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The system of two nonlinear coupled oscillators is studied. As partial case this system of equation is reduced to the Duffing oscillator which has many applications for describing physical processes. It is well known that the inverse scattering transform is one of the most powerful methods for solving the Cauchy problems of partial differential equations. To solve the Cauchy problem for nonlinear differential equations we can use the Lax pair corresponding to this equation. The Lax pair for ordinary differential or systems or for system ordinary differential equations allows us to find the first integrals, which also allow us to solve the question of integrability for differential equations. In this report we present the Lax pair for the system of coupled oscillators. Using the Lax pair we get two first integrals for the system of equations. The considered system of equations can be also reduced to the fourth-order ordinary differential equation and the Lax pair can be used for the ordinary differential equation of fourth order. Some special cases of the system of equations are considered., Comment: 9 pages

We consider a family of non-autonomous second-order differential equations, which generalizes the Li\'enard equation. We explicitly find the necessary and sufficient conditions for members of this family of equations to admit quadratic, with the respect to the first derivative, first integrals. We show that these conditions are equivalent to the conditions for equations in the family under consideration to possess Lax representations. This provides a connection between the existence of a quadratic first integral and a Lax representation for the studied dissipative differential equations, which may be considered as an analogue to the theorem that connects Lax integrability and Arnold--Liouville integrability of Hamiltonian systems. We illustrate our results by several examples of dissipative equations, including generalizations of the Van der Pol and Duffing equations, each of which have both a quadratic first integral and a Lax representation., Comment: To appear in Journal of Mathematical Analysis and Applications

We study nonlinear dynamics of two coupled contrast agents that are micro-meter size gas bubbles encapsulated into a viscoelastic shell. Such bubbles are used for enhancing ultrasound visualization of blood flow and have other promising applications like targeted drug delivery and noninvasive therapy. Here we consider a model of two such bubbles interacting via the Bjerknes force and exposed to an external ultrasound field. We demonstrate that in this five-dimensional nonlinear dynamical system various types of complex dynamics can occur, namely, we observe periodic, quasi-periodic, chaotic and hypechaotic oscillations of bubbles. We study the bifurcation scenarios leading to the onset of both chaotic and hyperchaotic oscillations. We show that chaotic attractors in the considered system can appear via either Feigenbaum's cascade of period doubling bifurcations or Afraimovich--Shilnikov scenario of torus destruction. For the onset of hyperchaotic attractor we propose a new bifurcation scenario, which is based on the appearance of a homoclinic chaotic attractor containing a saddle-focus periodic orbit with its two-dimensional unstable manifold. Finally, we demonstrate that the bubbles' dynamics can be multistable, i.e. various combinations of co-existence of the above mentioned attractors are possible. These cases include co-existence of hyperchaotic regime with any of the other remaining types of dynamics for different parameter values. Thus, the model of two coupled gas bubbles provide a new examples of physically relevant system with multistable hyperchaos., Comment: 22 pages, 13 figures

We consider a system of equations for the description of nonlinear waves in a liquid with gas bubbles. Taking into account high order terms with respect to a small parameter, we derive a new nonlinear partial differential equation for the description of density perturbations of mixture in the two-dimensional case. We investigate integrability of this equation using the Painlev\'e approach. We show that travelling wave reduction of the equation is integrable under some conditions on parameters. Some exact solutions of the equation derived are constructed. We also perform numerical investigation of the nonlinear waves described by the derived equation.

We study a new nonlinear partial differential equation of the fifth order for the description of perturbations in the Fermi-Pasta-Ulam mass chain. This fifth-order equation is an expansion of the Gardner equation for the description of the Fermi--Pasta--Ulam model. We use the potential of interaction between neighbouring masses with both quadratic and cubic terms. The equation is derived using the continuous limit. Unlike the previous works, we take into account higher order terms in the Taylor series expansions. We investigate the equation using the Painlev\'e approach. We show that the equation does not pass the Painlev\'e test and can not be integrated by the inverse scattering transform. We use the logistic function method and the Laurent expansion method to find travelling wave solutions of the fifth--order equation. We use the pseudospectral method for the numerical simulation of wave processes, described by the equation.

The Li\'enard equation is used in various applications. Therefore, constructing general analytical solutions of this equation is an important problem. Here we study connections between the Li\'enard equation and some equations from the Painlev\'e--Gambier classification. We show that with the help of such connections one can construct general analytical solutions of the Li\'enard equation's subfamilies. In particular, we find three new integrable families of the Li\'enard equation. We also propose and discuss an approach for finding one--parameter families of closed--form analytical solutions of the Li\'enard equation.

