On the spectral instability of some cnoidal and snoidal waves of the full Klein-Gordon-Zakharov system

We present a comprehensive study of nonlinear periodic waves, namely, the Korteweg--de Vries (KdV) and modified KdV (MKdV) cnoidal waves, and snoidal waves in a two-electron-temperature plasma. In the limiting case, these periodic waves reduce to bounded nonlinear structures, namely, KdV compressive and rarefactive solitons, MKdV compressive and rarefactive solitons, and double layers. The existence regions for these waves in the parameter space (${\mathrm{\ensuremath{\mu}}}_{\mathit{e}}$,${\mathrm{\ensuremath{\sigma}}}_{\mathit{e}}$), where ${\mathrm{\ensuremath{\mu}}}_{\mathit{e}}$ and ${\mathrm{\ensuremath{\sigma}}}_{\mathit{e}}$ are the density and temperature ratios of two electron species, are discussed in detail. The different nonlinear periodic waves and bounded structures have been explained in terms of physical parameters depicting the phase curves. It is found that the frequencies of the MKdV cnoidal and snoidal waves have different amplitude dependence behaviors than that of KdV cnoidal waves. The effect of other parameters on the characteristics of the nonlinear periodic waves are also discussed.

This paper studies the D(m, n) equation, which is the generalized version of the Drinfeld-Sokolov equation. The traveling wave hypothesis and exp-function method are applied to integrate this equation. The mapping method and the Weierstrass elliptic function method also display an additional set of solutions. The kink, soliton, shock waves, singular soliton solution, cnoidal and snoidal wave solutions are all obtained by these varieties of integration tools. Mathematics Subject Classification 37K10 · 35Q51 · 35Q55

We consider the instability and stability of periodic stationary solutions to the classical ϕ^{4} equationnumerically. In the superluminal regime, the model possesses dnoidal and cnoidal waves. The former are modulationally unstable and the spectrum forms a figureeight intersecting at the origin of the spectral plane. The latter can be modulationally stable, and the spectrum near the origin in that case is represented by vertical bands along the purely imaginary axis. The instability of the cnoidal states in that case stems from elliptical bands of complex eigenvalues far from the spectral plane origin. In the subluminal regime, there exist only snoidal waves which are modulationally unstable. Considering the subharmonic perturbations, we show that the snoidal waves in the subluminal regime are spectrally unstable with respect to all subharmonic perturbations, while for the dnoidal and cnoidal waves in the superluminal regime, the transition between the spectrally stable state and the spectrally unstable state occurs through a Hamiltonian Hopf bifurcation. The dynamical evolution of the unstable states is also considered, leading to some interesting localization events on the spatio-temporal backgrounds.

We report the existence of the cnoidal, dnoidal and snoidal waves in a nonlinear reflection grating with nonlinear modulation by means of direct substitution method. The characteristics of these nonlinear waves are briefly discussed. From the corresponding reality conditions, we found that the cnoidal and dnoidal waves can exist simultaneously for the same set of parameters and frequency.

The aim of this paper is to study three space-time (3 + 1)-dimensional modified Korteweg-de Vries equations. Nonlinear space-time (3 + 1)-dimensional partial differential equations model many realistic problems in the fields of engineering, wave propagation, fluids, etc. Firstly we construct exact closed-form solutions for the three (3 + 1)-dimensional modified Korteweg-de Vries equations using Lie symmetry method together with the extended Jacobi elliptic expansion method. The solutions obtained are soliton, cnoidal and snoidal waves. Secondly we determine conservation laws for the underlying equations using the multiplier method.

In this work, we study the generalized 2D equal-width equation which arises in various fields of science. With the aid of numerous methods which includes Lie symmetry analysis, power series expansion and Weierstrass method, we produce closed-form solutions of this model. The exact solutions obtained are the snoidal wave, cnoidal wave, Weierstrass elliptic function, Jacobi elliptic cosine function, solitary wave and exponential function solutions. Moreover, we give a graphical representation of the obtained solutions using certain parametric values. Furthermore, the conserved vectors of the underlying equation are constructed by utilizing two approaches: the multiplier method and Noether’s theorem. The multiplier method provided us with four local conservation laws, whereas Noether’s theorem yielded five nonlocal conservation laws. The conservation laws that are constructed contain the conservation of energy and momentum.

Multi-dimensional nonlinear evolution equations are studied in this paper. Jacobi’s elliptic function method, traveling wave hypothesis and Lie symmetry approaches are used to integrate these equations. The second approach only reveals toplogical 1-soliton solution while first approach displays an overwhelming number of solutions for these equations that include cnoidal waves, snoidal waves and others. In limiting cases, linear waves and solitary waves are revealed, depending on whether modulus of ellipticity approaches zero or one.

