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Three classes of exact solutions to Klein-Gordon-Schrodinger equation.

[section]1. Introduction

The investigation of exploring the exact traveling wave solutions of nonlinear mathematical physics equations plays an important role in the study of the solitary wave solution. Up to now, a variety of methods, such as inverse scattering method, Backhund, transformation method, Hirota, bilinear method, homogeneous balance method and tanh method [1-5]. The tanh method is one of the most direct and effective algebraic for computing the exact traveling wave solution. In this paper, we extend the method by replacing tanh function with some other functions f(x), such as polynomial function, trigonometric and elliptical function, where the choice of the function f(x) is different according to equation. Similarly to the steps introduced in [6], we can get the exact solution for equation, namely, assume that after simplifying the solution of the simplified nonlinear PDE, equation has the following form.

u([xi]) = [a.sub.m]f[([xi]).sup.m] + [a.sub.m - 1]f[([xi]).sup.m - 1] +... + [a.sub.0]/[b.sub.m]f[([xi]).sup.m] + [b.sub.m - 1]f[([xi]).sup.m - 1] + (x) + [b.sub.0]. (1)

Then the solution can be obtained by using the above method and this develops the tanh method.

[section]2. The extended tanh method

Consider the nonlinear PDE

F(u, [u.sub.t], [u.sub.x], [u.sub.xx],...) = 0, (2)

with two variables x, t. Let u(x, t) = U([xi]), [xi] = x - Vt be its travelling wave solutions, where the wave velocity V is a coefficient to be determined later, then equation (2) can be simplified to a nonlinear ODE

G(U, U', U",...) = 0. (3)

Assume that the solution of equation (3) has the form of (1). Substituting (1) into (3), then m can be determined by balancing the linear terms of the highest order in the resulting equation with the highest order nonlinear terms. This will give a system of algebra equations involving [a.sub.i], [b.sub.i](i = 0, 1, ..., 2m) and [alpha].

Let the function f(x) be [e.sup.[alpha][xi]], then the main steps for achieving the above method is as following.

Step 1 Suppose that equation (2) has the wave traveling solution u(x, t) = U([xi]) = U(x - VT). Then equation (2) can be simplified to a nonlinear ODE (3);

Step 2 assume that the solution of (3) has the form of (1), then m can be determined by using the balance method;

Step 3 substituting (1) into (3), we get a rational fractional equation

P([e.sup.[alpha][xi]]) = 0, (4)

which is just a polynomials of exponential [e.sup.[alpha][xi]];

Step 4 collecting all the terms with the same power of [e.sup.[alpha][xi]] yields a set of algebraic system for [a.sub.i], [b.sub.i](i = 0, 1, ..., 2m), [alpha], where [a.sub.i], [b.sub.i](i = 0, 1, ..., 2m), [alpha] are coefficients to be determined later;

Step 5 applying some mathematical package, for example, Mathematica, Maple, and etc., we can deal with the above tedious algebra equations and output directly the required solution. Further, substituting the solution into (1) and letting, [xi] = x - Vt, we can get the exact solution of equation (2).

[section]3. The exact solutions of Klein-Gordon-Schrodinger equation

KGS equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

is a classical model which reflects the interaction of nucleon-field and meson-field, where [DELTA] is a n, dimensional Laplace operator, [psi] is scalar complete nucleon-field and [phi] is meson-field [7]. In

[8] the author had proved that the steady-state solution of equations (5) has the form

([psi](x, t), [phi](x, t)) = ([e.sup.i[omega]t]u(x), v(x)), (6)

where x [member of] [R.sup.3], [omega] [member of] R. We will take into account the exact solution of equation (5) by using the technique basing on the expansion of the rational function method.

Here we only focus on the (1 + 1) dimensional equation, the discussion for (1 + n) dimensional equation is similar and is omitted. Consider the following equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

where [psi] is complex. Assume that

[psi] = [e.sup.i[eta]]u(x, t), (8)

where [eta] = [alpha]x + [beta]t and [alpha], [beta] are coefficients which will be determined later. Substituting (8) into equations (7) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

Let u(x, t) = U([xi]) = U(kx + [omega]t), V([xi]) = [phi](kx + ut), where k and [omega] are coefficients to be determined later. Then we obtain a simplified ODE.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)

Suppose that the solution, U([xi]) and V([xi]), for equations (10) are expansion of a second order differential function, then we can get the exact solution of equations (9) basing on the homogeneous balance principle.

