Transcendents defined by nonlinear fourth-order ordinary differential equations
|
|
- Penelope Merritt
- 5 years ago
- Views:
Transcription
1 J. Phys. A: Math. Gen ) Printed in the UK PII: S ) Transcendents defined by nonlinear fourth-order ordinary differential equations Nicolai A Kudryashov Department of Applied Mathematics, Moscow Engineering Physics Institute, 3 Kashirskoe Shosse, Moscow 5409, Russia Received 5 July 998 Abstract. Three ordinary differential equations are considered. The general solutions of these equations are shown to be the essentially transcendental functions with respect to their initial conditions. Irreducibility of these equations is discussed.. Introduction The problem of defining new functions by means of nonlinear ordinary differential equations ODEs) was stated by Fuchs and Poincare. These ODEs must possess two important properties: irreducibility and uniformiation of their solutions. The first property means that there exists no transformation, again within a precise class, reducing any of these equations either to a linear equation or to another order equation, ]. The second property corresponds to the Painlevé property of an ODE because the absence of movable critical singularities in its general solution leads to the single-valued function. Almost a century ago Painlevé and his school began a study the second-order ODE class 3]. They had two related objectives: to classify the second-order equations of a certain form on the basis of their possible singularities, and to identity equations of second order that essentially define new functions. Painlevé and his collaborators showed that out of all possible equations of a certain form, there are only 50 types which have the property of having no movable critical points. Furthermore, they showed that of these 50 equations, 44 were integrable in terms of previously known functions such as elliptic functions and linear equations) or were reducible to one of six new nonlinear ODEs. They also showed that there are exactly six second-order equations that define new functions 3]. The functions defined by them are now called the six Painlevé transcendents. Later, Bureau extended Painlevé s first objective, and gave a partial classification of third-order equations 4 6]. The results of Painlevé and his collaborators led to the problem of finding other new functions that could be defined by nonlinear ODEs like the Painlevé transcendents. However, despite huge efforts, no new function has yet been found. In fact, no irreducible equation has been discovered since 906 ]. Although the six Painlevé equations were first discovered from strictly mathematical considerations, they have recently appeared in several physical applications 3]. Current interest in the Painlevé property is known to stem from the observations made by Ablowit and Segur 7] and Ablowit et al 8, 9] that reductions of partial differential equations of the soliton type give rise to ODEs whose movable singularities are only poles. This circumstance reduced them to the famous Painlevé conjecture, the Painlevé ODE test /99/ $ IOP Publishing Ltd 999
2 000 N A Kudryashov 3]. Every ordinary differential equation which arises as a similarity reduction of a complete integrable partial differential equation is of Painlevé type, perhaps after a transformation of variables. The Painlevé ODE test is applied as follows: if a given partial differential equation reducible to an ODE is not of Painlevé type then the Painlevé ODE test predicts that the partial differential equation is not complete integrable 3]. This test allows us to also find new ODEs of Painlevé type if we have the nonlinear integrable partial differential equation. We will use this in this work. The aim of this paper is to show that the general solutions of three ODEs of fourth order, introduced as reductions of the integrable partial differential equations, are essentially transcendental functions with respect to their initial conditions. The outline of this work is as follows. Three ODEs are presented in section. The approach that we use at the proof, namely that the general solutions of equations studied are the transcendental functions with respect to their constants of integration, is discussed in section 3. The proofs that the general solutions of the three equations are transcendental functions with respect to constants of integration are given in sections 4 6. Irreducibility of studied equations is discussed in section 7.. Equations studied In a recent work 0] we presented a hierarchy which takes the form d n+ u) = n =,,...).) operator d n is determined by formula d d dn+ u) = d n +4udn +u d n.) d 0 = d =u and its hierarchy is in the form 0 3] ) d d +v d n v v ) v α = 0 n =,,...)..3) We have the first Painlevé equation u +3u =0.4) from equations.) at n =. If we take n = in equations.) we obtain the fourth-order equation in the form u +5u +0uu +0u3 =0..5) By analogy we obtain the second Painlevé equation v v 3 v α = 0.6) from equations.3) at n =. If we take n = in equations.3) we find the fourth-order equation which takes the form v 0v v 0vv +6v5 v α = 0..7) It is known that equations.4) and.6) determine new functions which are the Painlevé transcendents. The question arises as to whether there are new functions determined by equations.) and.3) at n. To answer this question we need in investigation of equations.) and.3) on the Painlevé property in the beginning and thereafter we have to show that the general solutions
