Periodic solutions to symmetric Newtonian systems in neighborhoods of orbits of equilibria

Anna Gołȩbiewska; Marta Kowalczyk; Sławomir Rybicki; Piotr Stefaniak; Anna Gołȩbiewska; Marta Kowalczyk; Sławomir Rybicki; Piotr Stefaniak

doi:10.3934/era.2022085

Electronic Research Archive

2022, Volume 30, Issue 5: 1691-1707. doi: 10.3934/era.2022085

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Periodic solutions to symmetric Newtonian systems in neighborhoods of orbits of equilibria

Faculty of Mathematics and Computer Science, Nicolaus Copernicus University in Toruń, PL-87-100 Toruń, ul. Chopina 12/18, Poland

Received: 13 August 2021 Revised: 15 January 2022 Accepted: 15 January 2022 Published: 25 March 2022

The aim of this paper is to prove the existence of periodic solutions to symmetric Newtonian systems in any neighborhood of an isolated orbit of equilibria. Applying equivariant bifurcation techniques we obtain a generalization of the classical Lyapunov center theorem to the case of symmetric potentials with orbits of non-isolated critical points. Our tool is an equivariant version of the Conley index. To compare the indices we compute cohomological dimensions of some orbit spaces.

Keywords:

Citation: Anna Gołȩbiewska, Marta Kowalczyk, Sławomir Rybicki, Piotr Stefaniak. Periodic solutions to symmetric Newtonian systems in neighborhoods of orbits of equilibria[J]. Electronic Research Archive, 2022, 30(5): 1691-1707. doi: 10.3934/era.2022085

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Abstract

1. Introduction

A key practice in nonlinear physics has been the study of both the formation and interaction of localized waves, including plasma physics, fluid dynamics, Bose-Einstein condensates and photonics, over the last few decades. Among different types of nonlinaer localized waves, solitons are the most representative and ideal testbed to investigate nonlinear wave interactions due to their intrinsic particle-like properties during propagation ^[1,2]. The position and phase shift interaction-induced displacement is typically independent of the relative phases of the solitons in the envelope. The dynamics of collisions in the region of interaction, however, depend strongly on the relative levels. The wave magnitude then evolves at the central collision point from the sum of the amplitudes of the two solitons to their distinction, respectively. A wide range of theoretical theories, computational simulations and experimental observations have already been recorded for such interactions with solitons and their possible synchronization ^[3,4,5]. In general, nonlinear Evolution Equations (NLEEs) are used to model social processes and structures of natural phenomena. In various fields of science and engineering, nonlinear wave phenomena exist, including fluid mechanics, chemical dynamics, plasma physics, solid-state physics, nonlinear optics, and population models. In nonlinear science, the precise wave outcomes of NLEEs play a crucial role, since they give us a lot of insight into the physical characteristics of the problem and can provide more physical details to support additional applications. In recent years, people have been fascinated by the precise and numerical solutions of NLEEs since they are increasingly becoming more used in various scientific fields to explain complex nonlinear phenomena. Via differential equations, some real-world problems are converted into mathematical equations.

The quest for soliton outcomes has been of significant importance in the understanding of nonlinear phenomena in recent years. Soliton's theory has been strengthened in recent decades in order to explain the meaning of impossibility in ordinary and partial differential equations ^{[6,7,8,9,10,11,12,13,14]}. The nonlinear Schrödinger equation (NLSEs) has been studied in various equations to demonstrate Soliton propagating molecules in an optical fiber, which has a variety of physical applications especially in plasma physics and nonlinear optics. There are several variants of the NLSEs; these forms, such as Kaup-Newell, Chen-Lee-Liu, and Gerdjikov-Ivanov equations, are the recently determined forms. The equation Kaup-Newell ^[15] has various forms. The equations can be modified to each other by Guage transformation ^[16]. Transformations can not sustain the reduction in the dispersion issue ^{[17,18,19,20]}. Conditions and integrations are highly dynamic and cannot be precisely planned. They need impartial investigation, therefore. In order to overcome these categorized NLSEs, a large number of numerical and exact moving wave mathematical schemes have been identified ^{[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43]}.

In this paper we consider the Hamiltonian amplitude equation with the M-truncated order " $\mu$ " given by ^[44]

$\begin{equation} i \mathscr{D}_{M, x}^{\mu, \zeta}\chi+\mathscr{D}_{M, t}^{2\mu, \zeta}\chi-2\sigma |\chi|^{2}\chi-\delta \mathscr{D}_{M, t}^{\mu, \zeta}\mathscr{D}_{M, x}^{\mu, \zeta}\chi_{xt} = 0,\;\;\;0 < \mu\le 1, \end{equation}$

(1.1)

where $\sigma\pm 1$ is the coefficient of the nonlinear term, $\delta < 1$ is the coefficients of the dispersionless term, $\zeta$ is a nonzero real number, and $\mathscr{D}_{M, *}^{\mu, \zeta}\chi$ represents the truncated $M$ -fractional derivative of $\chi$ with respect to $x$ and/or $t$ . Non-linearity arises when the change of the output is not proportional to the change of the input. Dispersion means that waves of different wavelength propagate at different phase velocities. The phase velocity of a wave is the rate at which the wave propagates in some medium. This is the velocity at which the phase of any one frequency component of the wave travels. Studies have shown that when the dispersion effect and nonlinear effect of the medium reach a stable equilibrium, the pulse can maintain its shape and velocity in the form of solitons during the transmission process ^[45].

