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This treatise analyzes the coupled nonlinear system of the model for the ion sound and Langmuir waves. The modified ( G ′ / G ) -expansion procedure is utilized to raise new closed-form wave solutions. Those solutions are investigated through hyperbolic, trigonometric and rational function. The graphical design makes the dynamics of the equations noticeable. It provides the mathematical foundation in diverse sectors of underwater acoustics, electromagnetic wave propagation, design of specific optoelectronic devices and physics quantum mechanics. Herein, we concluded that the studied approach is advanced, meaningful and significant in implementing many solutions of several nonlinear partial differential equations occurring in applied sciences.

We investigate generalized chaplygin gas for warm inflationary scenario in the context of locally rotationally symmetric Bianchi type I universe model. We assume two different cases of dissipative coefficient, i.e., constant as well as function of scalar field. We construct dynamical equations as well as a relationship between scalar and radiation energy densities under slow-roll approximation. We also derive slow-roll parameters, scalar and tensor power spectra, scalar spectral index, tensor to scalar ratio for analyzing inflationary background during high dissipative regime. We also use the WMAP7 data for the discussion of our parameters.

Cosmological models of a scalar field with dynamical equations containing fractional derivatives or derived from the Einstein-Hilbert action of fractional order, are constructed. A number of exact solutions to those equations of fractional cosmological models in both cases is given., 14 pages

The integrating factors and conservation theorems of nonholonomic dynamical system of relative motion are studied. First, the dynamical equations of relative motion of system are written. Next, the definition of integrating factors is given, and the necessary conditions for the existence of the conserved quantities are studied in detail. Then, the conservation theorem and its inverse of system are established. Finally, an example is given to illustrate the application of the result.

A new conservation theorem of the nonholonomic systems is studied. The conserved quantity is only constructed in terms of a general Lie group of transformation vector of the dynamical equations. Firstly, we establish the dynamical equations of the nonholonomic systems and the determining equations of Lie symmetry. Next, the theorem of non-Noether conserved quantity is deduced. Finally, we give an example to illustrate the application of the result.

The ocean rogue wave is one kind of puzzled destructive phenomenon that has not been understood thoroughly so far. The two-dimensional nature of this wave has inspired the vast endeavors on the recognizing new patterns of the rogue waves based on the dynamical equations with two-spatial variables and one-temporal variable, which is a very crucial step to prevent this disaster event at the earliest stage. Along this issue, we present twelve new patterns of the two-dimensional rogue waves, which are reduced from a rational and explicit formula of the solutions for a (2+1)-dimensional Maccari system. The extreme points (lines) of the first-order lumps (rogue waves) are discussed according to their analytical formulas. For the lower-order rogue waves, we show clearly in formula that parameter b 2 plays a significant role to control these patterns.

We investigate the mechanics ofa new double chain-model of DNA. The model consists of two long elastic homogeneous strands (or rods), which represent two polynucleotide chains of the DNA molecule, connected with each other by an elastic membrane (or some linear springs) representing the hydrogen bonds between the base pairs of the two chains. The nonlinear dynamical equations are derived and some solitary wave solutions are discussed. By the way, our model can be regarded as the mechanical model of cubic nonlinear Klein–Gordon equation and -field equation.

A single-spin transition critical dynamics is used to investigate the three-dimensional kinetic Ising model on an anisotropic cubic lattice. We first derive the fundamental dynamical equations, and then linearize them by a cutoff approximation. We obtain the approximate solutions of the local magnetization and equal-time pair correlation function in zero field. In which the axial-decoupling terms , and as higher infinitesimal quantity are ignored, where . We think that it is reasonable as the temperature of the system is very high. The result of what we obtain in this paper can go back to the one-dimensional Glauber's theory as long as .

A single-spin transitional critical dynamics based on the Glauber theory is used to investigate the d-dimensional kinetic S4 model. We derive the fundamental dynamical equations by a cumulant expansion, and then, a one-dirnensional model is treated. We obtain approximately the local magnetization and equal-time spin-pair correlation function, in which the terms of k3 and the higher orders are ignored (corresponding to high-temperature cutoff). These are finally extended to d-dimensional case. In addition, we obtain the critical point kc of d-dimensional S4 model, and give a relation satisfied by the dynamic critical exponent z and the correlation length critical exponent ν.

By analyzing the dynamical equations of dissipative diffusion systems with the help of Feynman diagrams, we explicitly show that Goldstone modes exist in these systems and how these Goldstone modes are related to the origin of self-organized criticality. The analysis gives a clear statistical field theory picture of self-organized criticality.

The exact equations of parallel dynamics for three-Ising neural networks at zero temperature are derived from a generating functional. The generating functional is proposed to be a -function to fit the dynamical rule of three-Ising model. The dynamical equations obtained in the case without self-coupling terms are in agreement with the published results derived with a probabilistic approach, which shows that this approach can be used to advantage in calculating the exact dynamical equations and other order parameters for multistate Ising neural networks.

The static study has shown that the excited state with electron-hole pair in C60 can cause the distortion of the bond structure to form a polaron-like exciton with symmetry D5d. This paper further reveals the relaxation process from the initial electron-hole pair state to the polaron-like exciton by solving the dynamical equations. The relaxation time of this dynamical process can be determined from the time-dependent bond distortion with time step 4 femtoseconds.

Vortex density current in the Gross–Pitaevskii theory is studied. It is shown that the inner structure of the topological vortices can be classified by Brouwer degrees and Hopf indices of -mapping. The dynamical equations of vortex density current have been given. The bifurcation behavior at the critical points of the current is discussed in detail.