# Hamiltonian Structure, Symmetries and Conservation Laws for a Generalized (2 + 1)-Dimensional Double Dispersion Equation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Potential Form and Hamiltonian Formulation

## 3. Lie Symmetries

#### 3.1. Point Symmetries

**Theorem**

**1.**

- (a)
- $a=0$,$d=0$, arbitrary$f\left({v}_{x}\right)$, b, and β,$$\begin{array}{c}{\mathbf{X}}_{5}=y{\partial}_{t}+\beta t{\partial}_{y},\hfill \end{array}$$$$\begin{array}{c}{(\tilde{t},\tilde{x},\tilde{y},\tilde{v})}_{5}=(cosh\left(\u03f5\sqrt{\beta}\right)t+{\textstyle \frac{1}{\sqrt{\beta}}}sinh\left(\u03f5\sqrt{\beta}\right)y,x,cosh\left(\u03f5\sqrt{\beta}\right)y+\sqrt{\beta}sinh\left(\u03f5\sqrt{\beta}\right)t,v),\hfill \\ \phantom{\rule{2.em}{0ex}}boost\text{}in\text{}the\text{}plane\text{}(y,t).\hfill \end{array}$$
- (b)
- $f\left({v}_{x}\right)=\alpha {({v}_{x}+c)}^{p+1}-{v}_{x}$,$a=0$, arbitrary b, d, and β,$$\begin{array}{c}{\mathbf{X}}_{6}=2pt{\partial}_{t}+px{\partial}_{x}+2py{\partial}_{y}+((p-2)v-2cx){\partial}_{v},\hfill \end{array}$$$$\begin{array}{c}{(\tilde{t},\tilde{x},\tilde{y},\tilde{v})}_{6}=({e}^{2p\u03f5}t,{e}^{p\u03f5}x,{e}^{2p\u03f5}y,{e}^{(p-2)\u03f5}v-2cx\u03f5),\phantom{\rule{2.em}{0ex}}scaling\text{}and\text{}shift.\hfill \end{array}$$
- (c)
- $f\left({v}_{x}\right)=\frac{1}{\alpha}ln\left(\alpha ({v}_{x}+c)\right)-{v}_{x}$,$a=0$, arbitrary b, d, and β,$$\begin{array}{c}{\mathbf{X}}_{7}=2t{\partial}_{t}+x{\partial}_{x}+2y{\partial}_{y}+(3v+2cx){\partial}_{v},\hfill \end{array}$$$$\begin{array}{c}{(\tilde{t},\tilde{x},\tilde{y},\tilde{v})}_{7}=({e}^{2\u03f5}t,{e}^{\u03f5}x,{e}^{2\u03f5}y,{e}^{3\u03f5}v+2cx\u03f5),\phantom{\rule{2.em}{0ex}}scaling\text{}and\text{}shift.\hfill \end{array}$$
- (d)
- $f\left({v}_{x}\right)=\alpha {({v}_{x}+c)}^{p+1}-{v}_{x}$,$b=0$,$d=0$, arbitrary a and β,$$\begin{array}{c}{\mathbf{X}}_{8}=pt{\partial}_{t}+py{\partial}_{y}-2(v+cx){\partial}_{v},\hfill \end{array}$$$$\begin{array}{c}{(\tilde{t},\tilde{x},\tilde{y},\tilde{v})}_{8}=({e}^{p\u03f5}t,x,{e}^{p\u03f5}y,{e}^{-2\u03f5}v-2cx\u03f5),\phantom{\rule{2.em}{0ex}}scaling\text{}and\text{}shift.\hfill \end{array}$$
- (e)
- $f\left({v}_{x}\right)=\alpha {e}^{p{v}_{x}}-{v}_{x}$,$b=0$,$d=0$, arbitrary a and β,$$\begin{array}{c}{\mathbf{X}}_{9}=pt{\partial}_{t}+py{\partial}_{y}-2x{\partial}_{v},\hfill \end{array}$$$$\begin{array}{c}{(\tilde{t},\tilde{x},\tilde{y},\tilde{v})}_{9}=({e}^{p\u03f5}t,x,{e}^{p\u03f5}y,v-2x\u03f5),\phantom{\rule{2.em}{0ex}}scaling\text{}and\text{}shift.\hfill \end{array}$$

