, where $\tilde{U}(z,t)$ satisfies:$${\tilde{U}}_{t}={z}^{2}\tilde{U}{\tilde{U}}_{zz}-{z}^{2}{\tilde{U}}_{z}^{2}+2z\tilde{U}{\tilde{U}}_{z}-2{\tilde{U}}^{2}\equiv {\tilde{J}}^{4}[\tilde{U}].$$
- (5)
- ${\tilde{X}}_{5}$. For ${\tilde{X}}_{5}$, its invariants are $\tilde{U}={y}^{-2}U$ and $z=y+{x}^{2}/y$. The invariant solutions of (21) are $U={y}^{2}\tilde{U}(z,t)$, where $\tilde{U}(z,t)$ satisfies ${\tilde{U}}_{t}={\tilde{J}}^{4}[\tilde{U}]$.
- (6)
- ${\tilde{X}}_{6}$. For ${\tilde{X}}_{6}$, its invariants are $\tilde{U}={sinh}^{-2}(ax)U$ and $z=cos(ay)/sinh(ax)$. The invariant solutions of (21) are $U={sinh}^{2}(ax)\tilde{U}(z,t)$, where $\tilde{U}(z,t)$ satisfies ${\tilde{U}}_{t}={a}^{2}{\tilde{J}}^{3}[\tilde{U}]$.
- (7)
- ${\tilde{X}}_{7}$. For ${\tilde{X}}_{7}$, its invariants are $\tilde{U}={sinh}^{-2}(ax)U$ and $z=sin(ay)/sinh(ax)$. The invariant solutions of (21) are $U={sinh}^{2}(ax)\tilde{v}(z,t)$, where $\tilde{U}(z,t)$ satisfies ${\tilde{U}}_{t}={a}^{2}{\tilde{J}}^{3}[\tilde{U}]$.
- (8)
- ${\tilde{X}}_{8}$. For ${\tilde{X}}_{8}$, its invariants are $\tilde{U}={cosh}^{-2}(ay)U$ and $z=sin(ax)/cosh(ay)$. The invariant solutions of (21) are $U={cosh}^{2}(ay)\tilde{U}(z,t)$, where $\tilde{U}(z,t)$ satisfies:$${\tilde{U}}_{t}={a}^{2}(1-{z}^{2})\tilde{U}{\tilde{U}}_{zz}+{a}^{2}({z}^{2}-1){\tilde{U}}_{z}^{2}-2{a}^{2}z\tilde{U}{\tilde{U}}_{z}+2{a}^{2}{\tilde{U}}^{2}\equiv {\tilde{J}}^{5}[\tilde{U}].$$
- (9)
- ${\tilde{X}}_{9}$. For ${\tilde{X}}_{9}$, its invariants are $\tilde{U}=U{cosh}^{-2}(ay)$ and $z=cos(ax)/cosh(ay)$. The invariant solutions of (21) are $U={cosh}^{2}(ay)\tilde{U}(z,t)$, where $\tilde{U}(z,t)$ satisfies ${\tilde{U}}_{t}={\tilde{J}}^{5}[\tilde{U}]$.