[section]4. Main result

1. Suppose that the solution for equations (10) has the following form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (11)

where [a.sub.i], [b.sub.i], [c.sub.i], [d.sub.i] (i = 0, 1, ..., m) are coefficients to be determined later. By comparing the highest derivative term with the highest nonlinear term in homogeneous balance equations, we get m = 2. Thus equation (10) has the solution of the following form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (12)

where a, b, c, f, g, h, 0, p, q, r, s, w are coefficients to be determined later. Substituting (12) into (10), we get a rational fractional equations which is just for [xi]. Collecting all the terms with the same power [[xi].sup.k] and letting their coefficients be zero yields a set of algebraic system for, a, b, c, f, g, h, 0, p, q, r, s, w, k, [alpha], [beta], [omega]. Using Maple package, we can deal with the above algebra equations and output directly the required solution. Further, substituting the solution into (12) and (8), we obtain the following exact solution for equation (7).

Case 1. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Case 2. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Case 3. [beta] = i, q = -[k.sup.2]r, b = ifkm, f = f, a = h = w = s = 0 = p = 0, c = ikmg, [omega] = -k, [alpha] = 1, m = m, r = r, k = k, g = g, [[psi].sub.3](x, t) = i[e.sup.ix-t]m/x - t, [[phi].sub.3](x, t) = -1/ [(x - t).sup.2].

Case 4. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Case 5. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Case 6. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

2. Suppose that the solution of equations (10) has the following form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (13)

where [a.sub.i], [b.sub.i], [c.sub.i], [d.sub.i] (i = 0, 1, ..., m) are coemcients to be determined later. By comparing the highest derivative term with the highest nonlinear term in homogeneous balance equations of equations (10), we get m = 2. Thus equation (10) has the solution of the following form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (14)

where a, b, c, f, g, h, 0, p, q, r, s, w are coemcients to be determined later. Substituting (14) into equations (10), we get a rational fractional equations which is just for tanh[xi].

Collecting all the terms with the same power of tanh[xi] and letting their coefficients be zero yields a set of algebraic system for a, b, c, f, g, h, 0, p, q, r, s, w, k, [alpha], [beta], [omega]. Using Maple, package, we can deal with the above algebra equations and output directly the required solution. Further, substituting the solution and [eta] = [alpha]x + [beta]t, [xi] = kx + [omega]t into (14) and (8), we obtain the following exact solution for equation (7).

Case 1. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Case 2. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Case 3. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

3. Suppose that the solution of equations (10) has the following form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (15)

where [a.sub.i], [b.sub.i], [c.sub.i], [d.sub.i](i = 0, 1, ..., m) are coefficients to be determined later. By comparing the highest derivative term with the highest nonlinear term in homogeneous balance equations for (10), we get m = 2, namely, equation (10) has the solution of the following form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (16)

where a, b, c, f, g, h, 0, p, q, r, s, w are coefficients to be determined later. Substituting (16) into equations (10), we get a rational fractional equations which is just for [e.sup.[xi]]. Collecting all the terms with the same power of [e.sup.[xi]] and letting their coefficients be zero yields a set of algebraic system for a, b, c, f, g, h, 0, p, q, r, s, w, k, [alpha], [beta], [omega]. Using Maple, package, we can deal with the above algebra equations and output directly the required solution. Further, substituting the solution and [eta] = [alpha]x + [beta]t, [xi] = kx + [omega]t into (15) and (8), we obtain the following exact solution for equation (7).

Case 1. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Case 2. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Remark 4.1. We claim that the tanh method can be extended by replacing tanh function with some generalized functions f(x), such as polynomial function, trigonometric function and Jacobi elliptical function. As an example, we obtain three classes exact solutions for KGS equation.

References

[1] M. J. Ablowitz and Solitons, Nonlinear evolution equations and incerse scattering, Cambridge University Press, New York, 1(1991).

[2] H. D. Wahlquist and F. B. Estabrook, Backhund transformations for solitons of the Korteweg-de Viru equation, Phys. Rev. Lett., 31(1971).

[3] R. Hirota and Solitons, Berlin: Springer, 1980.

[4] M. L. Wang, Exact solutions for a compound KdV-Burgers equation, Phys. Lett. A., (1996), 213.

[5] E. J. Parkes and B. R. Duffy, An automated tanh-function method for finding solitary wave solutions to nonlinear evolution equations, Comp. Phys Commun, 98(1996).

[6] H. M. Xia, An extension of tanh-function method and its application, J. of Gansu sciences, 4(2006).

[7] F. Tsutsumim, On coupled Klein-Gordon-Schrodinger equations, J. Math. Analysis Applic, 66(1978).

[8] M. Ohta, Stability of stationary states for the coupled Klein-Gordon-Schrodinger equations, Nonlinear Analysis, 27(1996).

Hongming Xia ([dagger]), Wansheng He ([double dagger]), Linxia Hu (#); and Zhongshe Gao (#);

College of Mathematics and Statistics, Tianshui Normal University, Tianshui-741001, P. R. China E-mail: xia-hm@sina.com

(1) This research is supported by the SF of education department of Gansu province (0608-04)
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Author:Xia, Hongming; He, Wansheng; Hu, Linxia; Gao, Zhongshe
Publication:Scientia Magna
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Date:Sep 1, 2013
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