相关论文: Special Polynomials and Exact Solutions of the Dis…
In this paper we are consider radiating solitary wave solutions of coupled regularised Boussinesq equations. This type of solution consists of a leading solitary wave with a small-amplitude co-propagating oscillatory tail, and emerges from…
New method for finding exact solutions of nonlinear differential equations is presented. It is based on constructing the polygon corresponding to the equation studied. The algorithms of power geometry are used. The method is applied for…
We will study special solutions of the fourth, fifth and sixth Painlev\'e equations with generic values of parameters whose linear monodromy can be calculated explicitly. We will show the relation between Umemura's classical solutions and…
We consider a system of equations for the description of nonlinear waves in a liquid with gas bubbles. Taking into account high order terms with respect to a small parameter, we derive a new nonlinear partial differential equation for the…
We revisit and present new linear spaces of explicit solutions to incompressible Euler and Navier-Stokes equations on $\mathbb{R}^n$, as well as the rotating Boussinesq equations on $\mathbb{R}^3$. We cast these solutions are superpositions…
In this paper, two kinds of the exact singular solutions are obtained by the improved homogeneous balance (HB) method and a nonlinear transformation. The two exact solutions show that special singular wave patterns exists in the classical…
We discuss the application of a variant of the method of simplest equation for obtaining exact traveling wave solutions of a class of nonlinear partial differential equations containing polynomial nonlinearities. As simplest equation we use…
B\"acklund transformations (BTs) for ordinary differential equations (ODEs), and in particular for hierarchies of ODEs, are a topic of great current interest. Here we give an improved method of constructing BTs for hierarchies of ODEs. This…
Proposed in this paper is a numerical procedure to generate periodic traveling wave solutions of some nonlinear dispersive wave equations. The method is based on a suitable modification of a fixed point algorithm of Petviahvili type and…
A generalized version of the $abcd$-Boussinesq class of systems is derived to accommodate variable bottom topography in two-dimensional space. This extension allows for the conservation of suitable energy functionals in some cases and…
A fundamental non-classical fourth-order partial differential equation to describe small amplitude linear oscillations in a rotating compressible fluid, is obtained. The dispersion relations for such a fluid, and the different regions of…
In this work, a novel Boussinesq system is put forward. The system is naturally nonlinearly entropy/energy-stable, and is designed for problems with sharply varying bathymetric features. The system is flexible and allows tuning of the…
This study deals with the analysis of the Cauchy problem of a general class of nonlocal nonlinear equations modeling the bi-directional propagation of dispersive waves in various contexts. The nonlocal nature of the problem is reflected by…
We utilise a recent approach via the so-called re-scaling method to derive a unified and comprehensive theory of the solutions to Painleve's differential equations (I), (II) and (IV), with emphasis on the most elaborate equation (IV).
Painleve transcendents are usually considered as complex functions of a complex variable, but in applications it is often the real cases that are of interest. Under a reasonable assumption (concerning the behavior of a dynamical system…
A nonlinear generalisation of Schrodinger's equation had previously been obtained using information-theoretic arguments. The nonlinearities in that equation were of a nonpolynomial form, equivalent to the occurence of higher-derivative…
Rational solutions of the Painleve IV equation are constructed in the setting of pseudo-differential Lax formalism describing AKNS hierarchy subject to the additional non-isospectral Virasoro symmetry constraint. Convenient Wronskian…
A nonlinear transformation of the dispersive long wave equations in (2+1) dimensions is derived by using the homogeneous balance method. With the aid of the transformation given here, exact solutions of the equations are obtained.
Extended shallow water wave equations are derived, using the method of asymptotic expansions, from the Euler (or water wave) equations. These extended models are valid one order beyond the usual weakly nonlinear, long wave approximation,…
Recurrence coefficients of semi-classical orthogonal polynomials (orthogonal polynomials related to a weight function $w$ such that $w'/w$ is a rational function) are shown to be solutions of non linear differential equations with respect…