- Email: [email protected]
Special symmetries to standard Riccati equations and applications
Applied Mathematics and Computation 216 (2010) 3089–3096
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Special symmetries to standard Riccati equations and applications Cesar A. Gómez S. a,*, Alvaro Salas b,c a b c
Department of Mathematics, Universidad Nacional de Colombia, Bogotá, Colombia Department of Mathematics, Universidad de Caldas, Manizales, Colombia Department of Mathematics, Universidad Nacional de Colombia, Manizales, Colombia
a r t i c l e
i n f o
Keywords: Riccati equation Lie groups Solitons
a b s t r a c t By using symmetries associated to Riccati equation in standard form (SRE), we obtain a family which can be integrated by quadratures. As a consequence, we get a new integrability condition for the generalized Riccati equation (GRE). We illustrate the result with some examples and we give some applications in the solitons theory. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction The simplest nonlinear ordinary differential equation is given by the general Riccati equation (GRE)
d/ðnÞ ¼ pðnÞ/2 ðnÞ þ qðnÞ/ðnÞ þ rðnÞ; dn
ð1:1Þ
where p(n), q(n) and r(n) are continuous functions, defined in some interval ½a; b # R. This equation is used in different fields of pure and applied mathematics, in theoretical physics, control theory and relaxation problems [1–3]. It is well known that solutions of particular cases of (1.1) are used in a great variety of computational methods to obtain exact solutions for nonlinear evolution equations (NEE) [4–8]. On the other hand, in spite of its apparent simplicity, the general solution of (1.1) cannot be expressed in a elementary form, except in some particular cases [3,9–14]. The principal objective of this paper is to obtain a new integrability condition for the generalized Riccati equation (1.1), starting from the Riccati equation in the standard form which is given by the equation
Eðn; uÞ ¼ u0 ðnÞ ¼ u2 ðnÞ þ nðnÞ;
ð1:2Þ
and making use of Lie groups theory [15–22]. More exactly, we obtain a group of symmetry for (1.2) and as a consequence, we obtain a family of Riccati equations in standard form which is integrable by quadratures. In accordance with the result, a new integrability condition for (1.1) is derived. 2. Symmetries to SRE (1.2) First, we need some preliminaries: Definition 2.1. The symmetries of (1.2) are given by the elements of a connected Lie group with parameter a
(
n ¼ n þ af ðn; uÞ þ oða2 Þ u ¼ u þ ahðn; uÞ þ oða2 Þ;
* Corresponding author. E-mail addresses: [email protected] (C.A. Gómez S.), [email protected] (A. Salas). 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.04.039
ð2:1Þ
3090
C.A. Gómez S., A. Salas / Applied Mathematics and Computation 216 (2010) 3089–3096
which transform solutions into solutions. Alternatively, the infinitesimal generators of the Lie algebra of (2.1), which are the components of the vector field associated to (2.1)
@ ¼ f ðn; uÞ@ n þ hðn; uÞ@ u ; @ , @n
ð2:2Þ
@ @u
where @ n ¼ and @ u ¼ are called symmetries of (1.2). The symmetry variables f(n, u), h(n, u) can be found by solving the following equation, which is called determinant equation [15–22]:
hn ðn; uÞ þ ðhu ðn; uÞ fn ðn; uÞÞEðn; uÞ fu ðn; uÞE2 ðn; uÞ f ðn; uÞEn ðn; uÞ hðn; uÞEu ðn; uÞ ¼ 0;
ð2:3Þ
where E(x, u) is as in (1.2). Eq. (2.3) does not split into an overdetermined system, therefore it has an infinite set of solutions. That is why we can hope find solutions of certain forms only. On the other hand, we can see that (2.3) can be written as
(
fu ðn; uÞu4 ðnÞ þ ð2nðnÞfu ðn; uÞ þ hu ðn; uÞ fx ðn; uÞÞu2 ðnÞ 2hðn; uÞuðnÞ þðnðnÞfhu ðn; uÞ fn ðn; uÞ fu ðn; uÞnðnÞg þ hn ðn; uÞ f ðn; uÞn0 ðnÞÞ ¼ 0:
ð2:4Þ
Respect to this last equation we have our first result which say us that f depend only on n and does not of u(n): Proposition 2.2. Let f ðn; uÞ ¼
Pm
Proof. Substituting f ðn; uÞ ¼
Pm
mþ2 X
i 0 fi ðnÞu ðnÞ,
0 fi ðnÞu
i
h(n, u) = h0(n) + h1(n)u(n), a solution of (2.4). Then f(n, u) = f(n).
ðnÞ and h(n, u) = h0(n) + h1(n)u(n) into (2.4), we obtain an expression in the form
F i ðnÞui ðnÞ mfm ðnÞumþ3 ðnÞ ¼ 0;
ð2:5Þ
i¼0
where Fi(n) does not depend of u(n). We give the proof by using induction: For m = 1, (2.5) reduces to
8 f1 ðnÞu4 ðnÞ > > > > 0 3 > > < f1 ðnÞu ðnÞ ðh1 ðnÞ þ 2f 1 ðnÞnðnÞ þ f00 ðnÞÞu2 ðnÞ > > 0 0 > > > þðk1 ðnÞ ðf1 ðnÞnðnÞÞ 2h0 ðnÞÞuðnÞ > : 0 þðh0 ðnÞ þ h1 ðnÞnðnÞ f1 ðnÞn2 ðnÞ ðf0 ðnÞnðnÞÞ0 Þ ¼ 0:
ð2:6Þ
Equaling the coefficients of (2.6) to zero, we have f1(n) = 0. Suppose that the assertion is true for m = k (f1(n) = , , = fk(n) = 0). We consider
