The Interactions of N

A generalized (2+1)-dimensional variable-coefficient KdV equation is introduced, which can describe the interaction between a water wave and gravity-capillary waves better than the (1+1)-dimensional KdV equation. The N-soliton solutions of the (2+1)-dimensional variable-coefficient fifth-order KdV equation are obtained via the Bell-polynomial method. Then the soliton fusion, fission, and the pursuing collision are analyzed depending on the influence of the coefficient eAij; when eAij=0, the soliton fusion and fission will happen; when eAij≠0, the pursuing collision will occur. Moreover, the Bäcklund transformation of the equation is gotten according to the binary Bell-polynomial and the period wave solutions are given by applying the Riemann theta function method

Patterns on liquid surfaces: cnoidal waves, compactons and scaling

Localized patterns and nonlinear oscillation formation on the bounded free surface of an ideal incompressible liquid are analytically investigated . Cnoidal modes, solitons and compactons, as traveling non-axially symmetric shapes are discused. A finite-difference differential generalized Korteweg-de Vries equation is shown to describe the three-dimensional motion of the fluid surface and the limit of long and shallow channels one reobtains the well known KdV equation. A tentative expansion formula for the representation of the general solution of a nonlinear equation, for given initial condition is introduced on a graphical-algebraic basis. The model is useful in multilayer fluid dynamics, cluster formation, and nuclear physics since, up to an overall scale, these systems display liquid free surface behavior.Comment: 14 pages RevTex, 5 figures in p

Be careful with variable separation solutions via the extended tanh-function method and periodic wave structures

We analyze the extended tanh-function method to realize variable separation, however, we find that various "different" solutions obtained by this method are seriously equivalent to the general solution derived by the multilinear variable separation approach. In order to illustrate this point, we take a general (2 + 1)-dimensional Korteweg–de Vries system in water for example. Eight kind of variable separation solutions for a general (2 + 1)-dimensional Korteweg–de Vries system are derived by means of the extended tanh-function method and the improved tanh-function method. By detailed investigation, we find that these seemly independent variable separation solutions actually depend on each other. It is verified that many of so-called "new" solutions are equivalent to one another. Based on the uniform variable separation solution, abundant localized coherent structures can be constructed. However, we must pay our attention to the solution expression of all components to avoid the appearance of some un-physical related and divergent structures: seemly abundant structures for a special component are obtained while the divergence of the corresponding other component for the same equation appears

On Soliton-type Solutions of Equations Associated with N-component Systems

The algebraic geometric approach to $N$-component systems of nonlinear integrable PDE's is used to obtain and analyze explicit solutions of the coupled KdV and Dym equations. Detailed analysis of soliton fission, kink to anti-kink transitions and multi-peaked soliton solutions is carried out. Transformations are used to connect these solutions to several other equations that model physical phenomena in fluid dynamics and nonlinear optics.Comment: 43 pages, 16 figure

New Traveling Wave Solutions For Some Nonlinear Fractional Differential Equations By Extensions Of Basic

Due to varied and important applications of nonlinear fractional differential equations in real world problems, it is often required to construct their exact analytical solutions. With the help of exact analytical solutions, if they exist, the modelled phenomena can be better understood. Generally, an important class of solutions of nonlinear evolution equations (EEs) is their travelling wave solutions

Ermakov-Painlevé II Reduction in Cold Plasma Physics. Application of a Bäcklund Transformation

A class of symmetry transformations of a type originally introduced in a nonlinear optics context is used here to isolate an integrable Ermakov-Painlev´e II reduction of a resonant NLS equation which encapsulates a nonlinear system in cold plasma physics descriptive of the uniaxial propagation of magneto-acoustic waves. A B¨acklund transformation is employed in the iterative generation of novel classes of solutions to the cold plasma system which involve either Yablonski-Vorob’ev polynomials or classical Airy function