New Families Of Exact Solutions Research Articles

New Family of Exact Solutions and Chaotic Soltions of Generalized Breor–KaupSystem in (2+1)-Dimensions via an Extended Mapping Approach**The projectsupported by the Natural Science Foundation of Zhejiang Province under GrantNo. Y604106, the Foundation of New Century ``151 Talent Engineering''of Zhejiang Province, the Scientific Research Foundation of KeyDiscipline of Zhejiang Province, and the Natural ScienceFoundation of Zhejiang Lishui University under Grant No.

Starting from an extended mapping approach, a new type of variableseparation solution with arbitrary functions of generalized(2+1)-dimensional Broer–Kaup system (GBK) system is derived. Thenbased on the derived solitary wave solution, we obtain some specificchaotic solitons to the (2+1)-dimensional GBK system.

Nonperturbative states in type II superstring theory from classical spinning membranes

We find a new family of exact solutions in membrane theory, representing toroidal membranes spinning in several planes. They have energy square proportional to the sum of the different angular momenta, generalizing Regge-type string solutions to membrane theory. By compactifying the eleven-dimensional theory on a circle and on a torus, we identify a family of new nonperturbative states of type IIA and type IIB superstring theory (which contains the perturbative spinning string solutions of type II string theory as a particular case). The solution represents a spinning bound state of D branes and fundamental strings. Then we find similar solutions for membranes on AdS 7 × S 4 and AdS 4 × S 7 . We also consider the analogous solutions in SU ( N ) Matrix-theory, and compute the energy. They can be interpreted as rotating open strings with D0 branes attached to their endpoints.

Auto-Bäcklund Transformation and Exact Solutions to the Generalized Kadomtsev-Petviashvili Equation with Variable Coefficients**The project supported by National Natural Science Foundation of China under Grant No. 10072013 and the State Key Basic Research Development Program under Grant No. G1998030600

By using the extended homogeneous balance method, a new auto-Bäcklund transformation(BT) to the generalized Kadomtsev–Petviashvili equation with variable coefficients (VCGKP) are obtained. And making use of the auto-BT and choosing a special seed solution, we get many families of new exact solutions of the VCGKP equations, which include single soliton-like solutions, multi-soliton-like solutions, and special-soliton-like solutions. Since the KP equation and cylindrical KP equation are all special cases of the VCGKP equation, and the corresponding results of these equations are also given respectively.

Exact semi-geostrophic flows in an elliptical ocean basin**Appendix and figures prepared by Maxim Trokhimtchouk.

A new family of exact solutions is analysed, which models two-dimensional circulations of an ideal fluid in a uniformly rotating elliptical tank, under the semi-geostrophic approximation from meteorology and oceanography. The fluid pressure and stream function remain quadratic functions of space at each instant in time, and their fluctuations are described by a single degree of freedom Hamiltonian system depending on two conserved parameters: domain eccentricity and the constant value of potential vorticity. These parameters determine the presence or absence of periodic orbits with arbitrarily long periods, fixed points of the dynamics, and aperiodic homoclinic orbits linking hyperbolic saddle points. The energy relative to these parameters selects the frequency and direction in which isobars nutate or precess, as well as the steady circulation direction of the fluctuating flow. The canonically conjugate variables are the moment of inertia and angle of inclination of an elliptical inverse-potential-vorticity patch evolving in dual coordinates.

A Computational Approach to the New Type Solutions of Whitham–Broer–Kaup Equation in Shallow Water

Based on computerized symbolic computation, a new method and its algorithm are proposed for searching for exact travelling wave solutions of the nonlinear partial differential equations. Making use of our approach, we investigate the Whitham–Broer–Kaup equation in shallow water and obtain new families of exact solutions, which include soliton-like solutions and periodic solutions. As its special cases, the solutions of classical long wave equations and modified Boussinesq equations can also be found.