8 Hðf Þ ¼ > > > > 4 > > < fu ðn; uÞu ðnÞ þð2nðnÞfu ðn; uÞ þ hu ðn; uÞ fx ðn; uÞÞu2 ðnÞ > > > > 2hðn; uÞuðnÞ > > : þðnðnÞfhu ðn; uÞ fn ðn; uÞ fu ðn; uÞnðnÞg þ hn ðn; uÞ f ðn; uÞn0 ðnÞÞ; P P i kþ1 GðnÞ ¼ ki¼0 fi ðnÞui ðnÞ, and f ðnÞ ¼ kþ1 ðnÞ. Taking into account (2.5), after simplifications we i¼0 fi ðnÞu ðnÞ ¼ GðnÞ þ fkþ1 ðnÞu obtain: H(f) = 0, if and only if
8 kþ2 > < P G ðnÞu ðnÞ kf ðnÞukþ3 ðnÞ fn2 ðnÞðk þ 1Þf ðnÞuk ðnÞ þ ðnðnÞf ðnÞÞ0 ukþ1 ðnÞ i i k kþ1 kþ1 i¼0 > : kþ2 0 kþ3 kþ4 þ2nðnÞðk þ 1Þfkþ1 ðnÞu ðnÞ þ fkþ1 ðnÞu ðnÞ þ ðk þ 1Þfkþ1 ðnÞu ðnÞg ¼ 0:
P k i This last equation implies fk+1(n) = 0. So that H(f) = 0, if and only if, HðGÞ ¼ H i¼0 fi ðnÞu ðnÞ ¼ 0. Using the induction hypothesis, fi(n) = 0 for i = 1, 2, . . . , k. So that f(n, u) = f0(n) and the proposition is proved. h As a consequence of the previous proposition, we have the more important result of this section: Theorem 2.3. The standard Riccati equation (1.2) admits the vector fields
f 00 ðnÞ @ ¼ f ðnÞ@ n f 0 ðnÞuðnÞ þ @u; 2
ð2:7Þ
where f(n) satisfies the third order ordinary differential equation
f 000 ðnÞ þ 4f 0 ðnÞnðnÞ þ 2n0 ðnÞf ðnÞ ¼ 0: Proof. In accordance with Proposition 2.2, we seek solutions to (2.4) in the form
ð2:8Þ
C.A. Gómez S., A. Salas / Applied Mathematics and Computation 216 (2010) 3089–3096
f ðn; uÞ ¼ f ðnÞ hðn; uÞ ¼ kðnÞ þ rðnÞuðnÞ:
3091
ð2:9Þ
Substituting (2.9) into (2.4) we get 0
ðrðnÞ f 0 ðnÞÞu2 ðnÞ þ ð2kðnÞ þ r0 ðnÞÞuðnÞ þ ðnðnÞrðnÞ ðnðnÞf ðnÞÞ0 þ k ðnÞÞ ¼ 0: Equaling the coefficients of this last equation to zero we obtain the following system
8 0 > < rðnÞ ¼ f ðnÞ 1 0 kðnÞ ¼ 2 r ðnÞ > : 0 nðnÞrðnÞ ðnðnÞf ðnÞÞ0 þ k ðnÞ ¼ 0:
ð2:10Þ
From the firsts two equations in (2.10) we have
1 hðx; uÞ ¼ f 00 ðnÞ f 0 ðnÞ: 2 Substituting the expressions for k(n) and r(n) in the third equation that appear in (2.10), finally Eq. (2.8) is obtained, and the proof is complete. h On the other hand, solving the system
@ðtðn; uÞÞ ¼ 0 @ðwðn; uÞÞ ¼ 1;
where @ is the vector field associate to Lie group (2.2), we obtain the so called canonical coordinates
(
t ¼ tðn; uÞ ¼ 12 f 0 ðnÞ þ f ðnÞuðnÞ R wðn; tÞ ¼ fdn : ðnÞ
ð2:11Þ
In this new variables, (1.2) reduces to following separable equation
dw 1 : ¼ 2 dt t þK
ð2:12Þ
Using the solutions of (2.12) and taking into account (2.11), after simplifications we obtain the following solutions of (1.2):
8 > 1 1 0 > R 1dn uðnÞ ¼ f ðnÞ þ > f ðnÞ 2 > > f ðnÞ > > > R > > ffiffi ffi ffiffi ffi p p > > K tan K fdn c 12f 0 ðnÞ > ðnÞ > uðnÞ ¼ > > f ðnÞ > > > R > < pffiffiffi pffiffiffi dn 1 0 K cot
K
K ¼ 0;
K > 0;
c 2f ðnÞ
f ðnÞ
uðnÞ ¼ K > 0; > f ðnÞ > > > > > pffiffiffiffiffi pffiffiffiffiffi R dn > > K tanh K f ðnÞc 12f 0 ðnÞ > > > uðnÞ ¼ K < 0; > f ðnÞ > > > > R > pffiffiffiffiffi pffiffiffiffiffi dn > > K coth K f ðnÞc 12f 0 ðnÞ > : uðnÞ ¼ K < 0; f ðnÞ
ð2:13Þ
where c is an integration constant and f(n) satisfies (2.8). We cannot find solutions of (2.8) in the case that n(n) is an arbitrary function. However, an analysis of (2.8) may be useful. In fact, if we consider (2.8) as an first order differential equation in the unknowns n(n) and we solve it, we obtain
nðnÞ ¼
ðf 0 ðnÞ2 Þ 2f ðnÞf 00 ðnÞ þ 4K ; 4f 2 ðnÞ
ð2:14Þ
where K is an integration constant. According with the previous results, Eq. (2.14) say us that the following family of Riccati equations in standard form
u0 ðnÞ ¼ u2 ðnÞ þ
ðf 0 ðnÞ2 Þ 2f ðnÞf 00 ðnÞ þ 4K ; 4f 2 ðnÞ
ð2:15Þ
is integrable by quadratures, and the respective solutions are obtained using (2.13). We have the following new integrability condition to generalized Riccati equation (1.1): Proposition 2.4. If in Eq. (1.1), the coefficients are defined in some interval ½a; b R and p(n) 2 C2[a, b], q(n) 2 C1[a, b], r(n) 2 C[a, b] are related as
3092
C.A. Gómez S., A. Salas / Applied Mathematics and Computation 216 (2010) 3089–3096
8 1 > > 4p2 ðnÞ > > > < fðpðnÞqðnÞÞ2 þ 4pðnÞ3 rðnÞ > 2pðnÞp0 ðnÞqðnÞ 3p02 ðnÞ þ 2p2 ðnÞq0 ðnÞ þ 2pðnÞp00 ðnÞg ¼ > > > > : f 02 ðnÞ2f ðnÞf 00 ðnÞþ4K ;
ð2:16Þ
4f 2 ðnÞ
2
with f(n) 2 C (a, b), f(n) – 0 a properly chosen function and K an arbitrary constant, then a solution to (1.1) can be obtain using elementary integration. Proof. With the change of variable
/ðnÞ ¼
( 0 ðnÞ) qðnÞ þ ppðnÞ 1 ; uðnÞ pðnÞ 2
ð2:17Þ
(1.1) reduces to (1.2), where
nðnÞ ¼
1 fðpðnÞqðnÞÞ2 þ 4pðnÞ3 rðnÞ 2pðnÞp0 ðnÞqðnÞ 3p02 ðnÞ þ 2p2 ðnÞq0 ðnÞ þ 2pðnÞp00 ðnÞg: 4p2 ðnÞ
Taking into account (2.14) we have the hypothesis given in the enunciate of the theorem. In this case, by the conclusions after Theorem 2.3, the solutions of (1.2) are obtained using (2.13), and by (2.17), solutions to (1.1) are derived by elementary integration as we want to show. h
3. Integrable families of Riccati equations In accordance with the previous results, we can construct a variety of families of Riccati equations in standard form that are integrable by quadratures [23]. In fact, if we take any f(n) 2 C2(a, b), f(n) – 0 and K arbitrary constant, the family of Riccati equations