Equation (1.1) was introduced in ^[46]. The term $\chi(x, t)$ is the unknown complex envelope function of $x$ and $t$ . This equation governs certain instabilities of modulated wave trains; the addition of the term $-\delta\chi_{xt}$ overcomes the ill-posedness of the unstable NLSEs. The equation is apparently not integrable, but a Hamiltonian analogue of the Kuramoto–Sivashinsky equation, which arises in dissipative system.

2. Applications

Here, we acquire the optical solitons of the Hamiltonian amplitude equation with M-truncated derivative by using the extended sinh-Gordon equation expansion method ^[47] and the and the extended rational sine-cosine/sinh-cosh methods ^[48].

Consider the following fractional wave transformation:

$\begin{equation} \begin{split} \chi(x,t) = \phi(\xi)e^{i\varphi},\;\;\xi = \frac{\eta \Gamma (\zeta +1) \left(q t^{\mu }+P x^{\mu }\right)}{\mu },\;\;\varphi = \frac{\Gamma (\zeta +1) \left(r x^{\mu }+\rho t^{\mu }\right)}{\mu }. \end{split} \end{equation}$

(2.1)

Plugging Eq (2.1) into (1.1), provides the following nonlinear ordinary differential equation

$\begin{equation} \begin{split} \eta ^2 q (\delta P-q)\phi''+2 \sigma\phi^{3}+(\rho ^2-\delta \rho r+r)\phi = 0 \end{split} \end{equation}$

(2.2)

from the real part, and the constraint condition

$\begin{equation} \begin{split} q = \frac{\delta \rho P-P}{2 \rho -\delta r}. \end{split} \end{equation}$

(2.3)

from the imaginary part.

2.1. Application of the extended sinh-Gordon equation expansion method

Here, we present the application of the extended sinh-Gordon equation expansion method to (1.1).

Balancing the terms $\phi''$ and $\phi^3$ , we get $n = 1$ . With $n = 1$ , we have the following test functions ^[47]:

$\begin{equation} \phi(\theta) = \gamma_{1}\sinh(\theta)+\lambda_{1}\cosh(\theta)+\lambda_{0}, \end{equation}$

(2.4)

$\begin{equation} \phi(\xi) = \pm \gamma_{1}\;i\;\rm{sech}(\xi)\pm \lambda_{1}\tanh(\xi)+\lambda_{0}, \end{equation}$

(2.5)

and

$\begin{equation} \phi(\xi) = \pm \gamma_{1}\;\rm{csch}(\xi)\pm \lambda_{1}\coth(\xi)+\lambda_{0}. \end{equation}$

(2.6)

Plugging Eq (2.4) and its second derivative alongside $\theta' = \sinh(\theta)$ into Eq (2.2), provides an equation with hyperbolic functions power. We collect a set of algebraic equations by equating each summation of the coefficients of the hyperbolic functions of the same power to zero. We simplify the set of algebraic equations and secured the values of the parameters involved. To reach the solutions of Eq (1.1), we substitute the values of the parameters into any of Eqs (2.5) and (2.6).

Ⅰ: For

$\lambda_0 = 0,\;\lambda_1 = -\frac{\sqrt{-\left(\rho ^2-\delta \rho r+r\right)}}{\sqrt{2} \sqrt{\sigma }},\;\gamma_1 = -\frac{\sqrt{-\left(\rho ^2-\delta \rho r+r\right)}}{\sqrt{2} \sqrt{\sigma }},\;\eta = -\frac{\sqrt{-2 \rho ^2+2 \delta \rho r-2 r}}{\sqrt{q (q-\delta P)}},$

we reveal the combined dark-bright soliton

$\begin{equation} \begin{split} \chi_{1.1}(x,t) = \frac{1}{2} \sqrt{\frac{|-2 \rho ^2+2 \delta \rho r-2 r|}{\sigma }} \Bigg(\tanh \left(\frac{\Gamma (\zeta +1) \sqrt{|-2 \rho ^2+2 \delta \rho r-2 r|} \left(\frac{t^{\mu } (\delta \rho P-P)}{2 \rho -\delta r}+P x^{\mu }\right)}{\mu \sqrt{\Big|\frac{(\delta \rho P-P) \left(\frac{\delta \rho P-P}{2 \rho -\delta r}-\delta P\right)}{2 \rho -\delta r}\Big|}}\right)\\-i \rm{sech}\left(\frac{\sqrt{|-2 \rho ^2+2 \delta \rho r-2 r|} \left(\Gamma (\zeta +1) \left(\frac{t^{\mu } (\delta \rho P-P)}{2 \rho -\delta r}+P x^{\mu }\right)\right)}{\mu \sqrt{\Big|\frac{(\delta \rho P-P) \left(\frac{\delta \rho P-P}{2 \rho -\delta r}-\delta P\right)}{2 \rho -\delta r}\Big|}}\right)\Bigg)e^{\frac{i \Gamma (\zeta +1) \left(r x^{\mu }+\rho t^{\mu }\right)}{\mu }} \end{split} \end{equation}$