**Theorem**

**2.**

- (i)
- arbitrary$f\left({v}_{x}\right)$, a, b, d, and β,$${\mathbf{X}}_{1},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{2},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{3},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{4,g,h}\phantom{\rule{0.166667em}{0ex}};$$$$[{\mathbf{X}}_{1},{\mathbf{X}}_{4,g,h}]={\mathbf{X}}_{4,\sqrt{\beta}{g}^{\prime},-\sqrt{\beta}{h}^{\prime}}\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{1.em}{0ex}}[{\mathbf{X}}_{3},{\mathbf{X}}_{4,g,h}]={\mathbf{X}}_{4,{g}^{\prime},{h}^{\prime}}\phantom{\rule{0.166667em}{0ex}}.$$
- (ii)
- $a=0$,$d=0$, arbitrary$f\left({v}_{x}\right)$, b, and β,$${\mathbf{X}}_{1},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{2},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{3},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{4,g,h},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{5};$$$$[{\mathbf{X}}_{1},{\mathbf{X}}_{5}]=\beta {\mathbf{X}}_{3}\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{1.em}{0ex}}[{\mathbf{X}}_{3},{\mathbf{X}}_{5}]={\mathbf{X}}_{1}\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{1.em}{0ex}}[{\mathbf{X}}_{4,g,h},{\mathbf{X}}_{5}]={\mathbf{X}}_{4,{g}_{1},{h}_{1}}\phantom{\rule{0.166667em}{0ex}},$$
- (iii)
- $f\left({v}_{x}\right)=\alpha {({v}_{x}+c)}^{p+1}-{v}_{x}$,$a=0$, arbitrary b, d, and β,$${\mathbf{X}}_{1},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{2},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{3},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{4,g,h},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{6};$$$$[{\mathbf{X}}_{1},{\mathbf{X}}_{6}]=2p{\mathbf{X}}_{1}\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{1.em}{0ex}}[{\mathbf{X}}_{2},{\mathbf{X}}_{6}]=p{\mathbf{X}}_{2}+{\mathbf{X}}_{4,-2c,0}\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{1.em}{0ex}}[{\mathbf{X}}_{3},{\mathbf{X}}_{6}]=2p{\mathbf{X}}_{3}\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{1.em}{0ex}}[{\mathbf{X}}_{4,g,h},{\mathbf{X}}_{6}]={\mathbf{X}}_{4,{g}_{2},{h}_{2}}\phantom{\rule{0.166667em}{0ex}},$$
- (iv)
- $f\left({v}_{x}\right)=\frac{1}{\alpha}ln\left(\alpha ({v}_{x}+c)\right)-{v}_{x}$,$a=0$, arbitrary b, d, and β,$${\mathbf{X}}_{1},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{2},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{3},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{4,g,h},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{7};$$$$[{\mathbf{X}}_{1},{\mathbf{X}}_{7}]=2{\mathbf{X}}_{1}\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{1.em}{0ex}}[{\mathbf{X}}_{2},{\mathbf{X}}_{7}]={\mathbf{X}}_{2}+{\mathbf{X}}_{4,2c,0}\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{1.em}{0ex}}[{\mathbf{X}}_{3},{\mathbf{X}}_{7}]=2{\mathbf{X}}_{3}\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{1.em}{0ex}}[{\mathbf{X}}_{4,g,h},{\mathbf{X}}_{7}]={\mathbf{X}}_{4,{g}_{3},{h}_{3}}\phantom{\rule{0.166667em}{0ex}},$$
- (v)
- $f\left({v}_{x}\right)=\alpha {({v}_{x}+c)}^{p+1}-{v}_{x}$,$b=0$,$d=0$, arbitrary a and β,$${\mathbf{X}}_{1},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{2},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{3},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{4,g,h},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{8};$$$$[{\mathbf{X}}_{1},{\mathbf{X}}_{8}]=p{\mathbf{X}}_{1}\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{1.em}{0ex}}[{\mathbf{X}}_{2},{\mathbf{X}}_{8}]={\mathbf{X}}_{4,-2c,0}\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{1.em}{0ex}}[{\mathbf{X}}_{3},{\mathbf{X}}_{8}]=p{\mathbf{X}}_{3}\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{1.em}{0ex}}[{\mathbf{X}}_{4,g,h},{\mathbf{X}}_{8}]={\mathbf{X}}_{4,{g}_{4},{h}_{4}}\phantom{\rule{0.166667em}{0ex}},$$
- (vi)
- $f\left({v}_{x}\right)=\alpha {e}^{p{v}_{x}}-{v}_{x}$,$b=0$,$d=0$, arbitrary a and β,$${\mathbf{X}}_{1},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{2},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{3},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{4,g,h},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{9};$$$$[{\mathbf{X}}_{1},{\mathbf{X}}_{9}]=p{\mathbf{X}}_{1}\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{1.em}{0ex}}[{\mathbf{X}}_{2},{\mathbf{X}}_{9}]={\mathbf{X}}_{4,-2,0}\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{1.em}{0ex}}[{\mathbf{X}}_{3},{\mathbf{X}}_{9}]=p{\mathbf{X}}_{3}\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{1.em}{0ex}}[{\mathbf{X}}_{4,g,h},{\mathbf{X}}_{9}]={\mathbf{X}}_{4,{g}_{5},{h}_{5}}\phantom{\rule{0.166667em}{0ex}},$$
- (vii)
- $f\left({v}_{x}\right)=\alpha {({v}_{x}+c)}^{p+1}-{v}_{x}$,$a=0$,$d=0$, arbitrary b and β,$${\mathbf{X}}_{1},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{2},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{3},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{4,g,h},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{5},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{6}.$$
- (viii)
- $f\left({v}_{x}\right)=\frac{1}{\alpha}ln\left(\alpha ({v}_{x}+c)\right)-{v}_{x}$,$a=0$,$d=0$, arbitrary b and β,$${\mathbf{X}}_{1},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{2},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{3},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{4,g,h},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{5},\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{7}.$$