u0 ðnÞ ¼ u2 ðnÞ þ
f 02 ðnÞ 2f ðnÞf 00 ðnÞ þ 4K ; 4f 2 ðnÞ
is integrable by quadratures, and its solution are given by (2.13). Moreover, f(n, u) = f(n) and hðn; uÞ ¼ 12 f 00 ðnÞ f 0 ðnÞ are the components of the vector field associated to Lie group (2.1) admitted by this family. The following families may be considered as important examples: (1) We consider the Riccati equation
/0 ðnÞ ¼ cðtÞ/2 ðnÞ þ bðtÞ/ðnÞ þ aðtÞ;
ð3:1Þ
where a(t), c(t) – 0, b(t) are functions that does not depend of n. By means of the substitution (2.17), then (3.1) reduces to following SRE
u0 ðnÞ ¼ u2 ðnÞ þ
4aðtÞcðtÞ b2 ðtÞ : 4
The hypothesis in Proposition 2.4, are satisfied if we take K ¼ 4aðtÞcðtÞb 4 (2.17) we get the following set of solutions to (3.1): (a) If a(t) – 0, c(t) – 0 and b(t) – 0:
8 bðtÞ 1 1 > > > cðtÞ n 2 > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 4aðtÞcðtÞb2 ðtÞ 4aðtÞcðtÞb2 ðtÞ > tan n bðtÞ > > c ðtÞ 2 2 2 > > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > < 1 4aðtÞcðtÞb2 ðtÞ 4aðtÞcðtÞb2 ðtÞ bðtÞ cot n /ðnÞ ¼ cðtÞ 2 2 2 > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > 2 2 b ðtÞ4aðtÞcðtÞ b ðtÞ4aðtÞcðtÞ > bðtÞ 1 > tanh n > 2 2 2 > cðtÞ > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 2 2 > b ðtÞ4aðtÞcðtÞ b ðtÞ4aðtÞcðtÞ > 1 > coth n 2b : cðtÞ 2 2 (b) If a(t) = 0, c(t) – 0:
2
ðtÞ
and f(n) = 1. Therefore, by (2.13) with c = 0 and using
b2 ¼ 4cðtÞaðtÞ; 4aðtÞcðtÞ bðtÞ2 > 0; 4aðtÞcðtÞ b2 ðtÞ > 0; 4aðtÞcðtÞ b2 ðtÞ < 0; 4aðtÞcðtÞ b2 ðtÞ < 0;
ð3:2Þ
C.A. Gómez S., A. Salas / Applied Mathematics and Computation 216 (2010) 3089–3096
8 bðtÞ > > cðtÞþbðtÞebðtÞn > > > 1 > cðtÞn > > > > pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi > > > b2 ðtÞ b2 ðtÞ 1 > tan n bðtÞ > > c ðtÞ 2 2 2 < pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi /ðnÞ ¼ b2 ðtÞ b2 ðtÞ bðtÞ > 1 > cot n > cðtÞ 2 2 2 > > > > h i > 1 bðtÞ > bðtÞ bðtÞ > > > cðtÞ 2 tanh 2 n 2 > > h i > > : 1 bðtÞ coth bðtÞ n bðtÞ cðtÞ
2
2
2
3093
bðtÞ – 0; bðtÞ ¼ 0; b2 ðtÞ < 0; b2 ðtÞ < 0;
ð3:3Þ
b2 ðtÞ > 0; b2 ðtÞ > 0:
(c) If b(t) = 0, c(t) – 0
8 1 > ; cðtÞn > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
> > 1 > aðtÞcðtÞ tan½ aðtÞcðtÞn > cðtÞ > < pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 aðtÞcðtÞ cot½ aðtÞcðtÞn /ðnÞ ¼ cðtÞ > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > cðtÞ aðtÞcðtÞ tanh½ aðtÞcðtÞn > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > : 1 aðtÞcðtÞ coth½ aðtÞcðtÞn cðtÞ
aðtÞ ¼ 0; aðtÞcðtÞ > 0; aðtÞcðtÞ > 0;
ð3:4Þ
aðtÞcðtÞ < 0; aðtÞcðtÞ < 0:
(d) If c(t) = 0, b(t) – 0
/ðnÞ ¼
aðtÞ þ bðtÞebðtÞn : bðtÞ
ð3:5Þ
(2) A particular case of the previous example is given by equation
/0 ðnÞ ¼ lð/2 ðnÞ þ kÞ
ð3:6Þ
with k arbitrary constant and l – 0. By means of (2.17), we obtain de SRE
u0 ðnÞ ¼ u2 ðnÞ þ l2 k: The hypothesis in Proposition 2.4 are satisfied taking K = l2k and f(n) = 1. In this way, if we take c = 0 in (2.13) and we use (2.17), we obtain the following set of solutions to (3.6):
8 1 ; > > > pffiffiffiln pffiffiffi > > > k tanðl knÞ > < p pffiffiffi ffiffiffi /ðnÞ ¼ k cotðl knÞ > pffiffiffiffiffiffiffi p ffiffiffiffiffiffi ffi > > > k tanhðl knÞ > > > pffiffiffiffiffiffiffi : pffiffiffiffiffiffiffi k cothðl knÞ
k ¼ 0; k > 0; k > 0;
ð3:7Þ
k < 0; k < 0:
(3) To solve the Riccati equation [11]
/0 ðxÞ ¼ cos n ðsin n /ðnÞÞyðnÞ; we use (2.17) to obtain the SRE 2
u0 ðnÞ ¼ u2 ðnÞ þ
2 cos n sin n : 4
The conditions in Proposition 2.4 are satisfied if we take K = 0 and f ðnÞ ¼ ecos n . So that, with c = 0 in (2.13) we obtain the solution
uðnÞ ¼ sin n
1 R : ecos n e cos n dn
Finally, using (2.17) with p(n) = 1, q(n) = sinn, a particular solution to initial equation is given by
/ðnÞ ¼ sin n
1 sin n R þ : 2 ecos n e cos n dn
(4) The Riccati equation [10]
/0 ¼ Anm ð/2 ðnÞ þ 1Þ; has the standard form
ð3:8Þ
3094
C.A. Gómez S., A. Salas / Applied Mathematics and Computation 216 (2010) 3089–3096
u0 ðnÞ ¼ u2 ðnÞ þ
m2 2m þ 4A2 n2mþ2 4n2
:
The hypothesis in Proposition 2.4 are satisfied if we take K = A2 and f(n) = nm. Therefore, by (2.13) we have
mþ1 8 < mþ2jAjnmþ1 tan jAj nmþ1 c m – 1; 2n uðnÞ ¼ : jAj12þtanðjAjðln jnjcÞÞ m ¼ 1: x Finally, by (2.17),
( /ðnÞ ¼
mþ1 tan jAj c nmþ1 m – 1; tanðjAjðc ln jnjÞÞ
m ¼ 1;
are solutions to (3.8). In this case, c an arbitrary constant. (5) The Riccati equation
u0 ðnÞ ¼ u2 ðnÞ þ
K ðAn2 þ Bn þ CÞ2
;
can be solve taking K – 0 and f(n) = An2 + Bn + C. (6) In the same way, the Riccati equation
u0 ðnÞ ¼ u2 ðnÞ þ
K An2 þ Bn þ C
;
can be solve with K – 0 and f ðnÞ ¼ (7) We consider the system
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi An2 þ Bn þ C .