(2.7)

and the combined singular soliton

$\begin{equation} \begin{split} \chi_{1.2}(x,t) = \frac{1}{2} \left(-\sqrt{\frac{|-2 \rho ^2+2 \delta \rho r-2 r|}{\sigma }}\right) \Bigg(-\coth \left(\frac{\Gamma (\zeta +1) \sqrt{|-2 \rho ^2+2 \delta \rho r-2 r|} \left(\frac{t^{\mu } (\delta \rho P-P)}{2 \rho -\delta r}+P x^{\mu }\right)}{\mu \sqrt{\Big|\frac{(\delta \rho P-P) \left(\frac{\delta \rho P-P}{2 \rho -\delta r}-\delta P\right)}{2 \rho -\delta r}\Big|}}\right)\\-\rm{csch}\left(\frac{\Gamma (\zeta +1) \sqrt{|-2 \rho ^2+2 \delta \rho r-2 r|} \left(\frac{t^{\mu } (\delta \rho P-P)}{2 \rho -\delta r}+P x^{\mu }\right)}{\mu \sqrt{\Big|\frac{(\delta \rho P-P) \left(\frac{\delta \rho P-P}{2 \rho -\delta r}-\delta P\right)}{2 \rho -\delta r}\Big|}}\right)\Bigg)e^{\frac{i \Gamma (\zeta +1) \left(r x^{\mu }+\rho t^{\mu }\right)}{\mu }}. \end{split} \end{equation}$

(2.8)

Ⅱ: For

$\lambda_0 = 0,\;\lambda_1 = 0,\;\gamma_1 = -\frac{\sqrt{\rho ^2-\delta \rho r+r}}{\sqrt{\sigma }},\;\eta = -\frac{\sqrt{\rho ^2-\delta \rho r+r}}{\sqrt{q (q-\delta P)}},$

we reveal the bright soliton

$\begin{equation} \begin{split} \chi_{2.1}(x,t) = -\sqrt{\frac{-\rho ^2+\delta \rho r-r}{\sigma }} \rm{sech}\left(\frac{\Gamma (\zeta +1) \sqrt{|\rho ^2-\delta \rho r+r|} \left(\frac{t^{\mu } (\delta \rho P-P)}{2 \rho -\delta r}+P x^{\mu }\right)}{\mu \sqrt{\Big|\frac{(\delta \rho P-P) \left(\frac{\delta \rho P-P}{2 \rho -\delta r}-\delta P\right)}{2 \rho -\delta r}\Big|}}\right)e^{\frac{i \Gamma (\zeta +1) \left(r x^{\mu }+\rho t^{\mu }\right)}{\mu }} \end{split} \end{equation}$

(2.9)

and the singular soliton

$\begin{equation} \begin{split} \chi_{2.2}(x,t) = \sqrt{\frac{\rho ^2-\delta \rho r+r}{\sigma }} \rm{csch}\left(\frac{\Gamma (\zeta +1) \sqrt{|\rho ^2-\delta \rho r+r|} \left(\frac{t^{\mu } (\delta \rho P-P)}{2 \rho -\delta r}+P x^{\mu }\right)}{\mu \sqrt{\Big|\frac{(\delta \rho P-P) \left(\frac{\delta \rho P-P}{2 \rho -\delta r}-\delta P\right)}{2 \rho -\delta r}\Big|}}\right)e^{\frac{i \Gamma (\zeta +1) \left(r x^{\mu }+\rho t^{\mu }\right)}{\mu }}. \end{split} \end{equation}$

(2.10)

Ⅲ: For

$\lambda_0 = 0,\;\lambda_1 = \frac{i \sqrt{\rho ^2-\delta \rho r+r}}{\sqrt{2} \sqrt{\sigma }},\;;\gamma_1 = 0;\eta = \frac{\sqrt{\rho ^2-\delta \rho r+r}}{\sqrt{2 \delta q P-2 q^2}},$

we get the dark soliton

$\begin{equation} \begin{split} \chi_{3.1}(x,t) = \sqrt{-\frac{\rho ^2-\delta \rho r+r}{2 \sigma }} \tanh \left(\frac{\Gamma (\zeta +1) \sqrt{|\rho ^2-\delta \rho r+r|} \left(\frac{t^{\mu } (\delta \rho P-P)}{2 \rho -\delta r}+P x^{\mu }\right)}{\mu \sqrt{\Big|\frac{2 \delta P (\delta \rho P-P)}{2 \rho -\delta r}-\frac{2 (\delta \rho P-P)^2}{(2 \rho -\delta r)^2}\Big|}}\right)e^{\frac{i \Gamma (\zeta +1) \left(r x^{\mu }+\rho t^{\mu }\right)}{\mu }} \end{split} \end{equation}$

(2.11)

and the singular soliton

$\begin{equation} \begin{split} \chi_{3.2}(x,t) = \sqrt{-\frac{\rho ^2-\delta \rho r+r}{2 \sigma }} \coth \left(\frac{\Gamma (\zeta +1) \sqrt{|\rho ^2-\delta \rho r+r|} \left(\frac{t^{\mu } (\delta \rho P-P)}{2 \rho -\delta r}+P x^{\mu }\right)}{\mu \sqrt{\Big|\frac{2 \delta P (\delta \rho P-P)}{2 \rho -\delta r}-\frac{2 (\delta \rho P-P)^2}{(2 \rho -\delta r)^2}\Big|}}\right)e^{\frac{i \Gamma (\zeta +1) \left(r x^{\mu }+\rho t^{\mu }\right)}{\mu }}. \end{split} \end{equation}$

(2.12)

2.2. Application of the extended rational sine-cosine/sinh-cosh methods

Here, we present the application of the extended rational sine-cosine/sinh-cosh methods to (1.1).