3 Transcendents defined by nonlinear fourth-order ODEs 00 of these equations are the essentially transcendentical functions with respect to their constants of integration. Recently we studied some properties of equations.) and.3) and we now know that these equations possess the Painlevé property because of the following reasons 4]. First, these equations were obtained as reductions of nonlinear partial differential equations which are solved by inverse scattering transform. Taking into account the conjecture of Ablowit et al 8, 9] one expects that equations.) and.3) possess the Painlevé property. Secondly, we checked equations.5) and.7) with the Painlevé test using the algorithm of Conte et al 5]. These equations passed the Painlevé test 4]. Thirdly, we found the Lax pairs for equations.) and.3) and we will now be able to solve these equations. It is known, 3] that good Lax pair is the sufficiency condition for the integrability of the original equation. As this takes place application of the Gelfand Levitan Marchenko integral equation gives the algorithm for the solution of the Cauchy problem and strict proof of the Painlevé property for nonlinear equations. We obtained the Lax pairs for equations.) and.3) which are good because they were used for solving equations.4) and.6) in the partial case. Taking into account the above-mentioned reasons we suppose that the property of uniformiation for the general solutions of equations.) and.3) is carried out. Let us now also consider the partial differential equation which takes the form q t + x q xxxx +5q x q xx 5q q xx 5qqx + q5 ) = 0..8) This equation was first written by Fordy and Gibbons 6] and can also be solved by the inverse scattering transform. Moreover, equation.8) passes the Painlevé test 7, 8]. Equation.8) admits the Lie group transformation 9] and therefore has the special solution in the form qx,t) = 5t) 5 w) = x5t) 5..9) As this takes place the equation for w) takes the form 0] w +5w w 5w w 5ww + w5 w γ = 0..0) Equation.0) may remind us of equation.7) but it is clear that they differ. This equation was obtained by the reduction of the integrable equation.8) and therefore one can expect that equation.0) possesses the Painlevé property as equations.5) and.7). It should be noted that there is a map for equation.8) which connects this equation with the singular manifold equation, ]. It takes the form q t + x q xxxxx +5q x q xx 5q q xx 5qqx + q5 ) = ) ϕ t + ϕ x {ϕ;x} xx +4{ϕ;x} )].) x ϕ x x q = ϕ xx.) ϕ x and {ϕ; x} is the Schwarian derivative 3] {ϕ; x} = ϕ xxx ϕ x 3 ϕxx..3) ϕx Taking into account variables.9) and ϕx,t) = ϕ) x,t) = ) = x5t) 5.4)
4 00 N A Kudryashov one can obtain some relation from identity.). This takes the form w +5w w 5w w 5ww + w5 w ) d = d + w F +4F ).5) F = w w..6) From relations.5) one can see that there are some special solutions of equation.0) at γ =. These solutions can be obtained by taking into account the solutions of the first Painlevé equations F +4F =0.7) and the Riccatti equations.6). We also have to note that there is some relation for equation.7) as.5). It takes the form 0] v 0v v 0vv +6v5 v ) d = d +v w +3w ).8) v v = w..9) The relation.8) shows that there is a special solution of equation.7) which can be found from the Riccatti equation.9), taking into account the solution of the first Painlevé equation.4). 3. Approach applied The solution of the problem of finding new functions determined by nonlinear ODEs.5),.7) and.0) reduces to the investigation of the functional dependence of their general solutions on the constants of integration. As this takes place three different cases are possible 4 6]. In the first case the general solution of the equation has rational or algebraic dependence on arbitrary constants. This case does not give any new function. In the second case the general solution of the equation does not have any rational or algebraic dependence on the arbitrary constants but the arbitrary constant can enter the first integral in algebraic form. This case leads to the semi-transcendental function of the general solution with respect to the constants of integration and does not give any new function. The third case corresponds to the special dependence of the general solution of the equation on constants of integration. This case contradicts the dependence of the general solution of the equation on the constants of integration which were in the first and second cases. They say that this case gives the essentially transcendental function with respect to the constants of integration. Let us note that the six Painlevé transcendents correspond to this case. Later we wish to show that the general solutions of equations.5),.7) and.0) are the essentially transcendental functions with respect to their constants of integration. We need to prove that equations.5),.7) and.0) have no first integrals in the polynomial form. To prove this we use the same approach as for the three equations. Let us consider this one. One can see that equations.5),.7) and.0) can be written in the following form y + F i y, y )y + G i y, y,)=0 i =,, 3) 3.)