Generalized transformations and abundant new families of exact solutions for (2 + 1)-dimensional dispersive long wave equations

A new generalized transformation based upon the well-known Riccati equation is presented and applied to find exact solutions of the (2 + 1)-dimensional dispersive long wave equation. Many explicit exact solutions are obtained that include new solitary and periodic wave solutions, and combined formal solitary and periodic wave solutions. The variant Boussinesq equation, as a special case of the (2 + 1)-dimensional dispersive long wave equation, is also solved. These exact solutions are obtained computationally using the Wu elimination method with the aid of Mathematica to solve the large system of algebraic equations produced by the solution ansatz.

New types of exact solutions for a breaking soliton equation

In this paper based on a system of Riccati equations, we present a newly generally projective Riccati equation expansion method and its algorithm, which can be used to construct more new exact solutions of nonlinear differential equations in mathematical physics. A typical breaking soliton equation is chosen to illustrate our algorithm such that more families of new exact solutions are obtained, which contain soliton-like solutions and periodic solutions. This algorithm can also be applied to other nonlinear differential equations.

Soliton-like solutions for a (2+1) -dimensional nonintegrable KdV equation and a variable-coefficient KdV equation

Based on a Riccati equation and a symbolic computation system-Maple, a generalized Riccati equation expansion method is presented for constructing soliton-like solutions and periodic form solutions for some nonlinear evolution equations (NEEs) or NEEs with variable coefficients. Compared with most of the existing tanh methods, the extended tanh-function method, the modified extended tanh-function method and the generalized hyperbolic-function method, the proposed method is more powerful. We study a (2+1)-dimensional general nonintegrable KdV equation, a KdV equation with variable coefficients. As a result, rich new families of exact solutions, including the non-travelling wave's and coefficient functions' soliton-like solutions, singular soliton-like solutions, periodic form solutions, are obtained. When setting the arbitrary functions in some solutions be equal to special constants or special functions, the solitary wave solutions can be recovered.

Symbolic Computation and Construction of Soliton-like Solutions of some Nonlinear Evolution Equations

Based on the computerized symbolic computation system Maple and a Riccati equation, a new generalized Riccati equation expansion method for constructing non-travelling waves and coefficient functions’ soliton-like solutions of nonlinear evolution equations (NEEs) is presented by a general ansatz. Compared with most of the existing tanh methods, namely the extended tanh-function method, the modified extended tanh-function method and the generalized hyperbolic-function method, the proposed method is more powerful. By use of the method one can not only successfully recover the previously known formal solutions but also construct new and more general formal solutions for some NEEs. The cylindrical Korteweg-de Vries (CkdV) equation, a Potential Kadomstev-Petviashvili (PKP) equation, the two-dimensional KdV-Burgers equation are chosen to illustrate our method such that rich new families of exact solutions, including the non-travelling wave soliton-like solutions, singular soliton-like solutions, periodic form solutions are obtained. When taking arbitrary functions of the solutions as some special constants, the known travelling wave solutions of the PKP equation, two-dimensional KdV-Burgers equation can be recovered.

Symbolic computation and construction of soliton-like solutions for a breaking soliton equation

Based on the symbolic computation system–– Maple and a Riccati equation, by introducing a new more general ansätz than the ansätz in the tanh method, extended tanh-function method, modified extended tanh-function method, generalized tanh method and generalized hyperbolic-function method, we propose a generalized Riccati equation expansion method for searching for exact soliton-like solutions of nonlinear evolution equations and implemented in computer symbolic system––Maple. Making use of our method, we study a typical breaking soliton equation and obtain new families of exact solutions, which include the nontravelling wave’ and coefficient function’ soliton-like solutions, singular soliton-like solutions and periodic solutions. The arbitrary functions of some solutions are taken to be some special constants or functions, the known solutions of this equation can be recovered.

Vortex stretching driven by Jeffery‐Hamel flow

AbstractWe present a new family of exact solutions of the Navier‐Stokes equations. The new flows, which result from vortex stretching, are generated by the well‐known class of Jeffery‐Hamel flows. These new solutions represent three‐dimensional flows and can be regarded as exact solutions or as perturbations of Jeffery‐Hamel flows. We also consider the effects of confining side‐walls on a particular Jeffery‐Hamel flow.