r0 ðnÞ ¼ erðnÞsðnÞ s0 ðnÞ ¼ es2 ðnÞ lrðnÞ þ r;
ð3:9Þ
with e, l and r arbitrary parameters. If l = 0, using the previous results we get the solutions
8 1 n > > > pffiffiffiffiffi pffiffiffiffiffi > > > er tanð er cÞ 1 < pffiffiffiffiffi pffiffiffiffiffi sðnÞ ¼ er cotð er cÞ e> pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi > > > er tanhð er cÞ > > pffiffiffiffiffiffiffiffiffi : pffiffiffiffiffiffiffiffiffi er cothð er cÞ
r ¼ 0; er > 0; er > 0;
ð3:10Þ
er < 0; er < 0;
and the solutions to r(n) are respectively
81 > n > > pffiffiffiffiffi > pffiffiffiffiffi > > < er secð ern cÞ pffiffiffiffiffi pffiffiffiffiffi er cscð ern cÞ > p ffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffi > > > ersechð ern cÞ > > pffiffiffiffiffiffiffiffiffi : pffiffiffiffiffiffiffiffiffi ercschð ern cÞ
r ¼ 0; er > 0; er > 0;
ð3:11Þ
er < 0; er < 0:
For l – 0, r – 0 the solutions are given by
pffiffiffi pffiffiffi 88 ernþcÞ < s1 ðnÞ ¼ 1e er tanð pffiffiffi > > secð er nþcÞþ1 > > pffiffiffi > > tanð er nþcÞ > pffiffiffi > : r1 ðnÞ ¼ l rsecð er > 0; > er nþcÞþ1 > > pffiffiffi pffiffiffi > >8 er cotð er nþcÞ > 1 < s2 ðnÞ ¼ e pffiffiffi > > cscð er nþcÞþ1 > > > pffiffiffi > r cscð er nþcÞ > < : r2 ðnÞ ¼ pffiffiffi er > 0; l cscð ernþcÞþ1 pffiffiffiffiffiffi 8 pffiffiffiffiffiffi ernþcÞ > > < s3 ðnÞ ¼ 1e er tanhð pffiffiffiffiffiffi > sechð er nþcÞþ1 > > > p ffiffiffiffiffi ffi >: > rsechð er nþcÞ > pffiffiffiffiffiffi r3 ðnÞ ¼ lsechð er < 0; > ernþcÞþ1 > >8 > pffiffiffiffiffiffi pffiffiffiffiffiffi > > er cothð ernþcÞ > ffi > < s3 ðnÞ ¼ 1e cschðpffiffiffiffiffi > er nþcÞþ1 > > pffiffiffiffiffiffi > :: rcschð er nþcÞ pffiffiffiffiffiffi r3 ðnÞ ¼ lcschð er < 0: ernþcÞþ1 Note that the solutions of the system (3.9) satisfies the relation (first integral)
ð3:12Þ
C.A. Gómez S., A. Salas / Applied Mathematics and Computation 216 (2010) 3089–3096
1 e
s2 ðnÞ ¼ ðr 2lrðnÞ þ
l2 þ d r
r2 ðnÞÞ;
3095
ð3:13Þ
with d = ±1. 4. Applications to solitons theory We consider a system of two coupled PDE’s in the variables x and t
Pðu; w; ux ; wx ; ut ; wt ; uxt ; wx;t ; uxx ; wxx ; . . .Þ ¼ 0; Q ðu; w; ux ; wx ; ut ; wt ; uxt ; wx;t ; uxx ; wxx ; . . .Þ ¼ 0:
ð4:1Þ
The traveling wave transformation
n ¼ lðx þ kt þ n0 Þ; converts (4.1) to a system of ordinary differential equations in the unknowns u(n), w(n)
P1 ðu; w; u0 ; w0 ; u00 ; w00 ; . . .Þ ¼ 0; Q 1 ðu; w; u0 ; w0 ; u00 ; w00 ; . . .Þ ¼ 0:
ð4:2Þ
We seek solutions to (4.2) by using the expansions
8 M 2M P P > > > uðnÞ ¼ ai /ðnÞi þ ai /ðnÞMi ; > < i¼0 i¼Mþ1 > M 2M > P P i > > bi /ðnÞMi ; : wðnÞ ¼ bi /ðnÞ þ i¼0
ð4:3Þ
i¼Mþ1
where M is a positive integer that will be determined and / = /(n) satisfies the Riccati equation (3.1) or (3.6). In the case that we use (3.6), the method is called the generalized tanh–coth method to coupled system. It is possible to consider only the first part in the right side of (4.3). In this case, the method is called generalized tanh method to coupled systems. However, if we use (3.1) the method is called improved generalized tanh–coth method to coupled systems (IGTCMCS). In this last case, substituting (4.3) into (4.2) and using (3.1) results in an algebraic system of two equations in powers of /(n). Balancing the linear terms of highest order in the resulting equations with the highest order nonlinear term to obtain M, will yields a set of algebraic equations for l, a, b, c, k, a0, . . . , a2M, b0, . . . , b2M because all coefficients of /(n)i (i = 1, 2, . . .) have to vanish. Solving the algebraic system, and reversing, we obtain exact solutions to (4.1) in the original variables. In the same way, the generalized projective Riccati equation method (using (3.9)) [24], can be adapted for solve coupled systems. Some applications of the mentioned methods can be found in [5,7,25,26]. 5. Conclusions In this work, we have presented some important results on Riccati equation. In particular, using the Lie groups theory, a new integrability condition to generalized Riccati equation has been obtained. Using this condition, we have obtained solutions for several important Riccati equations, which have been used to construct exact solutions for nonlinear evolution wave equations NLEE. We have given a description about some computational methods, used to construct traveling wave solutions for NLEE’s or systems of NLEE’s which use solutions of some Riccati equations. References [1] J.F. Cariñena, G. Marmo, J. Nasare, The non-linear superposition principle and the Wei-Norman method, Int. J. Mod. Phys 1 (1998) 601–3627. [2] J.F. Cariñena, A. Ramos, Lie systems and connections in fibre bundles: applications in quantum mechanics, differential geometry and its applications, Conf. Praga 2004, Charles University, Prague (Zech Republic), 2005, pp. 437–452. [3] V.M. Strelchenya, A new case of integrability of the general Riccati equation and its application to relaxation problems, J. Phys. A. Math Gen 24 (1991) 4965–4967. [4] D. Baldwin, U. Goktas, W. Hereman, L. Hong, R.S. Martino, J.C. Miller, Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDFs, J. Symbolic Compt. 37 (6) (2004) 669–705. Prepint version: nlin.SI/0201008(arXiv.org). [5] E. Fan, Y.C. Hon, Generalized tanh method extended to special types of nonlinear equations, Z. Naturforsch. A 57 (8) (2002) 692–700. [6] A.M. Wazwaz, The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations, Appl. Math. Comput 84-2 (2007) 1002–1014. [7] C.A. Gomez, Special forms of the fifth-order KdV equation with new periodic and soliton solutions, Appl. Math Comput. 189 (2007) 1066–1077. [8] Z. Yan, The Riccati equation with variable coefficients expansion algorithm to find more exact solutions of nonlinear differential equation, Comput. Phys. Comm. 152 (1) (2003) 1–8. Prepint version available at . [9] H.T. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover, New York, 1962. [10] E. Kamke, Differential Gleichungen, Chelsea Publishing Company, New York, 1959. [11] G. Murphy, Ordinary Differential Equations and their Solutions, D. Van Nostrand Company Inc., Princenton New Jersey, 1960. [12] P.R.P. Rao, V.H. Ukidave, Some separable forms of the Riccati equation, Am. Math. Mon. 75 (10) (1968) 1113–1114. [13] D.R. Haaheim, F.M. Stein, Methods of solution of the Riccati differential equation, Math. Mag. 42 (5) (1969) 233–240. [14] J.L. Allen, F.M. Stein, On solutions of certain Riccati differential equations, Am. Math. Mon. 71 (1964) 1113–1115.