5 Transcendents defined by nonlinear fourth-order ODEs 003 F = 0y G = 5y +0y3 y=u 3.) F = 0y G = 0yy +6y5 y α y = v 3.3) F 3 = 5y 5y G 3 = 5yy + y5 y γ y = w. 3.4) Let us assume that equations 3.) have the first integrals P i = P i y, y,y,y,)=c i i =,, 3). 3.5) Later we will use the following designations in this section y = y y = y y 3 = y y 4 = y P = P i C = C i, F = F i G = G i. 3.6) Then taking into account the definition of integral 3.5) we obtain the equation in the form E = P + P y y + P y + P y 3 + P y 4 = 0. y y y 3 3.7) Equation 3.7) on the other hand, has to correspond to equation 3.) so there is the identity E = Qy 4 + Fy +G) 3.8) Q is a polynomial of y, y, y, y 3 and. One can see from equation 3.8) the equality P = Q. y 3 3.9) Therefore, equation 3.8) takes the form P + P y y + P y + P y 3 Fy + G) P = 0. y y y 3 3.0) Now let us assume that the first integral of equations 3.) has the form m P = r k y m k 3 3.) k=0 r k = r k y, y,y,). 3.) Substituting 3.) into equation 3.0) and equating the same powers of y 3 to ero gives the following set of equations r 0 = 0 3.3) y r + r 0 y + r 0 y y + r 0 y = 0 3.4) y r + r y + r y y + r y = r 0 Fy + G) 3.5) y r k+ + r k y + r k y y + r k y = m k +)r k Fy + G) y k =,...,m ) 3.6) r m + r m y y + r m y = r m Fy +G) y 3.7)
6 004 N A Kudryashov One can see that all coefficients r k k = 0,,...,m ) can be found from equations 3.3) 3.6). As this takes the place the solutions r m and r m have to satisfy equation 3.7) if P is a integral of equations 3.). We have r 0 = r 0 y, y,) 3.8) from equation 3.3). The solution of equation 3.4) can be presented in the form r = r 0 y y b 0y + f y, y,) 3.9) b 0 = r 0 + r 0 y y. 3.0) Equating the same powers of y in equation 3.7) gives m+ r 0 y m+ = 0 m b 0 y m = 0. 3.) In fact, one can find the general form of polynomial dependence r 0 on y, y and by solving equations 3.3) 3.7) and taking into account 3.) but one notes that r 0 is contained in r k as a linear expression. Without loss of generality let us take that r 0 = y m. 3.) Now one can write r = m my y + f y, y,) 3.3) and the solution of equation 3.5) takes the form r = mm )ym y Fy m f ) y y Gy m f + f )] y y y + f y, y,). 3.4) One can assume that r k = a k y k + b ky k + c k y k ) the coefficients a k, b k and c k depend on y, y and. Substituting 3.5) into equation 3.6) and equating the same powers of y leads to the following recursion formulae a k+ = a k 3.6) k+ y b k+ = m k +)a k F b ] k 3.7) k y c k+ = m k +)a k G c )] k bk k y + y b k. 3.8) y We have, on the other hand, from equation 3.7) a m = 0 3.9) y b m = a m F 3.30) y c m + b m y + y b m y = a m G. 3.3)
