92 research outputs found

    Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs

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    Algorithms are presented for the tanh- and sech-methods, which lead to closed-form solutions of nonlinear ordinary and partial differential equations (ODEs and PDEs). New algorithms are given to find exact polynomial solutions of ODEs and PDEs in terms of Jacobi's elliptic functions. For systems with parameters, the algorithms determine the conditions on the parameters so that the differential equations admit polynomial solutions in tanh, sech, combinations thereof, Jacobi's sn or cn functions. Examples illustrate key steps of the algorithms. The new algorithms are implemented in Mathematica. The package DDESpecialSolutions.m can be used to automatically compute new special solutions of nonlinear PDEs. Use of the package, implementation issues, scope, limitations, and future extensions of the software are addressed. A survey is given of related algorithms and symbolic software to compute exact solutions of nonlinear differential equations.Comment: 39 pages. Software available from Willy Hereman's home page at http://www.mines.edu/fs_home/whereman

    A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo-Miwa equation

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    A direct approach to exact solutions of nonlinear partial differential equations is proposed, by using rational function transformations. The new method provides a more systematical and convenient handling of the solution process of nonlinear equations, unifying the tanh-function type methods, the homogeneous balance method, the exp-function method, the mapping method, and the F-expansion type methods. Its key point is to search for rational solutions to variable-coefficient ordinary differential equations transformed from given partial differential equations. As an application, the construction problem of exact solutions to the 3+1 dimensional Jimbo-Miwa equation is treated, together with a B\"acklund transformation.Comment: 13 page

    Integrability and Exact Solutions for a (2+1)-dimensional Variable-Coefficient KdV Equation

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    By using the WTC method and symbolic computation, we apply the Painlevé test for a (2+1)-dimensional variable-coefficient Kortweg-de Vries (KdV) equation, and the considered equation is found to possess the Painlevé property without any parametric constraints. The auto-Bǎcklund transformation and several types of exact solutions are obtained by using the Painlevé truncated expansion method. Finally, the Hirota’s bilinear form is presented and multi-soliton solutions are also constructed

    Traveling wave solutions for the Couple Boiti-Leon-Pempinelli System by using extended Jacobian elliptic function expansion method

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    In this work, an extended Jacobian elliptic function expansion method is pro-posed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to the Couple Boiti-Leon-Pempinelli System which plays an important role in mathematical physics

    Integrable systems with BMS3_{3} Poisson structure and the dynamics of locally flat spacetimes

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    We construct a hierarchy of integrable systems whose Poisson structure corresponds to the BMS3_{3} algebra, and then discuss its description in terms of the Riemannian geometry of locally flat spacetimes in three dimensions. The analysis is performed in terms of two-dimensional gauge fields for isl(2,R)isl(2,R). Although the algebra is not semisimple, the formulation can be carried out \`a la Drinfeld-Sokolov because it admits a nondegenerate invariant bilinear metric. The hierarchy turns out to be bi-Hamiltonian, labeled by a nonnegative integer kk, and defined through a suitable generalization of the Gelfand-Dikii polynomials. The symmetries of the hierarchy are explicitly found. For k≥1k\geq 1, the corresponding conserved charges span an infinite-dimensional Abelian algebra without central extensions, and they are in involution; while in the case of k=0k=0, they generate the BMS3_{3} algebra. In the special case of k=1k=1, by virtue of a suitable field redefinition and time scaling, the field equations are shown to be equivalent to a specific type of the Hirota-Satsuma coupled KdV systems. For k≥1k\geq 1, the hierarchy also includes the so-called perturbed KdV equations as a particular case. A wide class of analytic solutions is also explicitly constructed for a generic value of kk. Remarkably, the dynamics can be fully geometrized so as to describe the evolution of spacelike surfaces embedded in locally flat spacetimes. Indeed, General Relativity in 3D can be endowed with a suitable set of boundary conditions, so that the Einstein equations precisely reduce to the ones of the hierarchy aforementioned. The symmetries of the integrable systems then arise as diffeomorphisms that preserve the asymptotic form of the spacetime metric, and therefore, they become Noetherian. The infinite set of conserved charges is recovered from the corresponding surface integrals in the canonical approach.Comment: 34 pages, 2 figure

    Application of Bernoulli Sub-ODE Method For Finding Travelling Wave Solutions of Schrodinger Equation Power Law Nonlinearity

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    In this paper, the exact travelling wave solution of the Schr¨odinger equation with power law nonlinearity is studied by the Sub-ODE method. It is shown that the method is one of the most effective approaches for finding exact solutions of nonlinear differential equations

    Solitary, Explosive, Rational and Elliptic Doubly Periodic Solutions for Nonlinear Electron-Acoustic Waves in the Earth’s Magnetotail Region with Cold Electron Fluid and Isothermal Ions

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    A theoretical investigation has been made of electron acoustic wave propagating in unmagnetized collisionless plasma consisting of a cold electron fluid and isothermal ions with two different temperatures obeying Boltzmann type distributions. Based on the pseudo-potential approach, large amplitude potential structures and the existence of Solitary waves are discussed. The reductive perturbation method has been employed to derive the Korteweg-de Vries equation for small but finite amplitude electrostatic waves. An algebraic method with computerized symbolic computation, which greatly exceeds the applicability of the existing tanh, extended tanh methods in obtaining a series of exact solutions of the KdV equation, is used here. Numerical studies have been made using plasma parameters close to those values corresponding to Earth’s plasma sheet boundary layer region reveals different solutions i.e., bell-shaped solitary pulses and singularity solutions at a finite point which called “blowup” solutions, Jacobi elliptic doubly periodic wave, a Weierstrass elliptic doubly periodic type solutions, in addition to the propagation of an explosive pulses. The result of the present investigation may be applicable to some plasma environments, such as earth’s magnetotail region and terrestrial magnetosphere
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