Conservation Laws and Travelling Wave Solutions for a Negative-Order KdV-CBS Equation in 3+1 Dimensions

In this paper, we study a new negative-order KdV-CBS equation in (3 + 1) dimensions which is a combination of the Korteweg-de Vries (KdV) equation and Calogero-Bogoyavlenskii-Schiff (CBS) equation. Firstly, we determine the Lie point symmetries of the equation and conservation laws by using the multiplier method. The conservation laws will be used to obtain a triple reduction to a second order ordinary differential equation (ODE), which lead to line travelling waves and soliton solutions. Such solitons are obtained via the modified form of simple equation method and are displayed through three-dimensional plots at specific parameter values to lend physical meaning to nonlinear phenomena. It illustrates that these solutions might be extremely beneficial in understanding physical phenomena in a variety of applied mathematics areas

Symmetry reductions for thin film type equations

The lubrication equation ut = (u nuxxx)x plays an important role in the study of the interface movements. In this work we analyze the generalizations of the above equation given by ut = (u nuxxx)x − kumux. By using Lie classical method the corresponding reductions are performed and some solutions are characterized

Reductions and symmetries for a generalized Fisher equation with a diffusion term dependent on density and space

In this work, a generalized Fisher equation with a space–density diffusion term is analyzed by applying the theory of symmetry reductions for partial differential equations. The study of this equation is relevant in terms of its applicability in cell dynamics and tumor invasion. Therefore, classical Lie symmetries admitted by the equation are determined. In addition, by using the multipliers method, we derive some nontrivial conservation laws for this equation. Finally we obtain a direct reduction of order of the ordinary differential equations associated and a particular solution

Symmetry reductions and conservation laws of a modified-mixed KdV equation: exploring new interaction solutions

This article represented the investigation of the modified mixed Korteweg-de Vries equation using di erent versatile approaches. First, the Lie point symmetry approach was used to determine all possible symmetry generators. With the help of these generators, we reduced the dimension of the proposed equation which leads to the ordinary di erential equation. Second, we employed the unified Riccati equation expansion technique to construct the abundance of soliton dynamics. A group of kink solitons and other solitons related to hyperbolic functions were among these solutions. To give the physical meaning of these theoretical results, we plotted these solutions in 3D, contour, and 2D graphs using suitable physical parameters. The comprehend outcomes were reported, which can be useful and beneficial in the future investigation of the studied equation. The results showed that applied techniques are very useful to study the other nonlinear physical problems in nonlinear sciences

Conservation laws and line soliton solutions of a family of modified KP equations

A family of modi ed Kadomtsev-Petviashvili equations (mKP) in 2+1 dimensions is studied. This family includes the integrable mKP equa- tion when the coe cients of the nonlinear terms and the transverse dispersion term satisfy an algebraic condition. The explicit line soliton solution and all conservation laws of low order are derived for all equations in the family and compared to their counterparts in the integrable

Conservation laws, symmetries, and line soliton solutions of generalized KP and Boussinesq equations with p-power nonlinearities in two dimensions

Nonlinear generalizations of integrable equations in one dimension, such as the Korteweg–de Vries and Boussinesq equations with p-power nonlinearities, arise in many physical applications and are interesting from the analytic standpoint because of their critical behavior. We study analogous nonlinear p-power generalizations of the integrable Kadomtsev–Petviashvili and Boussinesq equations in two dimensions. For all p 0, we present a Hamiltonian formulation of these two generalized equations. We derive all Lie symmetries including those that exist for special powers p 0. We use Noether’s theorem to obtain conservation laws arising from the variational Lie symmetries. Finally, we obtain explicit line soliton solutions for all powers p > 0 and discuss some of their propertie

Line-solitons, line-shocks, and conservation laws of a universal KP-like equation in 2+1 dimensions

A universal KP-like equation in 2+1 dimensions, which models general nonlinear wave phenomena exhibiting p-power nonlinearity, dispersion, and small transver-sality, is studied. Special cases include the integrable KP (Kadomtsev-Petviashvili) equation and its modified version, as well as their p-power generalizations. Two main results are obtained. First, all low-order conservation laws are derived, includ-ing ones that arise for special powers p. The conservation laws comprise momenta, energy, and Galilean-type quantities, as well as topological charges. Their physical meaning and properties are discussed. The topological charges are shown to give rise to integral constraints on initial data for the Cauchy problem. Second, all line-soliton solutions are obtained in an explicit form. A parameterization is given using the speed and the direction angle of the line-soliton, and the allowed kinematic region is determined in terms of these parameters. Basic kinematical properties of the line-solitons are also discussed. These properties differ significantly compared to those for KP line-solitons and their p-power generalizations. A line-shock solution is shown to emerge when a special limiting case of the kinematic region is considered.35 página

Lie Symmetries and Conservation Laws for the Viscous Cahn-Hilliard Equation

In this paper, we study a viscous Cahn-Hilliard equation from the point of view of Lie symmetries in partial differential equations. The analysis of this equation is motivated by its applications since it serves as a model for many problems in physical chemistry, developmental biology, and population movement. Firstly, a classification of the Lie symmetries admitted by the equation is presented. In addition, the symmetry transformation groups are calculated. Afterwards, the partial differential equation is transformed into ordinary differential equations through symmetry reductions. Secondly, all low-order local conservation laws are obtained by using the multiplier method. Furthermore, we use these conservation laws to determine their associated potential systems and we use them to investigate nonlocal symmetries and nonlocal conservation laws. Finally, we apply the multi-reduction method to reduce the equation and find a soliton solution

Simetrías potenciales de un modelo matemático que describe las vibraciones de una viga

En este trabajo presentamos un estudio, desde el punto de vista de la teoría de las simetrías potenciales clásicas y no clásicas para ecuaciones en derivadas parciales, del modelo que describe las vibraciones de una viga

On symmetries and conservation laws of a Gardner equation involving arbitrary functions

Mathematics Subject Classification: 35C07, 35Q53In this work we study a generalized variable-coefficient Gardner equation from the point of view of Lie symmetries in partial differential equations. We find conservation laws by using the multipliers method of Anco and Bluman which does not require the use of a variational principle. We also construct conservation laws by using Ibragimov theorem which is based on the concept of adjoint equation for nonlinear differential equations.10 página