Two Sub-Ordinary Differential Equations And Their Applications For Finding The Exact Solutions Of Some Nonlinear Evolution Equations

In the present paper we study two sub-ordinary differential equations (Sub-ODEs for short):(where A, B, D and E are real constants, A≠ 0, u =u(ξ) and(where ι, m and n are real constants). We study their solutions and use them to solve some nonlinear evolution equations. (That is say, we obtain the solutions of the nonlinear evolution equations through the solutions of two sub-ordinary differential equation (1) directly and obtain the solutions of complicate nonlinear evolution equations in simple method using the sub-ordinary differential equation (2).)1. The exact solutions of Eq.(1) and its applications(1.1) Used methods(1.1.1) The extended F-expansion methodNow, given a nonlinear PDE:generally speaking, the left-hand side of Eq.(3) is a polynomial u and its various partial derivatives.Seeking its travelling wave solutions of the form:where c is a constant to be determined, inserting(4)into Eq.(3) yields an ODE for u(ξ):Supposing that u (ï¿¡) can be expanded as follows:j=-N i=0where a^j - -N, -N + 1, ? ? ?, 0,1, ? ? ? N, i = 0,1, ? â–  ?, N) are constants to be determined later and clon ^ 0, N is the balance number determined by the subtle balance mechanism in nonlinear PDE which demands the balance of the nonlinear term and the highest order partial derivative term.F(ï¿¡)andG(ï¿¡)satisfy the following relations:F'2 = PFA + QF2 + R,₃,f + Q) G> \*L + ? Uvl + n J It* V^where P, Q and R are constants, andG2(O = nF2(Q + v, (8)where /z and v are constants.So we get an equation for F(f)and G(ï¿¡)after the substitution of (6) into (5). Collecting the similar terms and setting the each coefficient of the polynomial term to zero, a series of equations for the parameters a^j = —AT, —N + 1, ? ? ?, 0,1, ? ? ? N, i = 0,1, ? ? ?, N) and c are obtained. Solving these algebraic equations to determine the parameters . Substituting different kinds of Jacobi elliptic functions into (7)to get the relations between the parameters P, Q and R and the modulus m of Jacobi elliptic function, thus we get different kinds of Jacobi elliptic functions solutions of (3).(1.1.2) Now suppose that ?(0m (5) can be expanded as follows:?(O = E [^(0 + WOF'-HO] + ao, (9)where ao,ai and bt(i = 1, â–  ? ?, N) are constants to be determined later, and on 7^ 0. This time use the projective Riccati equations as Sub-ODEs:(10)andG2 = AF2(O - 27F(0 + R, (11)where A, 7 and R are constants. We can find more solutions of the nonlinear evolution equation.(1.2) The solutions of the Sub-ODE(l)Using the method in (l.l.l),we got different kinds of Jacobi elliptic functions solutions of Eq.(l),including the single function solutions and the combined function solutions. For example: B , mV2AAâ– edit)cd(OB2D = — -1-m2 -6m oA1Ain & ? lwhere request ( D = — + — + -2AB(-2B2 + 9AD)In iti, it2, U3,m is modulu of Jacobi elliptic functions, A > 0, E =This solutions will degenerate as hyperbolic and trigonometric functions when m —? 1, and m —> 0,respectively . ( we list them in table 3 and table 4 ). With these tables one can get the corresponding hyperbolic function solutions and trigonometric function solutions of Eq.(l).Using the method in (1.1.2), the solutions with two arbitrary parameters of Eq.(l) can be expressed by the hyperbolic functions, the trigonometric functions and the rational functions respectively.B _, y/2AR{y/72 - (of - ap - [pn sinh Vl% + a2 cosh 3j4 2A I ai cosh \/#ï¿¡ + a2 sinh \/ft) + 71where R > 0, ai,a2 are arbitrary constants, A > 0,72 — (a2 — a2) > 0;B y/—2AR{ v/(o;2 -I- a2) — 72 + (a2 cos \/—Rï¿¡) — a\ sin v^W5 = -TTT^3^4 2>1 [c*i cos y/^Rï¿¡ + ^2 sin \/—Rï¿¡ + 7]where i? < 0, ai, a2 are arbitrary constants, A > 0,72 — (a2 + a2) > 0;B v^g+4^)-BU6 3i4 2A(-7ï¿¡2 + a^ + a2)where ati, a2 are arbitrary constants, A > 0,o:f + 47a2 > 0.(1.3) Applicating Eq.(l) to obtain the exact solutions of nonlinear evolution equationsBecause the mKdV equation, the Klein-Gordon equation, the new Hamilton amplitude equation, the Zakharov equations,etc. can be travelling wave reducted into the special form of the Eq.(l). For example the mKdV equation:ut + au2ux + (3uxxx = 0, (12)where a > 0;Looking for its travlling wave solutions of Eq.(12) in the form:u{x,t) = u{Z),S = x-ct, (13)where c is parameter to be determined later.Substituting (13) into (12) yields an ODE for ?(ï¿¡):-cu + \au3 + 0u" = 0. (14)oOf.Eq.(14)is the special form of the Eq.(l) when the coefficients A =-----, B =c 3^0, D = —, E = 0. So using the Eq.(l), we can get the solutions of the nonlinear evolution equations above.2.solutions of Eq.(2) and applications of Eq.(2) solving the nonlinear evolution equation with nonlinear terms of any order.(2.1)Solutions of Eq.(2):(2.1.1) when I > 0,n> 0,ra = —2\//n, Eq.(2) the solution as following:2Vn l±*?"l-<(15)(2.1.2)when I > 0,n = (a2 - l)l,m = -2al, where -1 < a < 1, the solution of Eq.(2) as following:(16)a + cosh In particularly, whencr — 0(/JV (17)(2.1.3)when I < 0,n = (a2 — l)/,m = -2al, where a > 1, the solutions of Eq.(2) as following:F(0 = _^ \ ,-,.. (18)(T ± COS (v —j\? (19)cr ± sm (v —k)(2.2)applications of Eq.(2) solving the nonlinear evolution equation with nonlinear terms of any order. Using the theory of F-expansion method (where F is a known solution of an ordinary different equation with nonlinear term of four orders) and the sub-ODE(2), we propose a method to solve the nonlinear evolution equation with nonlinear terms of any order:Given a PDE (3) with nonlinear terms of any order, we take Eq.(2) as the sub-ODE. In the extended F-expansion method, and look for the solutions of Eq.(5) in the form :?(0 = FN(0, (20)where F(ï¿¡) is a positive solution of Eq.(2). As an example, we solve the generalized BBM equation with nonlinear terms of any order, the generalizednonlinear Schrodinger equation with nonlinear terms of any order. Some solutions are obtained, including the bell and kink type solitary wave solutions, triangular periodic wave solutions.