Consider the following as the first test function to Eq (2.2):

$\begin{equation} \begin{split} \phi(\xi) = \frac{\lambda_{0}\sin( \xi)}{\lambda_{2}+\lambda_{1}\cos(\xi)}. \end{split} \end{equation}$

(2.13)

Putting Eq (2.13) into Eq (2.2), provides a polynomial functions. Setting the coefficients of the powers of $\cos(\xi)$ gives a system of algebraic equations. Solving the set of algebraic equations provides the following set of values to the unknown coefficients:

Ⅰ: For

$\lambda_0 = -\frac{i \lambda_1 \sqrt{r (\delta \rho -1)-\rho ^2}}{\sqrt{2} \sqrt{\sigma }},\;\lambda_2 = 0,\;\eta = -\frac{\sqrt{\rho ^2-\delta \rho r+r}}{\sqrt{2} \sqrt{q (q-\delta P)}},$

we reveal the singular periodic wave solution

$\begin{equation} \begin{split} \chi_{I}(x,t) = \sqrt{-\frac{r (\delta \rho -1)-\rho ^2}{2 \sigma }} \tan \left(\frac{\Gamma (\zeta +1) \sqrt{|\rho ^2-\delta \rho r+r|} \left(\frac{t^{\mu } (\delta \rho P-P)}{2 \rho -\delta r}+P x^{\mu }\right)}{\sqrt{2} \mu \sqrt{\Big|\frac{(\delta \rho P-P) \left(\frac{\delta \rho P-P}{2 \rho -\delta r}-\delta P\right)}{2 \rho -\delta r}\Big|}}\right)e^{\frac{i \Gamma (\zeta +1) \left(r x^{\mu }+\rho t^{\mu }\right)}{\mu }}. \end{split} \end{equation}$

(2.14)

Ⅱ: For

$\lambda_0 = \frac{i \lambda_1 \sqrt{r (\delta \rho -1)-\rho ^2}}{\sqrt{2} \sqrt{\sigma }},\;\lambda_2 = \lambda_1,\;\eta = -\frac{\sqrt{2} \sqrt{\rho ^2-\delta \rho r+r}}{\sqrt{q (q-\delta P)}},$

we reveal the singular periodic wave solution

$\begin{equation} \begin{split} \chi_{II}(x,t) = -\frac{\sqrt{-\frac{r (\delta \rho -1)-\rho ^2}{2 \sigma }} \sin \left(\frac{\sqrt{2} \Gamma (\zeta +1) \sqrt{|\rho ^2-\delta \rho r+r|} \left(\frac{t^{\mu } (\delta \rho P-P)}{2 \rho -\delta r}+P x^{\mu }\right)}{\mu \sqrt{\Big|\frac{(\delta \rho P-P) \left(\frac{\delta \rho P-P}{2 \rho -\delta r}-\delta P\right)}{2 \rho -\delta r}\Big|}}\right)}{\cos \left(\frac{\sqrt{2} \Gamma (\zeta +1) \sqrt{|\rho ^2-\delta \rho r+r|} \left(\frac{t^{\mu } (\delta \rho P-P)}{2 \rho -\delta r}+P x^{\mu }\right)}{\mu \sqrt{\Big|\frac{(\delta \rho P-P) \left(\frac{\delta \rho P-P}{2 \rho -\delta r}-\delta P\right)}{2 \rho -\delta r}\Big|}}\right)+1}e^{\frac{i \Gamma (\zeta +1) \left(r x^{\mu }+\rho t^{\mu }\right)}{\mu }}. \end{split} \end{equation}$

(2.15)

Consider the following as the second test function to Eq (2.2):

$\begin{equation} \begin{split} \phi(\xi) = \frac{\lambda_{0}\cos(\xi)}{\lambda_{2}+\lambda_{1}\sin(\xi)}. \end{split} \end{equation}$

(2.16)

Putting Eq (2.16) into Eq (2.2), provides a polynomial functions. Setting the coefficients of the powers of $\sin(\xi)$ gives a system of algebraic equations. Solving the set of algebraic equations provides the following set of values to the unknown coefficients:

Ⅰ: For

$\lambda_0 = -\frac{i \lambda_1 \sqrt{r (\delta \rho -1)-\rho ^2}}{\sqrt{2} \sqrt{\sigma }},\;\lambda_2 = 0,\;\eta = -\frac{\sqrt{\rho ^2-\delta \rho r+r}}{\sqrt{2} \sqrt{q (q-\delta P)}},$

we reveal the singular periodic wave solution

$\begin{equation} \begin{split} \chi_{I}(x,t) = \sqrt{-\frac{r (\delta \rho -1)-\rho ^2}{\sigma }} \cot \left(\frac{\Gamma (\zeta +1) \sqrt{|\rho ^2-\delta \rho r+r|} \left(\frac{t^{\mu } (\delta \rho P-P)}{2 \rho -\delta r}+P x^{\mu }\right)}{\sqrt{2} \mu \sqrt{\Big|\frac{(\delta \rho P-P) \left(\frac{\delta \rho P-P}{2 \rho -\delta r}-\delta P\right)}{2 \rho -\delta r}\Big|}}\right)e^{\frac{i \Gamma (\zeta +1) \left(r x^{\mu }+\rho t^{\mu }\right)}{\mu }}. \end{split} \end{equation}$

(2.17)

Ⅱ: For

$\lambda_0 = -\frac{i \lambda_1 \sqrt{r (\delta \rho -1)-\rho ^2}}{\sqrt{2} \sqrt{\sigma }},\;\lambda_2 = \lambda_1,\;\eta = -\frac{\sqrt{2} \sqrt{\rho ^2-\delta \rho r+r}}{\sqrt{q (q-\delta P)}},$

we reveal the singular periodic wave solution

$\begin{equation} \begin{split} \chi_{II}(x,t) = -\frac{\sqrt{-\frac{r (\delta \rho -1)-\rho ^2}{2 \sigma }} \cos \left(\frac{\sqrt{2} \Gamma (\zeta +1) \sqrt{|\rho ^2-\delta \rho r+r|} \left(\frac{t^{\mu } (\delta \rho P-P)}{2 \rho -\delta r}+P x^{\mu }\right)}{\mu \sqrt{\Big|\frac{(\delta \rho P-P) \left(\frac{\delta \rho P-P}{2 \rho -\delta r}-\delta P\right)}{2 \rho -\delta r}\Big|}}\right)}{1-\sin \left(\frac{\sqrt{2} \Gamma (\zeta +1) \sqrt{|\rho ^2-\delta \rho r+r|} \left(\frac{t^{\mu } (\delta \rho P-P)}{2 \rho -\delta r}+P x^{\mu }\right)}{\mu \sqrt{\Big|\frac{(\delta \rho P-P) \left(\frac{\delta \rho P-P}{2 \rho -\delta r}-\delta P\right)}{2 \rho -\delta r}\Big|}}\right)}e^{\frac{i \Gamma (\zeta +1) \left(r x^{\mu }+\rho t^{\mu }\right)}{\mu }}. \end{split} \end{equation}$

(2.18)

Consider the following as the third test function to Eq (2.2):

$\begin{equation} \begin{split} \phi(\xi) = \frac{\lambda_{0}\sinh(\xi)}{\lambda_{2}+\lambda_{1}\cosh(\xi)}. \end{split} \end{equation}$

(2.19)

Putting Eq (2.19) into Eq (2.2), provides a polynomial functions. Setting the coefficients of the powers of $\cosh(\xi)$ gives a system of algebraic equations. Solving the set of algebraic equations provides the following set of values to the unknown coefficients:

Ⅰ: For

$\lambda_0 = \frac{\lambda_1 \sqrt{r (\delta \rho -1)-\rho ^2}}{\sqrt{2} \sqrt{\sigma }},\;\lambda_2 = 0,\;\eta = -\frac{\sqrt{\rho ^2-\delta \rho r+r}}{\sqrt{2 \delta q P-2 q^2}},$

we reveal the dark soliton

$\begin{equation} \begin{split} \chi_{I}(x,t) = -\sqrt{\Big|\frac{r (\delta \rho -1)-\rho ^2}{2 \sigma }\Big|} \tanh \left(\frac{\Gamma (\zeta +1) \sqrt{|\rho ^2-\delta \rho r+r|} \left(\frac{t^{\mu } (\delta \rho P-P)}{2 \rho -\delta r}+P x^{\mu }\right)}{\mu \sqrt{\Big|\frac{2 \delta P (\delta \rho P-P)}{2 \rho -\delta r}-\frac{2 (\delta \rho P-P)^2}{(2 \rho -\delta r)^2}\Big|}}\right)e^{\frac{i \Gamma (\zeta +1) \left(r x^{\mu }+\rho t^{\mu }\right)}{\mu }}. \end{split} \end{equation}$

(2.20)

Ⅱ: For

$\lambda_0 = -\frac{\lambda_1 \sqrt{r (\delta \rho -1)-\rho ^2}}{\sqrt{2} \sqrt{\sigma }},\;\lambda_2 = \lambda_1;\eta = -\frac{\sqrt{-2 \rho ^2+2 \delta \rho r-2 r}}{\sqrt{q (q-\delta P)}},$