3096 [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
C.A. Gómez S., A. Salas / Applied Mathematics and Computation 216 (2010) 3089–3096 G. Bluman, S. Kumey, Symmetries and Differential Equations, Springer-Verlag, New York, 1989. P.J. Olver, Applications of Lie Group to Differential Equations, Springer-Verlag, 1980. G. Bauman, Symmetry Analysis of Differential Equations with Mathematica, Springer-Verlag, New York, 2000. L.V. Ovsianikov, Group Analysis of Differential Equations, Academic Press, New York, 1982. H. Stephani, Differential Equations: Their Solutions Using Symmetries, Cambridge University Press, Cambridge, 1989. J.M. Hill, Solutions of differential equations by means of one-parameter groups, University of Wollongong, vol. 1, 1982 S.P. Konovalov, Fifth-order equations with symmetries linear in the dependent variable, in: N.H. Ibragimov (Ed.), CRC Handbook of Lie group, 1, CRC Press, Boca Raton, 1994. C.A. Gomez, A. Salas, Grupos de Lie y ecuaciones diferenciales, Memorias XI encuentro de geometria y sus aplicaciones, Universidad Distrital Bogotá, 2000, pp. 1–22. C.A. Gomez, Ecuacion de Riccati en grupos de Lie, Memorias X encuentro de geometria y sus aplicaciones, Universidad Pedagógica Nacional Bogotá, 1999, pp. 149–153. C.A. Gomez, A. Salas, New exact solutions for the combined sinh–cosh Gordon equation, Lecturas Math. 21 (2006) 87–93. C.A. Gomez, A.H. Salas, The generalized tanh–coth method to special types of the fifth-order KdV equation, Appl. Math. Comput. 203 (2008) 873–880. A.H. Salas, C.A. Gomez, Computing exact solutions for some fifth KdV equations with forcing term, Appl. Math. Comput. 204 (2008) 257–260.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Special symmetries to standard Riccati equations and applications Cesar A. Gómez S. a,*, Alvaro Salas b,c a b c
Department of Mathematics, Universidad Nacional de Colombia, Bogotá, Colombia Department of Mathematics, Universidad de Caldas, Manizales, Colombia Department of Mathematics, Universidad Nacional de Colombia, Manizales, Colombia
a r t i c l e
i n f o
Keywords: Riccati equation Lie groups Solitons
a b s t r a c t By using symmetries associated to Riccati equation in standard form (SRE), we obtain a family which can be integrated by quadratures. As a consequence, we get a new integrability condition for the generalized Riccati equation (GRE). We illustrate the result with some examples and we give some applications in the solitons theory. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction The simplest nonlinear ordinary differential equation is given by the general Riccati equation (GRE)
d/ðnÞ ¼ pðnÞ/2 ðnÞ þ qðnÞ/ðnÞ þ rðnÞ; dn
ð1:1Þ
where p(n), q(n) and r(n) are continuous functions, defined in some interval ½a; b # R. This equation is used in different fields of pure and applied mathematics, in theoretical physics, control theory and relaxation problems [1–3]. It is well known that solutions of particular cases of (1.1) are used in a great variety of computational methods to obtain exact solutions for nonlinear evolution equations (NEE) [4–8]. On the other hand, in spite of its apparent simplicity, the general solution of (1.1) cannot be expressed in a elementary form, except in some particular cases [3,9–14]. The principal objective of this paper is to obtain a new integrability condition for the generalized Riccati equation (1.1), starting from the Riccati equation in the standard form which is given by the equation
Eðn; uÞ ¼ u0 ðnÞ ¼ u2 ðnÞ þ nðnÞ;
ð1:2Þ
and making use of Lie groups theory [15–22]. More exactly, we obtain a group of symmetry for (1.2) and as a consequence, we obtain a family of Riccati equations in standard form which is integrable by quadratures. In accordance with the result, a new integrability condition for (1.1) is derived. 2. Symmetries to SRE (1.2) First, we need some preliminaries: Definition 2.1. The symmetries of (1.2) are given by the elements of a connected Lie group with parameter a
(
n ¼ n þ af ðn; uÞ þ oða2 Þ u ¼ u þ ahðn; uÞ þ oða2 Þ;
* Corresponding author. E-mail addresses: [email protected] (C.A. Gómez S.), [email protected] (A. Salas). 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.04.039
ð2:1Þ
3090
C.A. Gómez S., A. Salas / Applied Mathematics and Computation 216 (2010) 3089–3096
which transform solutions into solutions. Alternatively, the infinitesimal generators of the Lie algebra of (2.1), which are the components of the vector field associated to (2.1)
@ ¼ f ðn; uÞ@ n þ hðn; uÞ@ u ; @ , @n
ð2:2Þ
@ @u
where @ n ¼ and @ u ¼ are called symmetries of (1.2). The symmetry variables f(n, u), h(n, u) can be found by solving the following equation, which is called determinant equation [15–22]:
hn ðn; uÞ þ ðhu ðn; uÞ fn ðn; uÞÞEðn; uÞ fu ðn; uÞE2 ðn; uÞ f ðn; uÞEn ðn; uÞ hðn; uÞEu ðn; uÞ ¼ 0;
ð2:3Þ
where E(x, u) is as in (1.2). Eq. (2.3) does not split into an overdetermined system, therefore it has an infinite set of solutions. That is why we can hope find solutions of certain forms only. On the other hand, we can see that (2.3) can be written as
(
fu ðn; uÞu4 ðnÞ þ ð2nðnÞfu ðn; uÞ þ hu ðn; uÞ fx ðn; uÞÞu2 ðnÞ 2hðn; uÞuðnÞ þðnðnÞfhu ðn; uÞ fn ðn; uÞ fu ðn; uÞnðnÞg þ hn ðn; uÞ f ðn; uÞn0 ðnÞÞ ¼ 0:
ð2:4Þ
Respect to this last equation we have our first result which say us that f depend only on n and does not of u(n): Proposition 2.2. Let f ðn; uÞ ¼
Pm
Proof. Substituting f ðn; uÞ ¼
Pm
mþ2 X
i 0 fi ðnÞu ðnÞ,
0 fi ðnÞu
i
h(n, u) = h0(n) + h1(n)u(n), a solution of (2.4). Then f(n, u) = f(n).
ðnÞ and h(n, u) = h0(n) + h1(n)u(n) into (2.4), we obtain an expression in the form
F i ðnÞui ðnÞ mfm ðnÞumþ3 ðnÞ ¼ 0;
ð2:5Þ
i¼0
where Fi(n) does not depend of u(n). We give the proof by using induction: For m = 1, (2.5) reduces to
8 f1 ðnÞu4 ðnÞ > > > > 0 3 > > < f1 ðnÞu ðnÞ ðh1 ðnÞ þ 2f 1 ðnÞnðnÞ þ f00 ðnÞÞu2 ðnÞ > > 0 0 > > > þðk1 ðnÞ ðf1 ðnÞnðnÞÞ 2h0 ðnÞÞuðnÞ > : 0 þðh0 ðnÞ þ h1 ðnÞnðnÞ f1 ðnÞn2 ðnÞ ðf0 ðnÞnðnÞÞ0 Þ ¼ 0:
ð2:6Þ
Equaling the coefficients of (2.6) to zero, we have f1(n) = 0. Suppose that the assertion is true for m = k (f1(n) = , , = fk(n) = 0). We consider
8 Hðf Þ ¼ > > > > 4 > > < fu ðn; uÞu ðnÞ þð2nðnÞfu ðn; uÞ þ hu ðn; uÞ fx ðn; uÞÞu2 ðnÞ > > > > 2hðn; uÞuðnÞ > > : þðnðnÞfhu ðn; uÞ fn ðn; uÞ fu ðn; uÞnðnÞg þ hn ðn; uÞ f ðn; uÞn0 ðnÞÞ; P P i kþ1 GðnÞ ¼ ki¼0 fi ðnÞui ðnÞ, and f ðnÞ ¼ kþ1 ðnÞ. Taking into account (2.5), after simplifications we i¼0 fi ðnÞu ðnÞ ¼ GðnÞ þ fkþ1 ðnÞu obtain: H(f) = 0, if and only if
8 kþ2 > < P G ðnÞu ðnÞ kf ðnÞukþ3 ðnÞ fn2 ðnÞðk þ 1Þf ðnÞuk ðnÞ þ ðnðnÞf ðnÞÞ0 ukþ1 ðnÞ i i k kþ1 kþ1 i¼0 > : kþ2 0 kþ3 kþ4 þ2nðnÞðk þ 1Þfkþ1 ðnÞu ðnÞ þ fkþ1 ðnÞu ðnÞ þ ðk þ 1Þfkþ1 ðnÞu ðnÞg ¼ 0:
P k i This last equation implies fk+1(n) = 0. So that H(f) = 0, if and only if, HðGÞ ¼ H i¼0 fi ðnÞu ðnÞ ¼ 0. Using the induction hypothesis, fi(n) = 0 for i = 1, 2, . . . , k. So that f(n, u) = f0(n) and the proposition is proved. h As a consequence of the previous proposition, we have the more important result of this section: Theorem 2.3. The standard Riccati equation (1.2) admits the vector fields
f 00 ðnÞ @ ¼ f ðnÞ@ n f 0 ðnÞuðnÞ þ @u; 2
ð2:7Þ
where f(n) satisfies the third order ordinary differential equation
f 000 ðnÞ þ 4f 0 ðnÞnðnÞ þ 2n0 ðnÞf ðnÞ ¼ 0: Proof. In accordance with Proposition 2.2, we seek solutions to (2.4) in the form
ð2:8Þ
C.A. Gómez S., A. Salas / Applied Mathematics and Computation 216 (2010) 3089–3096
f ðn; uÞ ¼ f ðnÞ hðn; uÞ ¼ kðnÞ þ rðnÞuðnÞ:
3091
ð2:9Þ
Substituting (2.9) into (2.4) we get 0
ðrðnÞ f 0 ðnÞÞu2 ðnÞ þ ð2kðnÞ þ r0 ðnÞÞuðnÞ þ ðnðnÞrðnÞ ðnðnÞf ðnÞÞ0 þ k ðnÞÞ ¼ 0: Equaling the coefficients of this last equation to zero we obtain the following system
8 0 > < rðnÞ ¼ f ðnÞ 1 0 kðnÞ ¼ 2 r ðnÞ > : 0 nðnÞrðnÞ ðnðnÞf ðnÞÞ0 þ k ðnÞ ¼ 0:
ð2:10Þ
From the firsts two equations in (2.10) we have
1 hðx; uÞ ¼ f 00 ðnÞ f 0 ðnÞ: 2 Substituting the expressions for k(n) and r(n) in the third equation that appear in (2.10), finally Eq. (2.8) is obtained, and the proof is complete. h On the other hand, solving the system
@ðtðn; uÞÞ ¼ 0 @ðwðn; uÞÞ ¼ 1;
where @ is the vector field associate to Lie group (2.2), we obtain the so called canonical coordinates
(
t ¼ tðn; uÞ ¼ 12 f 0 ðnÞ þ f ðnÞuðnÞ R wðn; tÞ ¼ fdn : ðnÞ
ð2:11Þ
In this new variables, (1.2) reduces to following separable equation
dw 1 : ¼ 2 dt t þK
ð2:12Þ
Using the solutions of (2.12) and taking into account (2.11), after simplifications we obtain the following solutions of (1.2):
8 > 1 1 0 > R 1dn uðnÞ ¼ f ðnÞ þ > f ðnÞ 2 > > f ðnÞ > > > R > > ffiffi ffi ffiffi ffi p p > > K tan K fdn c 12f 0 ðnÞ > ðnÞ > uðnÞ ¼ > > f ðnÞ > > > R > < pffiffiffi pffiffiffi dn 1 0 K cot
K
K ¼ 0;
K > 0;
c 2f ðnÞ
f ðnÞ
uðnÞ ¼ K > 0; > f ðnÞ > > > > > pffiffiffiffiffi pffiffiffiffiffi R dn > > K tanh K f ðnÞc 12f 0 ðnÞ > > > uðnÞ ¼ K < 0; > f ðnÞ > > > > R > pffiffiffiffiffi pffiffiffiffiffi dn > > K coth K f ðnÞc 12f 0 ðnÞ > : uðnÞ ¼ K < 0; f ðnÞ
ð2:13Þ
where c is an integration constant and f(n) satisfies (2.8). We cannot find solutions of (2.8) in the case that n(n) is an arbitrary function. However, an analysis of (2.8) may be useful. In fact, if we consider (2.8) as an first order differential equation in the unknowns n(n) and we solve it, we obtain
nðnÞ ¼
ðf 0 ðnÞ2 Þ 2f ðnÞf 00 ðnÞ þ 4K ; 4f 2 ðnÞ
ð2:14Þ
where K is an integration constant. According with the previous results, Eq. (2.14) say us that the following family of Riccati equations in standard form
u0 ðnÞ ¼ u2 ðnÞ þ
ðf 0 ðnÞ2 Þ 2f ðnÞf 00 ðnÞ þ 4K ; 4f 2 ðnÞ
ð2:15Þ
is integrable by quadratures, and the respective solutions are obtained using (2.13). We have the following new integrability condition to generalized Riccati equation (1.1): Proposition 2.4. If in Eq. (1.1), the coefficients are defined in some interval ½a; b R and p(n) 2 C2[a, b], q(n) 2 C1[a, b], r(n) 2 C[a, b] are related as
3092
C.A. Gómez S., A. Salas / Applied Mathematics and Computation 216 (2010) 3089–3096
8 1 > > 4p2 ðnÞ > > > < fðpðnÞqðnÞÞ2 þ 4pðnÞ3 rðnÞ > 2pðnÞp0 ðnÞqðnÞ 3p02 ðnÞ þ 2p2 ðnÞq0 ðnÞ þ 2pðnÞp00 ðnÞg ¼ > > > > : f 02 ðnÞ2f ðnÞf 00 ðnÞþ4K ;
ð2:16Þ
4f 2 ðnÞ
2
with f(n) 2 C (a, b), f(n) – 0 a properly chosen function and K an arbitrary constant, then a solution to (1.1) can be obtain using elementary integration. Proof. With the change of variable
/ðnÞ ¼
( 0 ðnÞ) qðnÞ þ ppðnÞ 1 ; uðnÞ pðnÞ 2
ð2:17Þ
(1.1) reduces to (1.2), where
nðnÞ ¼
