Extended two dimensional equation for the description of nonlinear waves in gas–liquid mixture

Kudryashov, Nikolay A.; Sinelshchikov, Dmitry I.; Volkov, Alexandr K.

doi:10.1016/j.amc.2015.06.095

Cited by 3 publications

(5 citation statements)

References 24 publications

(51 reference statements)

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“…Note that δ, µ, and g 3 are parameters in the solution. One can consider solution (26) with constants (27) and exclude poles from the real line. This solution is presented on figure 3 at δ = 1, µ = 0.5, C 0 = 5.…”

Section: Elliptic Solution For Equation (15)mentioning

confidence: 99%

“…To correlate the pole's order with the order found above, we look for the solution of equation (15) in the following form [27]: where H, A, B, C are some constants, ℘ is the elliptic Weierstrass ℘function with invariants g 2 and g 3 . We expand expression (26) in the Laurent series, substitute this expansion for v in equation (15), equate coefficients at the same powers of z to zero, and solve the derived system of algebraic equations. As a result we find the following relations:…”

Section: Elliptic Solution For Equation (15)mentioning

confidence: 99%

“…However for equation (10) the poles stay on the real line. Thus, solution (26) with coefficients (27) can not be reduced to the form of the solitary wave solution.…”

Section: Elliptic Solution For Equation (15)mentioning

confidence: 99%

“…We use the pseudospectral method [28,29] for the numerical simulation of the boundary value problem for equation (10) with periodic boundary conditions. This method was successfully used for the numerical simulation of different problems of mathematical physics [26]. We chose the pseudospectral method because it allows us to solve boundary problems for the equations of high order.…”

Section: Elliptic Solution For Equation (15)mentioning

confidence: 99%

“…We use the logistic function method. This method often allows to use the results of the Painlevé analysis for a partial differential equation for the construction of its exact solutions [24][25][26]. Let us search for the solution of equation ( 15) in the following form:…”

Section: Introductionmentioning

confidence: 99%

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The fifth-order partial differential equation for the description of theα+βFermi–Pasta–Ulam model

Kudryashov

Volkov

2017

Communications in Nonlinear Science and Numerical Simulation

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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 Painlevé approach. We show that the equation does not pass the Painlevé 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.set of approaches was used to explain it (see for example [2][3][4][5][6][7]). The main ideas of these approaches and their results can be found in work [8].Usually α or β FPU model is considered in literature, but in paper [9] the α + β FPU model is investigated. In this paper, we also use the α + β model but we use the continuous limit approximation for the investigation.This approach was used for the first time by N. Zabusky and M. Kruskal in work [10], where the authors derived the Korteweg-de Vries (KdV) equation for the description of the α FPU model. This equation is integrable by the inverse scattering transform and has been widely studied by now ( [11, 12]). Zabusky and Kruskal explained the FPU paradox (recurrence of initial conditions) using the numerical simulation of the KdV equation. Another result of work [10] was an introduction of soliton. In article [13] the author took into account high order (in comparison with [10]) terms in the Taylor series expansions for the continuous limit approximation of the α FPU model. The generalized KdV equation was obtained for a more accurate description of wave processes in the FPU model. Some exact solutions of the derived equation were found . It is shown [13] that if one takes into account more terms in the Taylor series and acts in analogy to paper [10] then he can not derive an integrable equation for the description of wave processes in the FPU mass chain. Recurrence of initial statement is also unrepresentative for wave processes, described by the fifth-order equation with arbitrary initial conditions. In paper [14] wave processes for the generalized KdV equation are simulated numerically.This brings up the question, whether the situation is similar for the β and the α + β FPU models? It is known [11] that one can obtain the Gardner equation for the description of the α + β FPU model using the continuous limit. The Gardner equation is integrable by the inverse scatterring transform and can be reduced to the modified KdV equation [15]. If we take more terms in the Taylor series expansion, we can derive the fifth-order partial differential ...

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Section: Elliptic Solution For Equation (15)mentioning

confidence: 99%

Section: Elliptic Solution For Equation (15)mentioning

confidence: 99%

“…However for equation (10) the poles stay on the real line. Thus, solution (26) with coefficients (27) can not be reduced to the form of the solitary wave solution.…”

Section: Elliptic Solution For Equation (15)mentioning

confidence: 99%

Section: Elliptic Solution For Equation (15)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The fifth-order partial differential equation for the description of theα+βFermi–Pasta–Ulam model

Kudryashov

Volkov

2017

Communications in Nonlinear Science and Numerical Simulation

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Acoustic bright solitons propagation in bubbly liquids

Yu,

Zhang

2024

The Journal of the Acoustical Society of America

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We study the propagation rules of acoustic bright solitons in bubble-containing media, as well as the strong anti-interference ability of acoustic solitons; and the effects of nonlinearity, dispersion, and dissipation on the dynamic properties of acoustic solitons are also analyzed. Based on the bubble–liquid mixture model, a lossy nonlinear Schrödinger equation is obtained. The analytical expression of the enveloped bright-acoustic solitons in the bubbly liquids is derived, which can accurately capture the propagation law of the acoustic bright solitons in the physical system, even if there is viscous loss in the medium. The dissipation-induced dynamics of acoustic solitons is studied through analytical and numerical methods, and the balancing effects of nonlinearity and dispersion in the propagation of bright solitons are analyzed. Furthermore, the particle nature and dynamic stability of bright-acoustic solitons in bubble-containing media are emphasized through fully elastic collisions between solitons moving in the same and opposite directions. This process obeys the energy and momentum conservation laws. After the collision, solitons can maintain their original amplitude, speed, and shape and continue to propagate undisturbed.

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Derivation of Weakly Nonlinear Wave Equations for Pressure Waves in Bubbly Flows with Different Types of Nonuniform Distribution of Initial Flow Velocities of Gas and Liquid Phases

Maeda

Kanagawa

2020

J. Phys. Soc. Jpn.

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Extended two dimensional equation for the description of nonlinear waves in gas–liquid mixture

Cited by 3 publications

References 24 publications

The fifth-order partial differential equation for the description of theα+βFermi–Pasta–Ulam model

The fifth-order partial differential equation for the description of theα+βFermi–Pasta–Ulam model

Acoustic bright solitons propagation in bubbly liquids

Derivation of Weakly Nonlinear Wave Equations for Pressure Waves in Bubbly Flows with Different Types of Nonuniform Distribution of Initial Flow Velocities of Gas and Liquid Phases

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