#### 3.2. Contact Symmetries

**Theorem**

**3.**

## 4. Variational Symmetries

**Theorem**

**4.**

## 5. Conservation Laws

**Theorem**

**5.**

- (a)
- $a=0$, arbitrary$f\left({v}_{x}\right)$, b and β,$$\begin{array}{c}{T}_{5}=-{\textstyle \frac{1}{2}}by{v}_{xx}^{2}+{\textstyle \frac{1}{2}}y{v}_{t}^{2}+{\textstyle \frac{1}{2}}y{v}_{x}^{2}+\beta {\textstyle \frac{1}{2}}y{v}_{y}^{2}+\beta t{v}_{t}{v}_{y}+yF\left({v}_{x}\right),\hfill \\ {X}_{5}=-b\beta t{v}_{xxx}{v}_{y}-by{v}_{xxx}{v}_{t}+by{v}_{tx}{v}_{xx}+b\beta t{v}_{xx}{v}_{xy}-\beta t{v}_{y}f\left({v}_{x}\right)-y{v}_{t}f\left({v}_{x}\right)-\beta t{v}_{x}{v}_{y}-y{v}_{t}{v}_{x},\hfill \\ {Y}_{5}=-{\textstyle \frac{1}{2}}b\beta t{v}_{xx}^{2}-\beta y{v}_{t}{v}_{y}-{\textstyle \frac{1}{2}}\beta t{v}_{t}^{2}-{\textstyle \frac{1}{2}}\beta t{v}_{x}^{2}-{\textstyle \frac{1}{2}}{\beta}^{2}t{v}_{y}^{2}+\beta tF\left({v}_{x}\right),\hfill \end{array}$$
- (b)
- $f\left({v}_{x}\right)=\alpha {e}^{p{v}_{x}}-{v}_{x}$,$b=0$, arbitrary a and β,$$\begin{array}{c}{T}_{6}={\textstyle \frac{1}{4}}apt{v}_{tx}^{2}+{\textstyle \frac{1}{2}}apy{v}_{xx}{v}_{ty}+{\textstyle \frac{1}{2}}ap{v}_{xx}{v}_{t}+{\textstyle \frac{1}{4}}pt{{v}_{t}}^{2}+{\textstyle \frac{1}{4}}\beta pt{{v}_{y}}^{2}+{\textstyle \frac{1}{2}}py{v}_{y}{v}_{t}+x{v}_{t}+\alpha t{\textstyle \frac{1}{2}}{e}^{p{v}_{x}},\hfill \\ {X}_{6}=-{\textstyle \frac{1}{2}}a(pt{v}_{t}+py{v}_{y}+2x)({v}_{ttx}+\alpha {e}^{p{v}_{x}})+{\textstyle \frac{1}{2}}a(py{v}_{xy}+2a){v}_{tt}-{\textstyle \frac{1}{2}}a(p{v}_{t}+py{v}_{ty}){v}_{tx},\hfill \\ {Y}_{6}=-{\textstyle \frac{1}{2}}apy{v}_{tt}{v}_{xx}+{\textstyle \frac{1}{4}}apy{{v}_{tx}}^{2}-{\textstyle \frac{1}{2}}\beta pt{v}_{t}{v}_{y}-{\textstyle \frac{1}{4}}\beta py{{v}_{y}}^{2}-\beta x{v}_{y}-{\textstyle \frac{1}{4}}py{v}_{t}^{2}+{\textstyle \frac{1}{2}}\alpha y{e}^{p{v}_{x}}.\hfill \end{array}$$