- (1)
- The operator ${\tilde{J}}^{1}[\tilde{U}]$ admits the invariant subspaces:$${\tilde{W}}_{3}=\left\{\begin{array}{cc}\mathcal{L}\{1,z,{z}^{2}\},\hfill & b=0,\hfill \\ \mathcal{L}\{1,cos(cz),sin(cz)\},\hfill & b={c}^{2},\hfill \\ \mathcal{L}\{1,exp(cz),exp(-cz)\},\hfill & b=-{c}^{2},\hfill \end{array}\right.$$$$\frac{{d}^{3}w}{d{z}^{3}}+b\frac{dw}{dz}=0.$$
- (2)
- The operator ${\tilde{J}}^{2}[\tilde{U}]$ admits the invariant subspaces:$${W}_{3}=\left\{\begin{array}{cc}\mathcal{L}\{z,zlnz,z{(lnz)}^{2}\},\hfill & b=-1,\hfill \\ \mathcal{L}\{z,{z}^{1-c},{z}^{1+c}\},\hfill & b=-1+{c}^{2},\hfill \\ \mathcal{L}\{z,zsin(clnz),zcos(clnz)\},\hfill & b=-1-{c}^{2},\hfill \end{array}\right.$$$$\frac{{d}^{3}w}{d{z}^{3}}+\frac{b}{{z}^{2}}\frac{dw}{dz}-\frac{b}{{z}^{3}}w=0.$$
- (3)
- The operator ${\tilde{J}}^{3}[\tilde{U}]$ admits the invariant subspaces:$${\tilde{W}}_{3}=\left\{\begin{array}{cc}\mathcal{L}\{(1+{z}^{2}),(1+{z}^{2})arctanz,(1+{z}^{2}){(arctanz)}^{2}\},\hfill & b=-4,\hfill \\ \mathcal{L}\{(1+{z}^{2}),(1+{z}^{2})sin(carctanz),(1+{z}^{2})cos(carctanz)\},\hfill & b=-4+{c}^{2},\hfill \\ \mathcal{L}\{(1+{z}^{2}),(1+{z}^{2})exp(carctanz),(1+{z}^{2})exp(-carctanz)\},\hfill & b=-4-{c}^{2},\hfill \\ \mathcal{L}\{1,z,{z}^{2}\},\hfill & b=0,\hfill \end{array}\right.$$$$\frac{{d}^{3}w}{d{z}^{3}}+\frac{b}{{(1+{z}^{2})}^{2}}\frac{dw}{dz}-\frac{2bz}{{(1+{z}^{2})}^{3}}w=0.$$
- (4)
- The operator ${\tilde{J}}^{4}[\tilde{U}]$ admits the invariant subspaces:$${\tilde{W}}_{3}=\left\{\begin{array}{cc}\mathcal{L}\{{z}^{2},{z}^{2}exp(-\frac{c}{z}),{z}^{2}exp(\frac{c}{z})\},\hfill & b=2{c}^{2},\hfill \\ \mathcal{L}\{{z}^{2},{z}^{2}sin(\frac{c}{z}),{z}^{2}cos(\frac{c}{z})\},\hfill & b=-2{c}^{2},\hfill \\ \mathcal{L}\{1,z,{z}^{2}\},\hfill & b=0,\hfill \end{array}\right.$$$$\frac{{d}^{3}w}{d{z}^{3}}-\frac{b}{2{z}^{4}}\frac{dw}{dz}+\frac{b}{{z}^{5}}w=0.$$
- (5)
- The operator ${\tilde{J}}^{5}[\tilde{U}]$ admits the invariant subspaces:$${\tilde{W}}_{3}=\left\{\begin{array}{cc}\mathcal{L}\{({z}^{2}-1),({z}^{2}-1)ln(\frac{z+1}{z-1}),({z}^{2}-1){(ln(\frac{z+1}{z-1}))}^{2}\},\hfill & b=8,\hfill \\ \mathcal{L}\{({z}^{2}-1),({z}^{2}-1)exp(c\mathrm{arctanh}z),({z}^{2}-1)exp(-c\mathrm{arctanh}z)\},\hfill & b=-8+8{c}^{2},\hfill \\ \mathcal{L}\{({z}^{2}-1),({z}^{2}-1)sin(c\mathrm{arctanh}z),({z}^{2}-1)cos(c\mathrm{arctanh}z)\},\hfill & b=-8-8{c}^{2},\hfill \\ \mathcal{L}\{1,z,{z}^{2}\},\hfill & b=0,\hfill \end{array}\right.$$$$\frac{{d}^{3}w}{d{z}^{3}}-\frac{b}{2{({z}^{2}-1)}^{2}}\frac{dw}{dz}+\frac{bz}{{({z}^{2}-1)}^{3}}w=0,$$

**Example**

**5.**

**Example**

**6.**

**Proposition**

**5.**

## 5. Concluding Remarks

**Proposition**

**6.**

**Proposition**

**7.**

- (1)
- ${\widehat{W}}_{3+3-1}^{xy}={W}_{3}^{1}\cup {W}_{3}^{2}$, with:$$\begin{array}{c}{W}_{3}^{1}=\mathcal{L}\{{({x}^{2}+{y}^{2})}^{2},{({x}^{2}+{y}^{2})}^{2}cos({b}_{1}\frac{x}{{x}^{2}+{y}^{2}}),{({x}^{2}+{y}^{2})}^{2}sin({b}_{1}\frac{x}{{x}^{2}+{y}^{2}})\},\hfill \\ {W}_{3}^{2}=\mathcal{L}\{{({x}^{2}+{y}^{2})}^{2},{({x}^{2}+{y}^{2})}^{2}exp({b}_{1}\frac{y}{{x}^{2}+{y}^{2}}),{({x}^{2}+{y}^{2})}^{2}exp(-{b}_{1}\frac{y}{{x}^{2}+{y}^{2}})\};\hfill \end{array}$$
- (2)
- ${\widehat{\widehat{W}}}_{3+3-1}^{xy}={W}_{3}^{3}\cup {W}_{3}^{4}$, with:$$\begin{array}{c}{W}_{3}^{3}=\mathcal{L}\{{({x}^{2}+{y}^{2})}^{2},{({x}^{2}+{y}^{2})}^{2}cos({b}_{1}\frac{y}{{x}^{2}+{y}^{2}}),{({x}^{2}+{y}^{2})}^{2}sin({b}_{1}\frac{y}{{x}^{2}+{y}^{2}})\},\hfill \\ {W}_{3}^{4}=\mathcal{L}\{{({x}^{2}+{y}^{2})}^{2},{({x}^{2}+{y}^{2})}^{2}exp({b}_{1}\frac{x}{{x}^{2}+{y}^{2}}),{({x}^{2}+{y}^{2})}^{2}exp(-{b}_{1}\frac{x}{{x}^{2}+{y}^{2}})\}.\hfill \end{array}$$

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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