# Symmetries and Reductions of Integrable Nonlocal Partial Differential Equations

## Abstract

**:**

## 1. Introduction

## 2. The Linearized Symmetry Condition for Nonlocal Differential Equations

**v**is the infinitesimal generator (6).

## 3. The Nonlocal NLS Equation

#### 3.1. Lie Point Symmetries

#### 3.2. Symmetry Reductions

- If $d=b=0$ (and ${a}^{2}+{e}^{2}\ne 0$), we have$$\begin{array}{cc}\hfill y& =t,\hfill \\ \hfill q(x,t)& =exp\left\{\frac{x(4ic-ex)}{4(iet+a)}\right\}p\left(t\right).\hfill \end{array}$$When $a=0$ and $e\ne 0$, the reduced equation is$$i{e}^{2}{p}^{\prime}\left(t\right)+\frac{ie}{2t}p\left(t\right)+\frac{{c}^{2}}{{t}^{2}}p\left(t\right)+2{\left|p\left(t\right)\right|}^{2}p\left(t\right)=0.$$
- If $d=0$ and $b\ne 0$, we have$$\begin{array}{cc}\hfill y& =bx-\frac{1}{2}ie{t}^{2}-at,\hfill \\ \hfill q(x,t)& =exp\left\{-\frac{ae}{4{b}^{2}}{t}^{2}-\frac{e}{2{b}^{2}}ty+i\left(\frac{c}{b}t-\frac{{e}^{2}}{12{b}^{2}}{t}^{3}\right)\right\}p\left(y\right).\hfill \end{array}$$As $b\ne 0$, we must choose $a=e=0$. Next, we consider the corresponding reductions to nonlocal and local ODEs separately (here and throughout).
- –
**Reduction to a nonlocal ODE.**If we choose $y=x$ and $q(x,t)=exp(ict)p\left(y\right)$, we obtain the nonlocal Painlevé-type equation as shown in [7]:$${p}^{\u2033}\left(y\right)-cp\left(y\right)+2{p}^{2}\left(y\right){p}^{\ast}(-y)=0.$$Note that, since $p\left(y\right)$ is invariant, and so is $p(-y)$; namely, the nonlocal invariant is$$p(-y)=exp(-ict)q(-x,t).$$- –
**Reduction to a local ODE.**Alternatively, we may choose the invariants as $y={x}^{2}$ and $q(x,t)=exp(ict)p\left(y\right)$. The reduced equation is a local ODE$$4y{p}^{\u2033}\left(y\right)=cp\left(y\right)-2{p}^{\prime}\left(y\right)-2{\left|p\left(y\right)\right|}^{2}p\left(y\right).$$If we assume that $p\left(y\right)$ is real, the solution of the above equation can be expressed, using the Jacobi elliptic function, as$$p\left(y\right)={C}_{2}\sqrt{\frac{c}{{C}_{2}^{2}+c-1}}sn\left(\sqrt{\frac{c}{{C}_{2}^{2}+c-1}}\left(\sqrt{-(c-1)y}+{C}_{1}\right),\frac{{C}_{2}}{\sqrt{c-1}}\right),$$$${\widehat{p}}^{\u2033}\left(z\right)=c\widehat{p}\left(z\right)-2{\left|\widehat{p}\left(z\right)\right|}^{2}\widehat{p}\left(z\right).$$

- If $d\ne 0$, we have$$\begin{array}{cc}\hfill y& =\frac{{d}^{2}x-ad+ie\left(dt-b\right)}{{d}^{2}\sqrt{|2dt-b|}},\hfill \\ \hfill q(x,t)& =exp\left\{\left(\frac{ae}{4{d}^{2}}-\frac{1}{2}\right)ln|2dt-b|+\frac{e}{2d}y\sqrt{|2dt-b|}\right\}\times \hfill \\ & exp\left\{-i\frac{{e}^{2}}{4{d}^{2}}t+i\left(\frac{b{e}^{3}}{8{d}^{3}}-\frac{c}{2d}\right)ln|2dt-b|\right\}p\left(y\right).\hfill \end{array}$$Now, we must set $a=e=0$.
- –
**Reduction to a nonlocal ODE.**Let$$\begin{array}{cc}\hfill y& =\frac{x}{\sqrt{|2dt-b|}},\hfill \\ \hfill q(x,t)& =exp\left\{-\left(\frac{1}{2}+\frac{ic}{2d}\right)ln|2dt-b|\right\}p\left(y\right).\hfill \end{array}$$The reduced equation is a nonlocal ODE$${p}^{\u2033}\left(y\right)=\left(id-c\right)p\left(y\right)+idy{p}^{\prime}\left(y\right)-2{p}^{2}\left(y\right){p}^{\ast}(-y).$$- –
**Reduction to a local ODE.**If we choose the invariant variables by$$\begin{array}{cc}\hfill y& =\frac{{x}^{2}}{2dt-b},\hfill \\ \hfill q(x,t)& =exp\left\{-\left(\frac{1}{2}+\frac{ic}{2d}\right)ln|2dt-b|\right\}p\left(y\right),\hfill \end{array}$$$$4y{p}^{\u2033}\left(y\right)=(id-c)p\left(y\right)+(2idy-2){p}^{\prime}\left(y\right)-2{\left|p\left(y\right)\right|}^{2}p\left(y\right).$$$${\widehat{p}}^{\u2033}\left(z\right)=(id-c)\widehat{p}\left(z\right)+idz\widehat{p}\left(z\right)-2{\left|\widehat{p}\left(z\right)\right|}^{2}\widehat{p}\left(z\right).$$