- (c)
- $f\left({v}_{x}\right)=\alpha {({v}_{x}+c)}^{7/3}-{v}_{x}$,$a=0$, arbitrary b and β,$$\begin{array}{c}{T}_{7}=-2bt{v}_{xx}^{2}+2t{v}_{t}^{2}+2\beta t{v}_{y}^{2}+2x{v}_{t}{v}_{x}+4y{v}_{t}{v}_{y}+v{v}_{t}+3xc{v}_{t}+{\textstyle \frac{6}{5}}\alpha t{({v}_{x}+c)}^{10/3},\hfill \\ {X}_{7}=-b(3cx+4t{v}_{t}+2x{v}_{x}+4y{v}_{y}+bv){v}_{xxx}+b(4t{v}_{tx}+x{v}_{xx}+4y{v}_{xy}+3c+4{v}_{x}){v}_{xx}\hfill \\ \text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}-\alpha ({\textstyle \frac{12}{5}}cx+4t{v}_{t}+{\textstyle \frac{7}{5}}x{v}_{x}+4y{v}_{y}+v){({v}_{x}+c)}^{7/3}+\beta x{v}_{y}^{2}-x{v}_{t}^{2},\hfill \\ {Y}_{7}=-2by{v}_{xx}^{2}-3xc\beta {v}_{y}-4\beta t{v}_{t}{v}_{y}-2\beta x{v}_{x}{v}_{y}-2\beta y{v}_{y}^{2}-\beta v{v}_{y}-2y{v}_{t}^{2}+{\textstyle \frac{6}{5}}\alpha y{({v}_{x}+c)}^{10/3}.\hfill \end{array}$$

## 6. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Appendix A

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**MDPI and ACS Style**

Recio, E.; Garrido, T.M.; de la Rosa, R.; Bruzón, M.S.
Hamiltonian Structure, Symmetries and Conservation Laws for a Generalized (2 + 1)-Dimensional Double Dispersion Equation. *Symmetry* **2019**, *11*, 1031.
https://doi.org/10.3390/sym11081031

**AMA Style**

Recio E, Garrido TM, de la Rosa R, Bruzón MS.
Hamiltonian Structure, Symmetries and Conservation Laws for a Generalized (2 + 1)-Dimensional Double Dispersion Equation. *Symmetry*. 2019; 11(8):1031.
https://doi.org/10.3390/sym11081031

**Chicago/Turabian Style**

Recio, Elena, Tamara M. Garrido, Rafael de la Rosa, and María S. Bruzón.
2019. "Hamiltonian Structure, Symmetries and Conservation Laws for a Generalized (2 + 1)-Dimensional Double Dispersion Equation" *Symmetry* 11, no. 8: 1031.
https://doi.org/10.3390/sym11081031