Narrow Results

In this work, exact solutions of the (3+1) dimensional Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation have been produced by applying the modified sub-equation method, which is reliable and effective among analytical methods. This equation is a wavelet equation that describes non-linear and dispersive wave propagation. The aim of considering this model is to model non-linear and dispersive interactions in an environment and enable their analysis by analytical methods. By applying this method, different wave solution classes in hyperbolic and rational forms are obtained. The graphs of solitary waves produced are in 2D, 3D and contour types. In this study, computer package programs have been used for complex operations and drawing graphics.

By using the solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct several exact travelling wave solutions for some nonlinear partial differential equations. By this method some physically important nonlinear equations are investigated and new exact travelling wave solutions are explicitly obtained with the aid of symbolic computation.

An approximate analytical solution of the Balitsky–Kovchegov (BK) equation using the homotopy perturbation method (HPM) is suggested in this work. We carried out our work in perturbative Quantum Chromodynamics (QCD) (pQCD) dipole picture of deep inelastic scattering (DIS). The BK equation in momentum space with some change of variables and truncation of the Balitsky–Fadin–Kuraev–Lipatov (BFKL) kernel can be reduced to Fisher–Kolmogorov–Petrovsky–Piscounov (FKPP) equation. The observed geometric scaling phenomena are similar to the travelling wave solution of the FKPP equation. We solved the BK equation using the HPM. The obtained solution in this work also suggests the travelling wave nature of the measured scattering amplitude $N(k,Y)$ plotted at various rapidities. We also extracted the saturation momentum, $Q_s^2(Y)$, from the obtained solution and plotted it against different rapidities. The result obtained in this work can be helpful for various phenomenological studies in high-density QCD.

The propagation of soliton through optical fibers has been studied by using nonlinear Schrödinger’s equation (NLSE). There are different types of NLSEs that study this physical phenomenon such as (GRKLE) generalized Radhakrishnan–Kundu–Lakshmanan equation. The generalized nonlinear RKL dynamical equation, which presents description of the dynamical of light pulses, has been studied. We used two formulas of the modified simple equation method to construct the optical soliton solutions of this model. The obtained solutions can be represented as bistable bright, dark, periodic solitary wave solutions.

To find periodic solutions of nonlinear equations, the Jacobi elliptic function expansion method is used. This strategy is more extensive than expanding the hyperbolic tangent series. This approach can produce periodic shock wave solutions such as solitary wave solutions.

In this paper, we consider the (3+1)-dimensional modified Korteweg–de Vries–Kadomtsev–Petviashvili (mKdV-KP) equation, which can be used to describe the nonlinear waves in plasma physics and fluid dynamics. By using solitary wave ansatz in the form of sech${}^p\tau$ function and a direct integrating way, we construct the exact bright soliton solutions and the travelling wave solutions of the equation, respectively. Moreover, we obtain its power series solutions with the convergence analysis. It is hoped that our results can provide the richer dynamical behavior of the KdV-type and KP-type equations.

This paper focuses on the instability modulation and new travelling wave solutions of the (2 + 1)-dimensional Kundu–Mukherjee–Naskar equation via the tanh function method. Dark, mixed dark–bright, complex solitons and periodic wave solutions are archived. Strain conditions for the validity of results are also reported. Instability modulation properties of the governing model are also extracted. Various wave simulations in 2D, 3D and contour graphs under the strain conditions are presented.

A direct algebraic method, which can be implemented on a computer with the help of symbolic computation software like Mathematica or Maple, is used to construct a series of travelling wave solutions of two extended coupled Ito systems. The obtained solutions include solitary wave solutions, rational solutions, triangular periodic solutions, Jacobi and Weierstrass doubly periodic wave solutions. Among them, the Jacobi elliptic periodic wave solutions exactly degenerate to the solitary wave solutions at a certain limit condition. Compared with most existing tanh methods and elliptic function method, the proposed method gives new and more general solutions. More importantly, the method provides a guideline to classify the various types of the solution according to some parameters.