The Riccati equation with variable coefficients expansion algorithm to find more exact solutions of nonlinear differential equations

In this paper based on a system of Riccati equations with variable coefficients, we present a new Riccati equation with variable coefficients expansion method and its algorithm, which are direct and more powerful than the tanh-function method, sine–cosine method, the generalized hyperbolic-function method and the generalized Riccati equation with constant coefficient expansion method to construct more new exact solutions of nonlinear differential equations in mathematical physics. A pair of generalized Hamiltonian equations is chosen to illustrate our algorithm such that more families of new exact solutions are obtained which contain soliton-like solution and periodic solutions. This algorithm can also be applied to other nonlinear differential equations.

Generalized Separation of Variables in Nonlinear Heat and Mass Transfer Equations

We outline generalized separation of variables as applied to nonlinear second-order partial differential equations (PDEs). In this context, we suggest a method for constructing exact solutions of nonlinear PDEs. The method involves searching for transformations that “reduce the dimensionality” of the equation. New families of exact solutions of 3D nonlinear elliptic and parabolic equations that govern processes of heat and mass transfer in inhomogeneous anisotropic media are described. Moreover, the method makes it possible to construct exact solutions of nonlinear wave equations. We also present solutions for three families of equations with logarithmic heat sources; the solutions are obtained by nonlinear separation of variables.

Autonomous Forms and Exact Solutions of Equations of Motion of Polytropic Gas

The system of motion of a polytropic gas can be reduced to an autonomous form by using group analysis. A new family of exact solutions are constructed.

Exact solutions for the problem of source location from measured time differences of arrival

Source location using the difference in time of arrival of a signal (TDOA) at an array of receivers is used in many fields including speaker location. Using three TDOA values from four noncollinear receivers one can, in principle, solve for the unknown source coordinates in terms of the receiver locations. However, the equations are nonlinear, and in practice signals are contaminated by noise. Practical systems often use multiple receivers for accuracy and robustness, and improved S/N, and solutions must be obtained via minimization. A new family of exact solutions, for the case of four receivers located in a plane, is presented. These solutions can be evaluated using a small number of arithmetic operations. The performance of these solutions for several practical situations is examined via simulation, and possible geometries of receiver locations suggested. For the case where multiple receivers (>4) are used, a new formulation is presented, that incorporates the present solutions, imposes additional constraints on source location, and enables use of a constrained L1 optimization procedure to achieve a robust estimate of the source location. The present estimator is compared with ones from the literature, and found to be robust and accurate, and more efficient. [Work partially supported by DARPA.]

New conditional symmetries and exact solutions of the nonlinear wave equation

We have constructed the new class of conditional symmetries of the nonlinear wave equation in 1 + 3 dimensions. These symmetries enable us to obtain broad families of new exact solutions of the nonlinear wave equation containing up to the four arbitrary functions.

THE HOMOGENEOUS BALANCE METHOD FOR SOLVING NONLINEAR SOLITON EQUATIONS

The homogeneous balance method is very effective for finding exact solutions of nonlinear soliton equations.But only soliton-type solutions can be obtained.In this paper we improve some key steps in this method,then by using modified homogeneous balance method we are able to obtain some new families of exact solutions to nonliear soliton equations.Solitary wave solutions are merely a special case of one famility.

New families of exact solutions to the integrable dispersive long wave equations in 2+1-dimensional spaces

The integrable dispersive long wave equations, especially the higher dimensional ones, are of current interest in both physics and mathematics. Obtained in this paper, via a symbolic-computation-based method, are new families of exact solutions to the 2+1-dimensional integrable dispersive long wave equations. Sample solutions from those families are presented. Solitary waves are merely a special case in one family.

Conditional symmetry and new classical solutions of the Yang-Mills equations

We suggest an effective method for reducing the Yang-Mills equations to systems of ordinary differential equations. With the use of this method we construct extensive families of new exact solutions of the Yang-Mills equations. Analysis of the solutions thus obtained shows that they correspond to the conditional (non-classical) symmetry of the equations under study.

Exact solutions for the Bogoyavlenskii coupled KdV equations

The application of the Painlevé analysis and computer algebra leads to a new family of exact solutions to the recently-proposed Bogoyavlenskii coupled KdV equations, via an auto-Bäcklund transformation. Sample solutions are presented.