7 Transcendents defined by nonlinear fourth-order ODEs 005 Solutions of equations 3.6) and 3.9) take the form k mm )...m k+) a k = ) y m k k 3.3) k! so that a m = ) m m y m a m = ) m. 3.33) m The formulae 3.33) and the sets of equations 3.7), 3.8) and 3.30), 3.3) will be used in the proof of dependences of the solution on constants of integration for equations.5),.7) and.0). 4. Transcendents defined by equation.5) Let us consider the dependence of the general solution of equation.5) on the constants of integration. In this section we prove the following theorem. Theorem 4.. The general solutions of equation.5) are the essentially transcendental functions with respect to their constants of integration. Proof. The proof falls into two parts. First, we need to prove that the general solution has transcendental dependence on the initial conditions. Secondly, we need to prove that the general solution of equation.5) is not a semi-transcendental function. Using the following variables 4 6] u = λ u = λ 4.) λ is some parameter, one can transform equation.5) into the following one: u +0uu +5u +0u3 λ7 =0 4.) the primes of the variables are omitted). It is easy to see that equation 4.) at λ = 0is transformed into the stationary Korteveg-de Vries equation of the fifth order, which takes the form u +0uu +5u +0u3 =0. 4.3) The solution of equation 4.3) was studied in detail by Drach 7] and by Dubrovin 8]. Dubrovin found that the solution of equation 4.3) can be expressed by the theta function on the Riemann surface 8 30] u = d d lnθa + 0)] 4.4) θ) is the theta function on the Riemann surface, a is the vector of periods of some normalied differential and 0 is the arbitrary two-dimensional vector 8]. Solution 4.4) has transcendental dependence on arbitrary constants 8]. Consequently, the general solution of equation 4.) at λ 0 also has transcendental dependence on the arbitrary constants. However, equation 4.3) has the first integral in the form P 4 = u u u +5uu + 5 u4 = C ) Consequently, the general solution of equation 4.3) is the semi-transcendental function with respect to constants of integration.
8 006 N A Kudryashov Let us show that equation.5) or equation 4.) at λ 0) has no first integrals in the polynomial form. For this purpose we use the set of equations 3.7) and 3.8) which can be written in the form m k +)0ua k b k u b k+ = ] 4.6) k c k+ = m k +) 5u k ) +0u3 ) a k c )] k bk u + u b k u k =,...,m ) 4.7) for equation.5). On the other hand, coefficients b m and c m have to satisfy the set of equations which can be obtained from equations 3.30) and 3.3) b m = 0ua m 4.8) u c m + b m u + u b m 5u u = +0u3 ) a m. 4.9) One obtains b m = ) m m m 5uu + g u, )] 4.0) from equation 4.8). Substituting 4.0) into equation 4.9) and equating the same powers of u to ero gives g u, ) = 5 u4 u + p ) 4.) p ) is a function of. By the method of mathematical induction one obtains the coefficients b k in the form k mm )...m k+) b k = ) u m k k 5uu k )! + 5 u4 ] u + p ) 4.) from equation 4.6). We also have b = mu m 5uu + 5 u4 u + p )] 4.3) from equation 4.). Taking into account solution 4.3) one can obtain c = mu m p u 4.4) from equation 4.7). Assuming that c k = ) k A k u m k+ p u 4.5) A k is some positive constant) it leads by mathematical induction to the solution for c k+ in the form c k+ = ) k A k+ u m k p u 4.6) A k+ = m k +) k ) A k + ] mm )...m k). 4.7) k k )!
