# The Cauchy Problem for the Generalized Hyperbolic Novikov–Veselov Equation via the Moutard Symmetries

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## Abstract

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## 1. Introduction

## 2. The Moutard Transformation

## 3. The Cauchy Problem for the Novikov–Veselov Equation

- Choose a differentiable function $T\left(t\right)$ and four numbers $\alpha $, $\beta $, $\gamma $ and $\delta $ that satisfy the condition,$$-2\frac{\gamma \xb7\delta}{{(\alpha +\beta )}^{2}}=C.$$
- Find two support function $\varphi \left(x\right)$ and $\Phi \left(y\right)$ via the formulas$$\begin{array}{cc}\hfill \varphi \left(x\right)& {\displaystyle =\frac{2\delta}{\underset{0}{\overset{x}{\int}}{A}_{1}\left(\xi \right)d\xi +\frac{2\delta}{\alpha +\beta}}-\beta}\hfill \\ \hfill \Phi \left(y\right)& {\displaystyle =\frac{2\gamma}{\underset{0}{\overset{y}{\int}}{B}_{1}\left(\zeta \right)d\zeta +\frac{2\gamma}{\alpha +\beta}}-\alpha .}\hfill \end{array}$$
- Substitute $\varphi \left(x\right)$ and $\Phi \left(y\right)$ into the equations$$\begin{array}{cc}\hfill A(x,t)& =\underset{0}{\overset{t}{\int}}\frac{T\left(\tau \right)}{\sqrt[3]{3(\tau -t)}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{Ai}\left(\frac{x}{\sqrt[3]{3(\tau -t)}}\right)d\tau +\frac{1}{\sqrt{2\pi}\sqrt[3]{3t}}\underset{-\infty}{\overset{\infty}{\int}}\varphi \left(\xi \right)\mathrm{Ai}\left(\frac{\xi -x}{\sqrt[3]{3t}}\right)d\xi \hfill \\ \hfill B(y,t)& =-\underset{0}{\overset{t}{\int}}\frac{T\left(\tau \right)}{\sqrt[3]{3(\tau -t)}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{Ai}\left(\frac{y}{\sqrt[3]{3(\tau -t)}}\right)d\tau +\frac{1}{\sqrt{2\pi}\sqrt[3]{3t}}\underset{-\infty}{\overset{\infty}{\int}}\Phi \left(\eta \right)\mathrm{Ai}\left(\frac{\eta -y}{\sqrt[3]{3t}}\right)d\eta .\hfill \end{array}$$
- Substitute the new functions $A(x,t)$ and $B(y,t)$ into the equation$$u=-2\frac{{\partial}_{x}A\xb7{\partial}_{y}B}{{(A+B)}^{2}}.$$

## 4. Generalization of the Method: The Higher-Order Equations

**Case 1: Odd n**. Let $n=2m+1$, where $m\ge 0$. Following our previous discussion, let us consider the special case $u\equiv 0$. Then the system (44) turns into

**Case 2: Even n.**Let $n=2m$, where $m\ge 0$. This time let us utilize not a Fourier but a Laplace transform:

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The dromion-type Cauchy problem (42) for ${u}_{0}(x,y)$ with ${x}_{0}={y}_{0}=-1$. There are two equipotential lines at $y-{y}_{0}=\pm (x-{x}_{0})$.

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Yurova, A.A.; Yurov, A.V.; Yurov, V.A.
The Cauchy Problem for the Generalized Hyperbolic Novikov–Veselov Equation via the Moutard Symmetries. *Symmetry* **2020**, *12*, 2113.
https://doi.org/10.3390/sym12122113

**AMA Style**

Yurova AA, Yurov AV, Yurov VA.
The Cauchy Problem for the Generalized Hyperbolic Novikov–Veselov Equation via the Moutard Symmetries. *Symmetry*. 2020; 12(12):2113.
https://doi.org/10.3390/sym12122113

**Chicago/Turabian Style**

Yurova, Alla A., Artyom V. Yurov, and Valerian A. Yurov.
2020. "The Cauchy Problem for the Generalized Hyperbolic Novikov–Veselov Equation via the Moutard Symmetries" *Symmetry* 12, no. 12: 2113.
https://doi.org/10.3390/sym12122113