1 fðpðnÞqðnÞÞ2 þ 4pðnÞ3 rðnÞ 2pðnÞp0 ðnÞqðnÞ 3p02 ðnÞ þ 2p2 ðnÞq0 ðnÞ þ 2pðnÞp00 ðnÞg: 4p2 ðnÞ
Taking into account (2.14) we have the hypothesis given in the enunciate of the theorem. In this case, by the conclusions after Theorem 2.3, the solutions of (1.2) are obtained using (2.13), and by (2.17), solutions to (1.1) are derived by elementary integration as we want to show. h
3. Integrable families of Riccati equations In accordance with the previous results, we can construct a variety of families of Riccati equations in standard form that are integrable by quadratures [23]. In fact, if we take any f(n) 2 C2(a, b), f(n) – 0 and K arbitrary constant, the family of Riccati equations
u0 ðnÞ ¼ u2 ðnÞ þ
f 02 ðnÞ 2f ðnÞf 00 ðnÞ þ 4K ; 4f 2 ðnÞ
is integrable by quadratures, and its solution are given by (2.13). Moreover, f(n, u) = f(n) and hðn; uÞ ¼ 12 f 00 ðnÞ f 0 ðnÞ are the components of the vector field associated to Lie group (2.1) admitted by this family. The following families may be considered as important examples: (1) We consider the Riccati equation
/0 ðnÞ ¼ cðtÞ/2 ðnÞ þ bðtÞ/ðnÞ þ aðtÞ;
ð3:1Þ
where a(t), c(t) – 0, b(t) are functions that does not depend of n. By means of the substitution (2.17), then (3.1) reduces to following SRE
u0 ðnÞ ¼ u2 ðnÞ þ
4aðtÞcðtÞ b2 ðtÞ : 4
The hypothesis in Proposition 2.4, are satisfied if we take K ¼ 4aðtÞcðtÞb 4 (2.17) we get the following set of solutions to (3.1): (a) If a(t) – 0, c(t) – 0 and b(t) – 0:
8 bðtÞ 1 1 > > > cðtÞ n 2 > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 4aðtÞcðtÞb2 ðtÞ 4aðtÞcðtÞb2 ðtÞ > tan n bðtÞ > > c ðtÞ 2 2 2 > > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > < 1 4aðtÞcðtÞb2 ðtÞ 4aðtÞcðtÞb2 ðtÞ bðtÞ cot n /ðnÞ ¼ cðtÞ 2 2 2 > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > 2 2 b ðtÞ4aðtÞcðtÞ b ðtÞ4aðtÞcðtÞ > bðtÞ 1 > tanh n > 2 2 2 > cðtÞ > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 2 2 > b ðtÞ4aðtÞcðtÞ b ðtÞ4aðtÞcðtÞ > 1 > coth n 2b : cðtÞ 2 2 (b) If a(t) = 0, c(t) – 0:
2
ðtÞ
and f(n) = 1. Therefore, by (2.13) with c = 0 and using
b2 ¼ 4cðtÞaðtÞ; 4aðtÞcðtÞ bðtÞ2 > 0; 4aðtÞcðtÞ b2 ðtÞ > 0; 4aðtÞcðtÞ b2 ðtÞ < 0; 4aðtÞcðtÞ b2 ðtÞ < 0;
ð3:2Þ
C.A. Gómez S., A. Salas / Applied Mathematics and Computation 216 (2010) 3089–3096
8 bðtÞ > > cðtÞþbðtÞebðtÞn > > > 1 > cðtÞn > > > > pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi > > > b2 ðtÞ b2 ðtÞ 1 > tan n bðtÞ > > c ðtÞ 2 2 2 < pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi /ðnÞ ¼ b2 ðtÞ b2 ðtÞ bðtÞ > 1 > cot n > cðtÞ 2 2 2 > > > > h i > 1 bðtÞ > bðtÞ bðtÞ > > > cðtÞ 2 tanh 2 n 2 > > h i > > : 1 bðtÞ coth bðtÞ n bðtÞ cðtÞ
2
2
2
3093
bðtÞ – 0; bðtÞ ¼ 0; b2 ðtÞ < 0; b2 ðtÞ < 0;
ð3:3Þ
b2 ðtÞ > 0; b2 ðtÞ > 0:
(c) If b(t) = 0, c(t) – 0
8 1 > ; cðtÞn > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
> > 1 > aðtÞcðtÞ tan½ aðtÞcðtÞn > cðtÞ > < pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 aðtÞcðtÞ cot½ aðtÞcðtÞn /ðnÞ ¼ cðtÞ > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > cðtÞ aðtÞcðtÞ tanh½ aðtÞcðtÞn > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > : 1 aðtÞcðtÞ coth½ aðtÞcðtÞn cðtÞ
aðtÞ ¼ 0; aðtÞcðtÞ > 0; aðtÞcðtÞ > 0;
ð3:4Þ
aðtÞcðtÞ < 0; aðtÞcðtÞ < 0:
(d) If c(t) = 0, b(t) – 0
/ðnÞ ¼
aðtÞ þ bðtÞebðtÞn : bðtÞ
ð3:5Þ
(2) A particular case of the previous example is given by equation
/0 ðnÞ ¼ lð/2 ðnÞ þ kÞ
ð3:6Þ
with k arbitrary constant and l – 0. By means of (2.17), we obtain de SRE
u0 ðnÞ ¼ u2 ðnÞ þ l2 k: The hypothesis in Proposition 2.4 are satisfied taking K = l2k and f(n) = 1. In this way, if we take c = 0 in (2.13) and we use (2.17), we obtain the following set of solutions to (3.6):
8 1 ; > > > pffiffiffiln pffiffiffi > > > k tanðl knÞ > < p pffiffiffi ffiffiffi /ðnÞ ¼ k cotðl knÞ > pffiffiffiffiffiffiffi p ffiffiffiffiffiffi ffi > > > k tanhðl knÞ > > > pffiffiffiffiffiffiffi : pffiffiffiffiffiffiffi k cothðl knÞ
k ¼ 0; k > 0; k > 0;
ð3:7Þ
k < 0; k < 0:
(3) To solve the Riccati equation [11]
/0 ðxÞ ¼ cos n ðsin n /ðnÞÞyðnÞ; we use (2.17) to obtain the SRE 2
u0 ðnÞ ¼ u2 ðnÞ þ
2 cos n sin n : 4
The conditions in Proposition 2.4 are satisfied if we take K = 0 and f ðnÞ ¼ ecos n . So that, with c = 0 in (2.13) we obtain the solution
uðnÞ ¼ sin n
1 R : ecos n e cos n dn
Finally, using (2.17) with p(n) = 1, q(n) = sinn, a particular solution to initial equation is given by
/ðnÞ ¼ sin n
1 sin n R þ : 2 ecos n e cos n dn
(4) The Riccati equation [10]
/0 ¼ Anm ð/2 ðnÞ þ 1Þ; has the standard form
ð3:8Þ
3094
C.A. Gómez S., A. Salas / Applied Mathematics and Computation 216 (2010) 3089–3096
u0 ðnÞ ¼ u2 ðnÞ þ
m2 2m þ 4A2 n2mþ2 4n2
:
The hypothesis in Proposition 2.4 are satisfied if we take K = A2 and f(n) = nm. Therefore, by (2.13) we have
mþ1 8 < mþ2jAjnmþ1 tan jAj nmþ1 c m – 1; 2n uðnÞ ¼ : jAj12þtanðjAjðln jnjcÞÞ m ¼ 1: x Finally, by (2.17),
( /ðnÞ ¼
mþ1 tan jAj c nmþ1 m – 1; tanðjAjðc ln jnjÞÞ
m ¼ 1;
are solutions to (3.8). In this case, c an arbitrary constant. (5) The Riccati equation
u0 ðnÞ ¼ u2 ðnÞ þ
K ðAn2 þ Bn þ CÞ2
;
can be solve taking K – 0 and f(n) = An2 + Bn + C. (6) In the same way, the Riccati equation
u0 ðnÞ ¼ u2 ðnÞ þ
K An2 þ Bn þ C
;
can be solve with K – 0 and f ðnÞ ¼ (7) We consider the system
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi An2 þ Bn þ C .
