# A τ-Symmetry Algebra of the Generalized Derivative Nonlinear Schrödinger Soliton Hierarchy with an Arbitrary Parameter

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## Abstract

**:**

## 1. Introduction

## 2. Basic Notions

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

## 3. Isospectral and Nonisospcetral Hierarchies of the gDNLS Equations

- The isospectral Kaup–Newell (KN) equation$$\begin{array}{c}{q}_{t}={q}_{xx}+2{\left({q}^{2}r\right)}_{x},\hfill \\ {r}_{t}=-{r}_{xx}+2{\left(q{r}^{2}\right)}_{x},\hfill \end{array}$$
- The isospectal Chen–Lee–Liu (CLL) equation$$\begin{array}{c}{q}_{t}={q}_{xx}+2qr{q}_{x},\hfill \\ {r}_{t}=-{r}_{xx}+2qr{r}_{x},\hfill \end{array}$$
- The isospectral Gerdjikov–Ivanov (GI) equation$$\begin{array}{c}{q}_{t}={q}_{xx}-2{q}^{2}{r}_{x}-2{q}^{3}{r}^{2},\hfill \\ {r}_{t}=-{r}_{xx}-2{r}^{2}{q}_{x}+2{q}^{2}{r}^{3},\hfill \end{array}$$

- The nonisospectral KN equation$$\begin{array}{c}{q}_{t}=x{q}_{xx}+2x{\left({q}^{2}r\right)}_{x}+\frac{3}{2}{q}_{x}+2{q}^{2}r,\hfill \\ {r}_{t}=-x{r}_{xx}+2x{\left(q{r}^{2}\right)}_{x}-\frac{3}{2}{r}_{x}+2q{r}^{2},\hfill \end{array}$$
- The nonisospectral CLL equation$$\begin{array}{c}{q}_{t}=x{q}_{xx}+2xqr{q}_{x}+\frac{3}{2}{q}_{x}+q{\partial}^{-1}{q}_{x}r,\hfill \\ {r}_{t}=-x{r}_{xx}+2xqr{r}_{x}-\frac{3}{2}{r}_{x}+r{\partial}^{-1}q{r}_{x},\hfill \end{array}$$
- The nonisospectral GI equation$$\begin{array}{c}{q}_{t}=x{q}_{xx}-2x{q}^{2}{r}_{x}-2x{q}^{3}{r}^{2}+\frac{3}{2}{q}_{x}+2q{\partial}^{-1}{q}_{x}r-2{q}^{2}r-2q{\partial}^{-1}{q}^{2}{r}^{2},\hfill \\ {r}_{t}=-x{r}_{xx}-2x{r}^{2}{q}_{x}+2x{q}^{2}{r}^{3}-\frac{3}{2}{r}_{x}+2r{\partial}^{-1}q{r}_{x}-2q{r}^{2}+2r{\partial}^{-1}{q}^{2}{r}^{2},\hfill \end{array}$$

- The modified Korteweg-de Vries (mKdV) equation$$\begin{array}{c}\hfill {q}_{t}={q}_{xxx}-6{q}^{2}{q}_{x}\end{array}$$
- The Sharma-Tasso-Olever (STO) equation$$\begin{array}{c}\hfill {q}_{t}={q}_{xxx}+3{\left(q{q}_{x}\right)}_{x}+3{q}^{2}{q}_{x}\end{array}$$
- A new equation$$\begin{array}{c}\hfill {q}_{t}={q}_{xxx}-{6i\left|q\right|}^{2}{q}_{xx}-6i{q}^{*}{q}_{x}^{2}-6iq|{q}_{x}{|}^{2}-{18\left|q\right|}^{4}{q}_{x}-12{q}^{2}{\left|q\right|}^{2}{q}_{x}\end{array}$$As far as we known, Equation (20) is a new integrable equation with fifth-order nonlinear term. Since it is a special case of the system in Equation (11), it has infinite conservation laws and Hamilton Structure. It also has two symmetries and these symmetries have an infinite dimensional Lie algebra structure.

## 4. A $\mathit{\tau}$-Symmetry Algebra of the gDNLS Soliton Hierarchy

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**1.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Zhang, J.-b.; Gongye, Y.; Ma, W.-X.
A *τ*-Symmetry Algebra of the Generalized Derivative Nonlinear Schrödinger Soliton Hierarchy with an Arbitrary Parameter. *Symmetry* **2018**, *10*, 535.
https://doi.org/10.3390/sym10110535

**AMA Style**

Zhang J-b, Gongye Y, Ma W-X.
A *τ*-Symmetry Algebra of the Generalized Derivative Nonlinear Schrödinger Soliton Hierarchy with an Arbitrary Parameter. *Symmetry*. 2018; 10(11):535.
https://doi.org/10.3390/sym10110535

**Chicago/Turabian Style**

Zhang, Jian-bing, Yingyin Gongye, and Wen-Xiu Ma.
2018. "A *τ*-Symmetry Algebra of the Generalized Derivative Nonlinear Schrödinger Soliton Hierarchy with an Arbitrary Parameter" *Symmetry* 10, no. 11: 535.
https://doi.org/10.3390/sym10110535