we reveal the periodic wave solution

$\begin{equation} \begin{split} \chi_{II}(x,t) = \frac{\sqrt{\frac{r (\delta \rho -1)-\rho ^2}{2 \sigma }} \sinh \left(\frac{\Gamma (\zeta +1) \sqrt{|-2 \rho ^2+2 \delta \rho r-2 r|} \left(\frac{t^{\mu } (\delta \rho P-P)}{2 \rho -\delta r}+P x^{\mu }\right)}{\mu \sqrt{\Big|\frac{(\delta \rho P-P) \left(\frac{\delta \rho P-P}{2 \rho -\delta r}-\delta P\right)}{2 \rho -\delta r}\Big|}}\right)}{\cosh \left(\frac{\Gamma (\zeta +1) \sqrt{|-2 \rho ^2+2 \delta \rho r-2 r|} \left(\frac{t^{\mu } (\delta \rho P-P)}{2 \rho -\delta r}+P x^{\mu }\right)}{\mu \sqrt{\Big|\frac{(\delta \rho P-P) \left(\frac{\delta \rho P-P}{2 \rho -\delta r}-\delta P\right)}{2 \rho -\delta r}\Big|}}\right)+1}e^{\frac{i \Gamma (\zeta +1) \left(r x^{\mu }+\rho t^{\mu }\right)}{\mu }}. \end{split} \end{equation}$

(2.21)

Consider the following as the fourth test function to Eq (2.2):

$\begin{equation} \begin{split} \phi(\xi) = \frac{\lambda_{0}\cosh(\xi)}{\lambda_{2}+\lambda_{1}\sinh(\xi)}. \end{split} \end{equation}$

(2.22)

Putting Eq (2.22) into Eq (2.2), provides a polynomial functions. Setting the coefficients of the powers of $\sinh(\xi)$ gives a system of algebraic equations. Solving the set of algebraic equations provides the following set of values to the unknown coefficients:

Ⅰ: For

$\lambda_0 = -\frac{\lambda_1 \sqrt{r (\delta \rho -1)-\rho ^2}}{\sqrt{2} \sqrt{\sigma }},\;\lambda_2 = 0;\eta = -\frac{\sqrt{\rho ^2-\delta \rho r+r}}{\sqrt{2 \delta q P-2 q^2}},$

we reveal the singular soliton

$\begin{equation} \begin{split} \chi_{I}(x,t) = \sqrt{\frac{r (\delta \rho -1)-\rho ^2}{2 \sigma }} \coth \left(\frac{\Gamma (\zeta +1) \sqrt{|\rho ^2-\delta \rho r+r|} \left(\frac{t^{\mu } (\delta \rho P-P)}{2 \rho -\delta r}+P x^{\mu }\right)}{\mu \sqrt{\Big|\frac{2 \delta P (\delta \rho P-P)}{2 \rho -\delta r}-\frac{2 (\delta \rho P-P)^2}{(2 \rho -\delta r)^2}\Big|}}\right)e^{\frac{i \Gamma (\zeta +1) \left(r x^{\mu }+\rho t^{\mu }\right)}{\mu }}. \end{split} \end{equation}$

(2.23)

Ⅱ: For

$\lambda_0 = \frac{\lambda_1 \sqrt{r (\delta \rho -1)-\rho ^2}}{\sqrt{2} \sqrt{\sigma }},\;\lambda_2 = -i \lambda_1,\;\eta = -\frac{\sqrt{-2 \rho ^2+2 \delta \rho r-2 r}}{\sqrt{q (q-\delta P)}},$

we reveal the periodic wave solution

$\begin{equation} \begin{split} \chi_{II}(x,t) = \frac{\sqrt{\frac{r (\delta \rho -1)-\rho ^2}{2 \sigma }} \cosh \left(\frac{\Gamma (\zeta +1) \sqrt{|-2 \rho ^2+2 \delta \rho r-2 r|} \left(\frac{t^{\mu } (\delta \rho P-P)}{2 \rho -\delta r}+P x^{\mu }\right)}{\mu \sqrt{\Big|\frac{(\delta \rho P-P) \left(\frac{\delta \rho P-P}{2 \rho -\delta r}-\delta P\right)}{2 \rho -\delta r}\Big|}}\right)}{-\sinh \left(\frac{\Gamma (\zeta +1) \sqrt{|-2 \rho ^2+2 \delta \rho r-2 r|} \left(\frac{t^{\mu } (\delta \rho P-P)}{2 \rho -\delta r}+P x^{\mu }\right)}{\mu \sqrt{\Big|\frac{(\delta \rho P-P) \left(\frac{\delta \rho P-P}{2 \rho -\delta r}-\delta P\right)}{2 \rho -\delta r}\Big|}}\right)-i}e^{\frac{i \Gamma (\zeta +1) \left(r x^{\mu }+\rho t^{\mu }\right)}{\mu }}. \end{split} \end{equation}$

(2.24)

3. Physical interpretation

As it was presented in the preceding section, two analytical approaches were used in securing the optical solitons to the fractional Hamiltonian amplitude equation. To have a clear physical view to the features of the reported results in this study, numerical simulations are performed via the 3-dimensional and contour graphs by a careful choice of the parameters' values involved. The dynamics of the reported solutions is analyze based on the choice of the fractional values $\mu$ .

displays the shape of bright-soliton at $\mu = 1$ . When $\mu = 0.7$ , solution (2.9) maintains its dark-soliton shape with shork-shock, and changes to short-shock waves at $\mu = 0.5$ . displays the shape of dark-soliton at $\mu = 1$ . When $\mu = 0.9$ , solution (2.11) maintains its bright-soliton shape with shork-shock, and changes to short-shock waves at $\mu = 0.5$ . portrays the shock-periodic wave shape at three different fractional values of $\mu$ .