r0 ðnÞ ¼ erðnÞsðnÞ s0 ðnÞ ¼ es2 ðnÞ lrðnÞ þ r;
ð3:9Þ
with e, l and r arbitrary parameters. If l = 0, using the previous results we get the solutions
8 1 n > > > pffiffiffiffiffi pffiffiffiffiffi > > > er tanð er cÞ 1 < pffiffiffiffiffi pffiffiffiffiffi sðnÞ ¼ er cotð er cÞ e> pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi > > > er tanhð er cÞ > > pffiffiffiffiffiffiffiffiffi : pffiffiffiffiffiffiffiffiffi er cothð er cÞ
r ¼ 0; er > 0; er > 0;
ð3:10Þ
er < 0; er < 0;
and the solutions to r(n) are respectively
81 > n > > pffiffiffiffiffi > pffiffiffiffiffi > > < er secð ern cÞ pffiffiffiffiffi pffiffiffiffiffi er cscð ern cÞ > p ffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffi > > > ersechð ern cÞ > > pffiffiffiffiffiffiffiffiffi : pffiffiffiffiffiffiffiffiffi ercschð ern cÞ
r ¼ 0; er > 0; er > 0;
ð3:11Þ
er < 0; er < 0:
For l – 0, r – 0 the solutions are given by
pffiffiffi pffiffiffi 88 ernþcÞ < s1 ðnÞ ¼ 1e er tanð pffiffiffi > > secð er nþcÞþ1 > > pffiffiffi > > tanð er nþcÞ > pffiffiffi > : r1 ðnÞ ¼ l rsecð er > 0; > er nþcÞþ1 > > pffiffiffi pffiffiffi > >8 er cotð er nþcÞ > 1 < s2 ðnÞ ¼ e pffiffiffi > > cscð er nþcÞþ1 > > > pffiffiffi > r cscð er nþcÞ > < : r2 ðnÞ ¼ pffiffiffi er > 0; l cscð ernþcÞþ1 pffiffiffiffiffiffi 8 pffiffiffiffiffiffi ernþcÞ > > < s3 ðnÞ ¼ 1e er tanhð pffiffiffiffiffiffi > sechð er nþcÞþ1 > > > p ffiffiffiffiffi ffi >: > rsechð er nþcÞ > pffiffiffiffiffiffi r3 ðnÞ ¼ lsechð er < 0; > ernþcÞþ1 > >8 > pffiffiffiffiffiffi pffiffiffiffiffiffi > > er cothð ernþcÞ > ffi > < s3 ðnÞ ¼ 1e cschðpffiffiffiffiffi > er nþcÞþ1 > > pffiffiffiffiffiffi > :: rcschð er nþcÞ pffiffiffiffiffiffi r3 ðnÞ ¼ lcschð er < 0: ernþcÞþ1 Note that the solutions of the system (3.9) satisfies the relation (first integral)
ð3:12Þ
C.A. Gómez S., A. Salas / Applied Mathematics and Computation 216 (2010) 3089–3096
1 e
s2 ðnÞ ¼ ðr 2lrðnÞ þ
l2 þ d r
r2 ðnÞÞ;
3095
ð3:13Þ
with d = ±1. 4. Applications to solitons theory We consider a system of two coupled PDE’s in the variables x and t
Pðu; w; ux ; wx ; ut ; wt ; uxt ; wx;t ; uxx ; wxx ; . . .Þ ¼ 0; Q ðu; w; ux ; wx ; ut ; wt ; uxt ; wx;t ; uxx ; wxx ; . . .Þ ¼ 0:
ð4:1Þ
The traveling wave transformation
n ¼ lðx þ kt þ n0 Þ; converts (4.1) to a system of ordinary differential equations in the unknowns u(n), w(n)
P1 ðu; w; u0 ; w0 ; u00 ; w00 ; . . .Þ ¼ 0; Q 1 ðu; w; u0 ; w0 ; u00 ; w00 ; . . .Þ ¼ 0:
ð4:2Þ
We seek solutions to (4.2) by using the expansions
8 M 2M P P > > > uðnÞ ¼ ai /ðnÞi þ ai /ðnÞMi ; > < i¼0 i¼Mþ1 > M 2M > P P i > > bi /ðnÞMi ; : wðnÞ ¼ bi /ðnÞ þ i¼0
ð4:3Þ
i¼Mþ1
where M is a positive integer that will be determined and / = /(n) satisfies the Riccati equation (3.1) or (3.6). In the case that we use (3.6), the method is called the generalized tanh–coth method to coupled system. It is possible to consider only the first part in the right side of (4.3). In this case, the method is called generalized tanh method to coupled systems. However, if we use (3.1) the method is called improved generalized tanh–coth method to coupled systems (IGTCMCS). In this last case, substituting (4.3) into (4.2) and using (3.1) results in an algebraic system of two equations in powers of /(n). Balancing the linear terms of highest order in the resulting equations with the highest order nonlinear term to obtain M, will yields a set of algebraic equations for l, a, b, c, k, a0, . . . , a2M, b0, . . . , b2M because all coefficients of /(n)i (i = 1, 2, . . .) have to vanish. Solving the algebraic system, and reversing, we obtain exact solutions to (4.1) in the original variables. In the same way, the generalized projective Riccati equation method (using (3.9)) [24], can be adapted for solve coupled systems. Some applications of the mentioned methods can be found in [5,7,25,26]. 5. Conclusions In this work, we have presented some important results on Riccati equation. In particular, using the Lie groups theory, a new integrability condition to generalized Riccati equation has been obtained. Using this condition, we have obtained solutions for several important Riccati equations, which have been used to construct exact solutions for nonlinear evolution wave equations NLEE. We have given a description about some computational methods, used to construct traveling wave solutions for NLEE’s or systems of NLEE’s which use solutions of some Riccati equations. References [1] J.F. Cariñena, G. Marmo, J. Nasare, The non-linear superposition principle and the Wei-Norman method, Int. J. Mod. Phys 1 (1998) 601–3627. [2] J.F. Cariñena, A. Ramos, Lie systems and connections in fibre bundles: applications in quantum mechanics, differential geometry and its applications, Conf. Praga 2004, Charles University, Prague (Zech Republic), 2005, pp. 437–452. [3] V.M. Strelchenya, A new case of integrability of the general Riccati equation and its application to relaxation problems, J. Phys. A. Math Gen 24 (1991) 4965–4967. [4] D. Baldwin, U. Goktas, W. Hereman, L. Hong, R.S. Martino, J.C. Miller, Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDFs, J. Symbolic Compt. 37 (6) (2004) 669–705. Prepint version: nlin.SI/0201008(arXiv.org). [5] E. Fan, Y.C. Hon, Generalized tanh method extended to special types of nonlinear equations, Z. Naturforsch. A 57 (8) (2002) 692–700. [6] A.M. Wazwaz, The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations, Appl. Math. Comput 84-2 (2007) 1002–1014. [7] C.A. Gomez, Special forms of the fifth-order KdV equation with new periodic and soliton solutions, Appl. Math Comput. 189 (2007) 1066–1077. [8] Z. Yan, The Riccati equation with variable coefficients expansion algorithm to find more exact solutions of nonlinear differential equation, Comput. Phys. Comm. 152 (1) (2003) 1–8. Prepint version available at
3096 [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
C.A. Gómez S., A. Salas / Applied Mathematics and Computation 216 (2010) 3089–3096 G. Bluman, S. Kumey, Symmetries and Differential Equations, Springer-Verlag, New York, 1989. P.J. Olver, Applications of Lie Group to Differential Equations, Springer-Verlag, 1980. G. Bauman, Symmetry Analysis of Differential Equations with Mathematica, Springer-Verlag, New York, 2000. L.V. Ovsianikov, Group Analysis of Differential Equations, Academic Press, New York, 1982. H. Stephani, Differential Equations: Their Solutions Using Symmetries, Cambridge University Press, Cambridge, 1989. J.M. Hill, Solutions of differential equations by means of one-parameter groups, University of Wollongong, vol. 1, 1982 S.P. Konovalov, Fifth-order equations with symmetries linear in the dependent variable, in: N.H. Ibragimov (Ed.), CRC Handbook of Lie group, 1, CRC Press, Boca Raton, 1994. C.A. Gomez, A. Salas, Grupos de Lie y ecuaciones diferenciales, Memorias XI encuentro de geometria y sus aplicaciones, Universidad Distrital Bogotá, 2000, pp. 1–22. C.A. Gomez, Ecuacion de Riccati en grupos de Lie, Memorias X encuentro de geometria y sus aplicaciones, Universidad Pedagógica Nacional Bogotá, 1999, pp. 149–153. C.A. Gomez, A. Salas, New exact solutions for the combined sinh–cosh Gordon equation, Lecturas Math. 21 (2006) 87–93. C.A. Gomez, A.H. Salas, The generalized tanh–coth method to special types of the fifth-order KdV equation, Appl. Math. Comput. 203 (2008) 873–880. A.H. Salas, C.A. Gomez, Computing exact solutions for some fifth KdV equations with forcing term, Appl. Math. Comput. 204 (2008) 257–260.