Applications of cnoidal and snoidal wave solutions via optimal system of subalgebras for a generalized extended (2+1)-D quantum Zakharov-Kuznetsov equation with power-law nonlinearity in oceanography and ocean engineering

The concept of partially invariant solutions is discussed in the framework of the group analysis of models derived from the Nambu–Goto action. In particular, we consider the nonrelativistic Chaplygin gas and the relativistic Born–Infeld theory for a scalar field. Using a general systematic approach based on subgroup classification methods, nontrivial partially invariant solutions with defect structure δ=1 are constructed. For this purpose, a classification of the subgroups of the Lie point symmetry group, which have generic orbits of dimension 2, has been performed. These subgroups allow us to introduce the corresponding symmetry variables and next to reduce the initial equations to different nonequivalent classes of partial differential equations and ordinary differential equations. The latter can be transformed to standard form and, in some cases, solved in terms of elementary and Jacobi elliptic functions. This results in a large number of new partially invariant solutions, which are determined to be either reducible or irreducible with respect to the symmetry group. Some physical interpretation of the results in the area of fluid dynamics and field theory are discussed. The solutions represent traveling and centered waves, algebraic solitons, kinks, bumps, cnoidal and snoidal waves.

In this paper we study the modified equal-width equation, which is used in handling simulation of a single dimensional wave propagation in nonlinear media with dispersion processes. Lie point symmetries of this equation are computed and used to construct an optimal system of one-dimensional subalgebras. Thereafter using an optimal system of one-dimensional subalgebras, symmetry reductions and new group-invariant solutions are presented. The solutions obtained are cnoidal and snoidal waves. Furthermore, conservation laws for the modified equal-width equation are derived by employing two different methods, the multiplier method and Noether approach.

Symmetry reductions, exact solutions and conservation laws of a new coupled KdV system

We analytically investigated two-dimensional localized nonlinear waves in Kerr media with radial and azimuthal modulation of the nonlinearity and in the presence of an external potential. The solutions have been derived through the similarity transformation. We demonstrate that the properties of nonlinear waves are determined by two parameters: a whole number n (the index of the Jacobi elliptical waves) and an integer m (the topological charge). Our results indicate that the dynamic evolution, including cnoidal and snoidal waves, can be strongly affected by these two parameters, providing an approach to controlling nonlinear waves by an appropriate radial–azimuthal modulation of the nonlinearity, with an appropriate external potential.

We study the stability of the cnoidal, dnoidal and snoidal elliptic functions as spatially-periodic standing wave solutions of the 1D cubic nonlinear Schrodinger equations. First, we give global variational characterizations of each of these periodic waves, which in particular provide alternate proofs of their orbital stability with respect to same-period perturbations, restricted to certain subspaces. Second, we prove the spectral stability of the cnoidal waves against same-period perturbations (in a certain parameter range), and provide an alternate proof of this (known) fact for the snoidal waves, which does not rely on complete integrability. Third, we give a rigorous version of a formal asymptotic calculation of Rowlands to establish the instability of a class of real-valued periodic waves in 1D, which includes the cnoidal waves of the 1D cubic focusing nonlinear Schrodinger equation, against perturbations with period a large multiple of their fundamental period. Finally, we develop a numerical method to compute the minimizers of the energy with fixed mass and momentum constraints. Numerical experiments support and complete our analytical results.

In a recent paper, the equation of motion for the complex order parameter of the Landau-Ginzburg Hamiltonian for first- and second-order phase transitions was derived. Several special solutions of this nonlinear Schr\"odinger equation were obtained which correspond to a zero value of the integration constant and involve a restriction on the carrier velocity. In the present paper, we systematically analyze this equation and its solutions in order to obtain a complete set of propagating solutions. We classify all the types of order-parameter envelopes and find among them both localized and periodic solutions. The former include various singular forms, as well as nonsingular forms such as bumps and kinks. We also find trigonometric and elliptic (cnoidal, dnoidal, and snoidal) waves. Physical interpretation of all the solutions is also provided. The velocities of the soliton solutions are restricted by certain specific temperature-dependent inequalities. We also derive a general expression for the energy of each solution and its limiting values for several particularly important cases. The obtained results are illustrated using the superconductor--normal-metal transition.

Conserved quantities and solutions of a [formula omitted]-dimensional H a ˇ r a ˇ gus-Courcelle–Il’ichev model