## 4. The Nonlocal mKdV Equation

- When $c=0$, it corresponds to the traveling-wave case.
- –
**Reduction to a nonlocal ODE.**The corresponding invariants are$$\begin{array}{c}\hfill y=bx-at\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}v\left(y\right)=u(x,t).\end{array}$$The reduced equation is$${b}^{3}{v}^{\u2034}\left(y\right)+bv\left(y\right)v(-y){v}^{\prime}\left(y\right)-a{v}^{\prime}\left(y\right)=0.$$When $b=0$, we obtain a constant solution; when $b\ne 0$, without loss of generality, it can be chosen as $b=1$; namely$${v}^{\u2034}\left(y\right)+v\left(y\right)v(-y){v}^{\prime}\left(y\right)-a{v}^{\prime}\left(y\right)=0.$$In principle, it can be integrated once, as it admits a symmetry generated by ${\partial}_{y}$, but will involve the inverse of nonlocal functions. We will show some of its special solutions with the assumption $a>0$.- ∗
- Exponential solutions:$$v\left(y\right)={C}_{1}exp\left({C}_{2}y\right)\phantom{\rule{4.pt}{0ex}}\mathrm{subject}\phantom{\rule{4.pt}{0ex}}\mathrm{to}\phantom{\rule{4.pt}{0ex}}{C}_{1}^{2}+{C}_{2}^{2}=a.$$
- ∗
- Soliton solutions:$$v\left(y\right)=\pm \frac{2\sqrt{6a}}{exp\left(\sqrt{a}y\right)+exp(-\sqrt{a}y)}.$$

- –
**Reduction to a local ODE.**We may, alternatively, introduce the invariants in another way; namely, $y={(bx-at)}^{2}$ and $v\left(y\right)=u(x,t)$. Now, the reduced equation reads$$4{b}^{3}y{v}^{\u2034}\left(y\right)+6{b}^{3}{v}^{\u2033}\left(y\right)+b{v}^{2}\left(y\right){v}^{\prime}\left(y\right)-a{v}^{\prime}\left(y\right)=0,$$$$4{b}^{3}y{v}^{\u2033}\left(y\right)+2{b}^{3}{v}^{\prime}\left(y\right)+\frac{b}{3}{v}^{3}\left(y\right)-av\left(y\right)+{C}_{1}=0.$$This equation can be further simplified by introducing $y={z}^{2}$ and $\widehat{v}\left(z\right)=v\left(y\right)$, amounting to$${b}^{3}{\widehat{v}}^{\u2033}\left(z\right)+\frac{b}{3}{\widehat{v}}^{3}\left(z\right)-a\widehat{v}\left(z\right)+{C}_{1}=0.$$The final equation is solvable by letting $\widehat{v}\left(z\right)=w\left(\widehat{v}\right)$; the general solution is$$z+{C}_{3}=\pm {\int}_{0}^{\widehat{v}\left(z\right)}\frac{\sqrt{6}{b}^{3/2}}{\sqrt{-b{s}^{4}+6a{s}^{2}-12{C}_{1}s+6{C}_{2}{b}^{3}}}d\phantom{\rule{-0.166667em}{0ex}}s,$$

- If $c\ne 0$, the invariants are$$\begin{array}{c}\hfill y=(cx-a){(3ct-b)}^{-1/3}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}v\left(y\right)={(3ct-b)}^{1/3}u(x,t).\end{array}$$Now, we must set $a=b=0$; namely, reduction related to the generator $-x{\partial}_{x}-3t{\partial}_{t}+u{\partial}_{u}$. The related invariants are $y={t}^{-1/3}x$ and $v\left(y\right)={t}^{1/3}u(x,t)$, and we obtain the reduced equation as a local ODE$${v}^{\u0204}\left(y\right)-{v}^{2}\left(y\right){v}^{\prime}\left(y\right)-\frac{v\left(y\right)+y{v}^{\prime}\left(y\right)}{3}=0.$$It can be integrated once to the second Painlevé equation$${v}^{\u2033}\left(y\right)=\frac{1}{3}{v}^{3}\left(y\right)+\frac{1}{3}yv\left(y\right)+C.$$

**Remark**

**1.**

## 5. A Remark on Transformations Between Nonlocal Differential Equations and Differential-Difference Equations

**Remark**

**2.**

## 6. Conclusions

## Funding

## Conflicts of Interest

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Peng, L.
Symmetries and Reductions of Integrable Nonlocal Partial Differential Equations. *Symmetry* **2019**, *11*, 884.
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Peng L.
Symmetries and Reductions of Integrable Nonlocal Partial Differential Equations. *Symmetry*. 2019; 11(7):884.
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**Chicago/Turabian Style**

Peng, Linyu.
2019. "Symmetries and Reductions of Integrable Nonlocal Partial Differential Equations" *Symmetry* 11, no. 7: 884.
https://doi.org/10.3390/sym11070884