9 Transcendents defined by nonlinear fourth-order ODEs 007 Thus we have c m = ) m p A m u u. 4.8) Substituting 3.33), 4.) and 4.8) into equation 4.9) gives the contradiction m ) + A p m m u ) This contradiction shows that the integral of equation.5) in the form 3.) does not exist. Consequently the general solutions of equation.5) are essentially transcendental functions with respect to their constants of integration. This proves theorem Transcendents defined by equation.7) Let us consider the dependence of the general solution of equation.7) on the constants of integration. We wish to prove the following theorem. Theorem 5.. The general solution of equation.7) are essentially transcendental functions with respect to their constants of integration. Proof. This proof also contains two parts. First, one uses the variables v = µ v = µ 5.) µ is some parameter) so that equation.7) is transformed to the following equation v 0vv 0vv +6v5 µ 3 v µ 5 α = 0 5.) the primes of the variables are omitted). Assuming µ = 0 in equation 5.) we obtain the equation v 0vv 0vv +6v5 =0 5.3) which corresponds to the stationary modified Korteveg-de Vries equation of the fifth order. The general solution of equation 5.3) can be presented via the theta function on the Riemann surface as the solution of equation 4.3) 3]. This solution has transcendental dependence on the constants of integration. Consequently, the general solution of equation 5.) at µ 0 also has transcendental dependence on the constants of integration. Thus equation.7) does not belong to the first case of dependence on the constants. However, equation 5.3) has the first integral in the form P 5 = v v v 0v v + v6 = C ) Consequently equation 5.3) has the general solution in the form of the semi-transcendental function with respect to the initial conditions. We show this is not the case for equation.7). For this we again use the set of equations 3.7), 3.8) which can be presented in the following form b k+ = 0v m k +)a k b ] k 5.5) k v c k+ = 0vv k ) +6v5 v β)m k +)a k c )] k bk v + u b k v k =,...,m ) 5.6) for equation.7).
10 008 N A Kudryashov On the other hand, the coefficients b m and c m have to satisfy the set of equations b m +0v a m =0 5.7) v c m + b m v + v b m v + 0vv 6v5 + v + β)a m = 0 5.8) which is obtained from equations 3.30) and 3.3). We have b m = ) m m m 5v v + g v, )] 5.9) from equation 5.7). Substituting solution 5.9) into equation 5.8) and equating the same powers of v to ero gives g u, ) = v 6 v βv + p ) 5.0) p ) is a function of integration. By the method of mathematical induction we obtain k+ mm )...m k+) b k = ) v k m k v 6 5v v k )! ] v βv + p ) and 5.) b = mv m v 6 5v v v βv + p )]. 5.) Now one can find the coefficient c from equation 5.6). It takes the form c = mv m p v. 5.3) Taking into account the method of mathematical induction we obtain c k+ = ) k A k+ v m k p v 5.4) A k+ can be expressed by the recursion formula 4.7). We have c m = ) m p A m v v 5.5) from equation 5.4). Substituting 3.33), 5.9) and 5.5) into equation 5.8) gives the contradiction m ) + A p m m v ) The integral of equation.7) in the form 3.) does not, therefore, exist. Consequently the general solution of equation.7) are the essentially transcendental functions with respect to their constants of integration.