\'Eliashberg theory generalized to the case of the electron-phonon (EP) systems with the not constant density of electronic states, as well as to the case of the frequency dependent renormalization of both the electron mass and the chemical potential, is used to study hydrogen sulphide phase under pressure. The frequency and the temperature dependences of the electron mass renormalization ReZ , the density of electronic states , renormalized by the EP interaction , the spectral function of the electron-phonon interaction, obtained by calculation are used to calculate the electronic abnormal GF. The generalized \'Eliashberg equations with the variable density of electronic states are resolved for the hydrogen sulphide phase under pressure. The dependence of both the real and the imaginary part of the order parameter on the frequency in the phase is obtained. The results of the solution of the Eliashberg equations for the Im-3m (170 GPa), Im-3m (200 GPa), Im-3m (225 GPa) and R3m (120 GPa) hydrogen sulphide phases are presented. The value in the hydrogen sulfide phase at the pressure P = 225 GPa has been defined. The value for the hydrogen sulfide phase at the pressure P = 170 GPa has been predicted., Comment: 43 pages, 15 figures. arXiv admin note: substantial text overlap with arXiv:1610.02688, arXiv:1607.02162

Using the continuous limit approximation in the dynamical system we study a nonlinear partial differential equation which corresponds to the generalization of both the Fermi-Pasta-Ulam and the Frenkel-Kontorova models. This generalized model can be considered as a model for the description nonlinear dislocation waves in the crystal lattice. We obtain the nonlinear partial differential equation for the description of dislocations. Taking into account the wave moving in one direction we obtain another nonlinear evolution equation. Using the Painlev\'e test we analyze the integrability of this equation. We find that there exist an integrable case of the partial differential equation for nonlinear dislocations. The Lax pair for the solution of the Cauchy problem is found. The solution of the Cauchy problem for nonlinear evolution equation is discussed. One and the two-soliton solutions for the nonlinear evolution equation are presented. The influence of parameters on the propagation of one and the two-soliton solutions is analyzed and demonstrated.

\'Eliashberg theory, being generalized to the case of the electron-phonon (EP) systems with the not constant density of electronic states, as well as to the frequency behavior of the renormalization of both the electron mass and the chemical potential, is used to study the normal properties and of the metal hydrogen phase under pressure of P = 500 GPa. The phonon contribution to the anomalous electron Green's function (GF) is considered. The pairing is considered within the overall width of the electron band, and not only in a narrow layer at the Fermi surface. The frequency dependence of the electron mass renormalization ReZ , the density of electronic states , the spectral function of the electron-phonon interaction, obtained by calculation are used to calculate the electronic abnormal GF. The \'Eliashberg generalized equations with the variable density of electronic states are resolved. The dependence of both the real and the imaginary parts of the order parameter on the frequency in the phase is obtained. The value for the metal hydrogen phase at the pressure P = 500 GPa has been defined., Comment: 28 pages, 9 figures. arXiv admin note: substantial text overlap with arXiv:1607.02162

Generalized theory of the normal properties of the metal in the case of the electron-phonon (EP) systems with not constant density of electronic states is used to examine the normal state of the SH3 and SH2 phase of the hydrogen sulfide at different pressures. The frequency dependence of the real and imaginary part of the self-energy part (SP) of the electron Green's function, the real and imaginary part of the complex renormalization of the electron mass, the real and imaginary part of the complex renormalization of the chemical potential, as well as the renormalized with the strong electron-phonon interaction the electron density of states are calculated. The calculations are performed for the stable orthorhombic IM-3M hydrogen sulfide SH3 structure at three pressures P = 170 GPa, P = 180 GPa, P = 225 GPa and SH2 structure with the symmetry I4/MMM (D4H-17) for the three values of pressure P = 150 GPa , P = 180 GPa, P = 225 GPa at the temperature of T = 200K. Key words: hydrogen sulfide, electron-phonon system, the pressure, the electronic properties., Comment: 19 pages, 11 figures

\'Eliashberg theory, being generalized to the electron-phonon (EP) systems with not constant density of electronic states, as well as to the frequency behavior of the renormalization of both the electron mass and the chemical potential, is used to study hydrogen sulphide phase under pressure. The phonon contribution to the anomalous electron Green's function (GF) is considered. The pairing is considered within the overall width of the electron band, and not only in a narrow layer at the Fermi surface. The frequency and the temperature dependences of the electron mass renormalization ReZ , the density of electronic states , renormalized by the EP interaction , the spectral function of the electron-phonon interaction, obtained by calculation are used to calculate the electronic abnormal GF. The \'Eliashberg generalized equations with the variable density of electronic states are resolved. The dependence of both the real and the imaginary part of the order parameter on the frequency in the phase is obtained. The value in the hydrogen sulfide phase at the pressure P = 225 GPa has been defined., Comment: 20 pages, 8 Figures

In this work we generalize the models for nonlinear waves in a gas--liquid mixture taking into account an interphase heat transfer, a surface tension and a weak liquid compressibility simultaneously at the derivation of the equations for nonlinear waves. We also take into consideration high order terms with respect to the small parameter. Two new nonlinear differential equations are derived for long weakly nonlinear waves in a liquid with gas bubbles by the reductive perturbation method considering both high order terms with respect to the small parameter and the above mentioned physical properties. One of these equations is the perturbation of the Burgers equation and corresponds to main influence of dissipation on nonlinear waves propagation. The other equation is the perturbation of the Burgers--Korteweg--de Vries equation and corresponds to main influence of dispersion on nonlinear waves propagation.