Figure 1. The 3-dimensional and contour graphs of solution (2.9) at

$r = 1, \; \delta = 0.9, \; \rho = 0.62, \; P = 0.86, \; \sigma = -1, \; \zeta = 3.57$ , and different fractional values of

$\mu$ , for (a, d)

$\mu = 1$ , for (b, c)

$\mu = 0.7$ , for (c, f)

$\mu = 0.5$ .

DownLoad: Full-Size Img PowerPoint

Figure 2. The 3-dimensional and contour graphs of solution (2.11) at

$r = 1.71, \; \delta = 0.5, \; \rho = -0.54, \; P = 0.44, \; \sigma = 1, \; \zeta = 1.33$ , and different fractional values of

$\mu$ , for (a, d)

$\mu = 1$ , for (b, c)

$\mu = 0.9$ , for (c, f)

$\mu = 0.5$ .

DownLoad: Full-Size Img PowerPoint

Figure 3. The 3-dimensional and contour graphs of solution (2.21) at

$r = 1.71, \; \delta = -2, \; \rho = -0.54, \; P = 0.44, \; \sigma = -1, \; \zeta = 1.33$ , and different fractional values of

$\mu$ , for (a, d)

$\mu = 1$ , for (b, c)

$\mu = 0.85$ , for (c, f)

$\mu = 0.56$ .

DownLoad: Full-Size Img PowerPoint

4. Results and discussion

We investigated the Hamiltonian amplitude equation that governs certain instabilities of modulated wave trains. We have successfully reached some important wave solutions to this nonlinear model such as the dark, bright, combined dark-bright, singular solitons, periodic and singular periodic wave solutions. It is known that dark soliton describes the solitary waves with lower intensity than the background, bright soliton describes the solitary waves whose peak intensity is larger than the background and the singular soliton solutions is a solitary wave with discontinuous derivatives; examples of such solitary waves include compactions, which have finite (compact) support, and peakons, whose peaks have a discontinuous first derivative ^[49,50].

Moreover, results presented in this study have some important physical meaning, for instance, the hyperbolic sine arises in the gravitational potential of a cylinder and the calculation of the Roche limit, the hyperbolic cosine function is the shape of a hanging cable (the so-called CATENARY), the hyperbolic tangent arises in the calculation of magnetic moment and rapidity of special relativity, the hyperbolic secant arises in the profile of a laminar jet and the hyperbolic cotangent arises in the Langevin function for magnetic polarization ^[51].

5. Conclusions

In this study the Hamiltonian amplitude equation have been investigated comprehensively with the aid of two efficient analytical schemes namely; the extended sinh-Gordon equation expansion method and the and the extended rational sine-cosine/sinh-cosh methods. To this end, so many different solitons and other solutions with an interesting behavior have been established. The physical characteristics of the acquired solutions are plotted in order to provide a good perspective of their features. To investigate the interpretation of complex physical phenomena, the results obtained can be used in different branches of science. It is also worth noting that the original equation has been satisfied by all the recorded solutions. The two techniques are very vehement in constructing novel solutions to nonlinear partial differential equations.

Acknowledgment

The authors would like to acknowledge the financial support of Taif University Researchers Supporting Project number (TURSP-2020/162), Taif University, Taif, Saudi Arabia.

Conflict of interest

The authors declare no conflict of interest.