11 Transcendents defined by nonlinear fourth-order ODEs Transcendents defined by equation.0) By way of the last example let us consider equation.0). Using the variables w = η w = η 6.) η is some parameter) one can transform equation.0) to the following one w +5w w 5w w 5ww + w5 η 3 w η 5 γ = 0 6.) the primes in the variables are also omitted). Equation 6.) at η = 0 takes the form of the stationary Fordy Gibbons equation.8) w +5w w 5w w 5ww + w5 = ) The general solution of this equation can also be expressed via the theta function on the Riemann surface 3]. This solution has transcendental dependence on the initial conditions. Consequently, one can expect that the general solution of equation 6.) has transcendental dependence on the constants of integration at η 0 as well. However, equation 6.3) has the first integral in the form P 6 = w w w3 5 w w + 6 w6 w = C 6 6.4) and consequently the solution of equation 6.3) is the semi-transcendental function with respect to their initial condition. Let us show that equation.0) does not have any integral in the form 3.). For this purpose we again use the set of equations 3.7), 3.8). These ones can be written in the form b k+ = m k +)5w 5w )a k b ] k 6.5) k w c k+ = m k +)w 5 5ww k ) w γ)a k c )] k bk w + w b k w k =,...,m ). 6.6) Moreover, the coefficients b m and c m have to satisfy the set of equations b m = 5w 5w )a m 6.7) w c m + b m w + w b m w = w5 5ww w γ)a m. 6.8) Solution of equation 6.7) can be presented in the form b m = ) m m 5 m 3 w3 5 ] w w + g 3 w, ). 6.9) Substituting 6.9) into equation 6.8) and equating of the same powers of w to ero leads to the solution for g 3 w, ) in the form g 3 w, ) = 6 w6 w γw+p 3 ). 6.0) By the method of mathematical induction we obtain k mm )...m k+) b k = ) w k m k k )! ] 5 3 w 5 w + 6 w6 w γw+p 3 ) k =,...,m ). 6.)
12 00 N A Kudryashov One can find c in the form c = mw m p3 w ) 6.) from equation 6.6). By again taking into account the method of the mathematical induction we have c k+ = ) k A k+ w m p3 w 6.3) A k+ can be also expressed by the recursion formula 4.7). Using c m = ) m p3 A m w w 6.4) and 3.33) and 6.9) we have m ) + A p 3 m m w 0 6.5) from equation 6.8). We showed that the integral of equation.0) in the form 3.) does not exist. This shows that the general solution of equation.0) is the transcendental function on the initial conditions. 7. On irreducibility of equations.5),.7) and.0) We proved in sections 4 6 that equations.5),.7) and.0) do not have any first integrals in the polynomial form and that their solutions are the essentially transcendental functions with respect to the constants of integration. We will discuss the problem of irreducibility of these equations in this section. It is known that the notions of irreducibility and the transcendental dependence on initial conditions are equivalent for the second algebraic differential equations ], but there is not such proof of equivalence at higher-order equations. The problem of finding new transcendents defined by nonlinear ODEs.5),.7) and.0) is only one in the investigation of dependence on the solutions of the Painlevé equations. Actually, one can see from relations.5) and.8) that the special solutions of equations.7) and.0) are expressed via the solutions of the first Painlevé equations and one can imagine that solutions of equations.5),.7) and.0) are expressed via the solutions of the Painlevé equations in the general case. It is known, for example, that the third-order equation which takes the form y 3 +6yy y y = 0 7.) y = dy y 3 = d3 y d d. 3 The solution of this equation is the transcendental function with respect to constants of integration but this one is not the new transcendent because the solution of equation 7.) is expressed by the formula y = v v 7.) v is the solution of the second Painlevé equation.6). The matter is that equation 7.) is reducible one.
