# Steady Solitary and Periodic Waves in a Nonlinear Nonintegrable Lattice

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Structure and Equation of Motion of the Lattice

## 3. Transition from a Lattice Equation to a Continuous Equation

## 4. Checking the Correctness of the Transition to a Continuous Equation

## 5. Approximate Solution of the Original Lattice Equation, Case 1

**v**is an $M\times 1$ matrix of unknowns,

**L**is a nonsingular $M\times M$ real matrix, and $f:{\mathbb{R}}^{M}\to {\mathbb{R}}^{M}$ is a nonlinear function, which is homogeneous with respect to the components of vector

**v**.

## 6. Approximate Solution of the Original Lattice Equation, Case 2

## 7. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Solution (18) for $\tilde{V}=1,\phantom{\rule{0.277778em}{0ex}}\tilde{\mathsf{\Omega}}=-\frac{5}{6}$ and $S=1$.

**Figure 3.**Wave propagation in periodic lattice structure at $\delta =0$ for initial conditions determined by the exact solution (18). (

**a**) $h=\frac{1}{2}$; (

**b**) $h=2$.

**Figure 4.**Approximate solution of Equation (13), combining two exact solutions for ${c}_{2}=-0.9,k=1$.

**Figure 7.**(

**a**) Initial wave profile ${y}_{0n}$ (blue) and wave profile ${y}_{n}$ (red) after 50 oscillation periods; (

**b**) absolute error $\left|{y}_{n}-{y}_{0n}\right|$.

**Figure 8.**Periodic solution of Equation (27) for: (

**a**) $V=1.00429,\phantom{\rule{0.222222em}{0ex}}\mathsf{\Omega}=-0.231$; (

**b**) $V=6.33,\phantom{\rule{0.222222em}{0ex}}\mathsf{\Omega}=-0.231$.

**Figure 9.**Weakly nonlinear periodic wave in the lattice. (

**a**) Coincidence of the initial wave profile ${y}_{n}\left(0\right)$ and the profile ${y}_{n}\left(t\right)$ after $t=100$ oscillation periods; (

**b**) absolute error $\mathsf{\Delta}{y}_{n}={y}_{n}\left(t\right)-{y}_{n}\left(0\right)$.

**Figure 10.**Strongly nonlinear periodic lattice wave. (

**a**) coincidence of the initial wave profile ${y}_{n}\left(0\right)$ and the profile ${y}_{n}\left(t\right)$ after $t=10$ oscillation periods; (

**b**) absolute error $\mathsf{\Delta}{y}_{n}={y}_{n}\left(t\right)-{y}_{n}\left(0\right)$.

**Figure 11.**(

**a**) Solitary wave solution of Equation (35) and (

**b**) corresponding wave in the original lattice for $V=2,\phantom{\rule{0.222222em}{0ex}}\mathsf{\Omega}=-0.231$.

**Figure 12.**Solitary wave in the original lattice for $V=-1.75,\phantom{\rule{0.222222em}{0ex}}\mathsf{\Omega}=-0.231$.

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**MDPI and ACS Style**

Andrianov, I.; Zemlyanukhin, A.; Bochkarev, A.; Erofeev, V.
Steady Solitary and Periodic Waves in a Nonlinear Nonintegrable Lattice. *Symmetry* **2020**, *12*, 1608.
https://doi.org/10.3390/sym12101608

**AMA Style**

Andrianov I, Zemlyanukhin A, Bochkarev A, Erofeev V.
Steady Solitary and Periodic Waves in a Nonlinear Nonintegrable Lattice. *Symmetry*. 2020; 12(10):1608.
https://doi.org/10.3390/sym12101608

**Chicago/Turabian Style**

Andrianov, Igor, Aleksandr Zemlyanukhin, Andrey Bochkarev, and Vladimir Erofeev.
2020. "Steady Solitary and Periodic Waves in a Nonlinear Nonintegrable Lattice" *Symmetry* 12, no. 10: 1608.
https://doi.org/10.3390/sym12101608