References

[1]	A. M. Lyapunov, Problème général de la stabili $\mathop {\rm{t}}\limits^{' }$ e du mouvement, Ann. Fac. Sci. Univ. Toulouse, 9 (1907), 203–474.
[2]	A. Weinstein, Normal modes for nonlinear Hamiltonian systems, Invent. Math., 20 (1973), 47–57. https://doi.org/10.1007/BF01405263 doi: 10.1007/BF01405263
[3]	J. Moser, Periodic orbits near an equilibrium and a theorem by Alan Weinstein, Comm. Pure Appl. Math., 29 (1976), 724–747. https://doi.org/10.1002/cpa.3160290613 doi: 10.1002/cpa.3160290613
[4]	E. Fadell, P. H. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math., 45 (1978), 139–174. https://doi.org/10.1007/BF01390270 doi: 10.1007/BF01390270
[5]	J. A. Montaldi, R. M. Roberts, I. N. Stewart, Periodic solutions near equilibria of symmetric Hamiltonian systems, Phil. Trans. R. Soc. Lond. A, 325 (1988), 237–293. https://doi.org/10.1098/rsta.1988.0053 doi: 10.1098/rsta.1988.0053
[6]	T. Bartsch, A generalization of the Weinstein-Moser theorems on periodic orbits of a Hamiltonian system near an equilibrium, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 691–718. https://doi.org/10.1016/S0294-1449(97)80130-8 doi: 10.1016/S0294-1449(97)80130-8
[7]	E. N. Dancer, S. Rybicki, A note on periodic solutions of autonomous Hamiltonian systems emanating from degenerate stationary solutions, Differ. Integral Equ., 12 (1999), 147–160.
[8]	A. Szulkin, Bifurcation for strongly indefinite functionals and a Liapunov type theorem for Hamiltonian systems, Differ. Integral Equ., 7 (1994), 217–234.
[9]	E. Pérez-Chavela, S. Rybicki, D. Strzelecki, Symmetric Liapunov center theorem, Calc. Var. Partial Differ. Equ., 56 (2017), 1–23. https://doi.org/10.1007/s00526-017-1120-1 doi: 10.1007/s00526-017-1120-1
[10]	E. Pérez-Chavela, S. Rybicki, D. Strzelecki, Symmetric Liapunov center theorem for minimal orbit, J. Differ. Equ., 265 (2018), 752–778. https://doi.org/10.1016/j.jde.2018.03.009 doi: 10.1016/j.jde.2018.03.009
[11]	D. Strzelecki, Periodic solutions of symmetric Hamiltonian systems, Arch. Ration. Mech. Anal., 237 (2020), 921–950. https://doi.org/10.1007/s00205-020-01522-6 doi: 10.1007/s00205-020-01522-6
[12]	M. Kowalczyk, E. Pérez-Chavela, S. Rybicki, Symmetric Lyapunov center theorem for orbit with nontrivial isotropy group, Adv. Differ. Equ., 25 (2020), 1–30.
[13]	M. Izydorek, Equivariant Conley index in Hilbert spaces and applications to strongly indefinite problems, Nonlinear Anal., 51 (2002), 33–66. https://doi.org/10.1016/S0362-546X(01)00811-2 doi: 10.1016/S0362-546X(01)00811-2
[14]	T. tom Dieck, Transformation groups, Walter de Gruyter & Co., Berlin, 1987. https://doi.org/10.1515/9783110858372
[15]	T. Kawasaki, Cohomology of twisted projective spaces and lens complexes, Math. Ann., 206 (1973), 243–248. https://doi.org/10.1007/BF01429212 doi: 10.1007/BF01429212
[16]	A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. https://doi.org/10.1017/S0013091503214620
[17]	K. H. Mayer, G-invariante Morse-funktionen, Manuscripta Math., 63 (1989), 99–114. http://dx.doi.org/10.1007/bf01173705 doi: 10.1007/bf01173705
[18]	J. Fura, A. Ratajczak, S. Rybicki, Existence and continuation of periodic solutions of autonomous Newtonian systems, J. Differ. Equ., 218 (2005), 216–252. https://doi.org/10.1016/j.jde.2005.04.004 doi: 10.1016/j.jde.2005.04.004
[19]	A. Gołȩbiewska, S. Rybicki, Equivariant Conley index versus degree for equivariant gradient maps, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 985–997. http://dx.doi.org/10.3934/dcdss.2013.6.985 doi: 10.3934/dcdss.2013.6.985
[20]	Z. Balanov, W. Krawcewicz, H. Steinlein, Applied Equivariant Degree, AIMS Series on Differential Equations & Dynamical Systems, 1, Springfield, 2006.
[21]	A. Gołȩbiewska, P. Stefaniak, Global bifurcation from an orbit of solutions to non-cooperative semi-linear Neumann problem, J. Differ. Equ., 268 (2020), 6702–6728. https://doi.org/10.1016/j.jde.2019.11.053 doi: 10.1016/j.jde.2019.11.053
[22]	J. P. Serre, Linear representations of finite groups, Graduate Texts in Mathematics, 42 (1977), Springer-Verlag, New York-Heidelberg. https://doi.org/10.1007/978-1-4684-9458-7 doi: 10.1007/978-1-4684-9458-7
[23]	T. Bartsch, Topological methods for variational problems with symmetries, Lect. Notes Math., 1560, Springer-Verlag, Berlin, 1993. https://doi.org/10.1007/BFb0073859
[24]	K. Gȩba, Degree for gradient equivariant maps and equivariant Conley index, Topological nonlinear analysis II, Birkhäuser, (1997), 247–272. https://doi.org/10.1007/978-1-4612-4126-3_5
[25]	C. Conley, Isolated invariants sets and the Morse index, CBMS Regional Conference Series in Mathematics, 38, American Mathematical Society, Providence, R. I., 1978. https://doi.org/10.1090/cbms/038
[26]	J. Smoller, A. Wasserman, Bifurcation and symmetry-breaking, Invent. Math., 100 (1990), 63–95. https://doi.org/10.1007/BF01231181 doi: 10.1007/BF01231181

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Periodic solutions to symmetric Newtonian systems in neighborhoods of orbits of equilibria

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Abstract

1. Introduction

2. Applications

2.1. Application of the extended sinh-Gordon equation expansion method

2.2. Application of the extended rational sine-cosine/sinh-cosh methods

3. Physical interpretation

4. Results and discussion

5. Conclusions

Acknowledgment

Conflict of interest

References

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Electronic Research Archive

Periodic solutions to symmetric Newtonian systems in neighborhoods of orbits of equilibria

Related Papers:

Abstract

1. Introduction

2. Applications

2.1. Application of the extended sinh-Gordon equation expansion method

2.2. Application of the extended rational sine-cosine/sinh-cosh methods

3. Physical interpretation

4. Results and discussion

5. Conclusions

Acknowledgment

Conflict of interest

References

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Reader Comments

通讯作者: 陈斌, bchen63@163.com

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