13 Transcendents defined by nonlinear fourth-order ODEs 0 However, regarding equations.5),.7) and.0) this is not the case because there are reasons to believe that equations.5),.7) and.0) are irreducible. For example, let us assume that the solution of equation.5) is expressed via the solution of the second Painlevé equation corresponding to the following transformation y = gv, v,) 7.3) y is a solution of equation.5) and v is a solution of equation.6). As this takes place one notes that we do not need to take the transformation 7.3) in more general form because it can be transformed to the transformation 7.3) taking into account equation.6). It is easy to check that the transformation 7.3) cannot be found for equation.5) and.6). One can suggest closely approximating arguments for transformations 7.3) which connect equations.5),.7) and.0) with other Painlevé equations. Now let us assume that there exists an expression W = Dy, y,y,) 7.4) in equation.5) so that W satisfies the first Painlevé equation. Taking into account 7.4) one can obtain W x = D x + D y y + D y y + D y y 3 7.5) W xx = D xx + D yx y + D y y + D y xy + D y y 3 + D y xy 3 + D y y 4 + D xy y + D yy y +D y yy y + D y yy 3 y + D xy y + D yy y y + D y y y + D y y y 3 y +D xy y 3 + D yy y y 3 + D y y y y 3 + D y y y3. 7.6) Substituting 7.4) and 7.6) into equation.4) we do not obtain equation.5). This contradiction shows that we cannot find the expression 7.4) in equation.5) which reduces equation.5) to.4). Finally, one can suggest more arguments why one can expect that the solutions of equations.5),.7) and.0) are new transcendents. Let us take the variables u = 3 ω x = ) then equation.5) can be presented in the form ω xxxx +5ωx +0ωω xx +0ω 3 λ7 + x ω xxx + 0 x ωω x 4 49 x ω xx x ω x ω 3 x 80 ω = ) 40 x4 One can see from equation 7.8) that this equation takes the form ω xxxx +5ωx +0ωω xx +0ω 3 λ7 =0 7.9) at x. The solution of equation 7.9) is expressed in terms of the theta function on the Riemann surface The asymptotic solution of equation.5) takes the form u) 3 ω ) 7.0) ω) is a solution of equation 7.9). The asymptotic solution of equation.5) corresponds to the solution of the irreducible equation and consequently one can expect that equation.5) is also the irreducible equation. Certainly the rigorous proof of the irreducibility of equations.5),.7) and.0) is derivable from group theory but the above-mentioned arguments allow us to expect that equations.5),.7) and.0) give new transcendents.
14 0 N A Kudryashov 8. Conclusion Thus we have shown that the general solutions of equations.5),.7) and.0) are essentially transcendentical functions with respect to their constants of integration. Actually, these equations possess the Painlevé property and their general solutions are the single-valued functions. In the approximate limit solutions of these equations can be presented via the theta functions on the Riemann surfaces which are the semi-transcendentical functions with respect to their constants of integration. Consequently, these solutions have transcendental dependences on their constants. We have shown that equations.5),.7) and.0) have no first integrals in the polynomial form. Consequently, their general solutions are the essentially transcendental functions with respect to their constants of integration. These solutions belong to the class of functions as the six Painlevé transcendents. We have discussed the irreducibility of equations.5),.7) and.0) and have presented some reasons why one can expect that these equations are irreducible ones. We believe that these equations can give new transcendents defined by nonlinear ODEs. In fact, we think that every general solution of equations.) and.3) are the transcendents defined by nonlinear ODE and therefore we hope to obtain the infinite number of such transcendents. Acknowledgments Results of this work were presented at the workshop Nonlinear Evolution Equations and Dynamical Systems in Leeds June, 998). I am very grateful to Allan Fordy and A V Michailov for the invitation to visit this Conference. I am also grateful to Martin Kruskal, V B Kunetsov, L A Bordag and V Z Enolsky, for useful discussions of this work. This work was supported by the International Science and Technology Centre under project B3-96. This material is partially based upon work supported by the US Civilian Research and Development Foundation under award no RG-44. References ] Conte R ed) 998 The Painlevé approach to nonlinear ordinary differential equations The Painlevé Property, One Century Later CRM Series in Mathematical Physics) Berlin: Springer) ] Umemura H 990 Nagoya Math. J. 9 3] Ablowit M J and Clarkson P A 99 Solitons, Nonlinear Evolution Equations and Inverse Scattering Cambridge: Cambridge University Press) 4] Bureau F J 964 Ann. Mat ] Bureau F J 964 Ann. Mat. 66 6] Bureau F J 97 Ann. Mat ] Ablowit M J and Segur H 977 Phys. Rev. Lett ] Ablowit M I, Ramani A and Segur H 978 Lett. Nuovo Cimento ] Ablowit M J, Ramani A and Segur H 980 J. Math. Phys. 75 Ablowit M J, Ramani A and Segur H 980 J. Math. Phys ] Kudryashov N A 997 Phys. Lett. A ] Flaschka H and Newell A C 980 Commun. Math. Phys ] Airault H 979 Stud. Appl. Math ] Gromak V I 984 Diff. Eqns 0 04 in Russian) 4] Kudryashov N A and Soukharev M B 998 Phys. Lett. A ] Conte R, Fordy A P and Pickering A 993 Physica D ] Fordy A P and Gibbons J 980 Phys. Lett. A ] Weiss J, Tabor M and Carnevale G 983 J. Math. Phys. 4 5
15 Transcendents defined by nonlinear fourth-order ODEs 03 8] Weiss J 984 J. Math. Phys ] Gromak V I and Tsigelnik V V 988 Painlevé equations, group analysis and nonlinear evolution equations Preprint Minsk in Russian) 0] Hone A W 998 Physica D 8 6 ] Kudryashov N A 997 J. Phys. A: Math. Gen ] Kudryashov N A 994 J. Phys. A: Math. Gen ] Weiss J 983 J. Math. Phys ] Ince F L 956 Ordinary Differential Equations New York: Dover) 5] Gromak V I and Luashevich N A 990 The Analytic Solutions of the Painlevé Equations Minsk: Universitetskoye Publishers) in Russian) 6] Golubev V V 953 Lectures on the Integration of the Equation of Motion of a Rigid Body about a Fixed Point Gostechidat Moscow: State Publishing House) in Russian) 7] Drach J 99 Comptes rendus Acad. Sci., Paris ] Dubrovin B A 98 Russ. Math. Surveys 36 9] Novikov S P, Manakov S V, Pitaevski L P and Zakharov V E 984 Theory of Solitons. The Inverse Scattering Method New York: Plenum) 30] Dubrovin B A, Matveev V B and Novikov S P 976 Russ. Math. Surveys ] Gabo R, Gavrillov L and Ravoson V 993 J. Math. Phys
arxiv:nlin/ v2 [nlin.si] 9 Oct 2002
Journal of Nonlinear Mathematical Physics Volume 9, Number 1 2002), 21 25 Letter On Integrability of Differential Constraints Arising from the Singularity Analysis S Yu SAKOVICH Institute of Physics, National
More informationBäcklund transformations for fourth-order Painlevé-type equations
Bäcklund transformations for fourth-order Painlevé-type equations A. H. Sakka 1 and S. R. Elshamy Department of Mathematics, Islamic University of Gaza P.O.Box 108, Rimae, Gaza, Palestine 1 e-mail: asakka@mail.iugaza.edu
More informationFirst-order Second-degree Equations Related with Painlevé Equations
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.8(2009) No.3,pp.259-273 First-order Second-degree Equations Related with Painlevé Equations Ayman Sakka 1, Uğurhan
More informationTHE LAX PAIR FOR THE MKDV HIERARCHY. Peter A. Clarkson, Nalini Joshi & Marta Mazzocco
Séminaires & Congrès 14, 006, p. 53 64 THE LAX PAIR FOR THE MKDV HIERARCHY by Peter A. Clarkson, Nalini Joshi & Marta Mazzocco Abstract. In this paper we give an algorithmic method of deriving the Lax
More informationTHE SECOND PAINLEVÉ HIERARCHY AND THE STATIONARY KDV HIERARCHY
THE SECOND PAINLEVÉ HIERARCHY AND THE STATIONARY KDV HIERARCHY N. JOSHI Abstract. It is well known that soliton equations such as the Korteweg-de Vries equation are members of infinite sequences of PDEs
More informationNonlinear Differential Equations with Exact Solutions Expressed via the Weierstrass Function
Nonlinear Differential Equations with Exact Solutions Expressed via the Weierstrass Function Nikolai A. Kudryashov Department of Applied Mathematics Moscow Engineering and Physics Institute State university),
More informationBäcklund transformation and special solutions for Drinfeld Sokolov Satsuma Hirota system of coupled equations
arxiv:nlin/0102001v1 [nlin.si] 1 Feb 2001 Bäcklund transformation and special solutions for Drinfeld Sokolov Satsuma Hirota system of coupled equations Ayşe Karasu (Kalkanli) and S Yu Sakovich Department
More informationThe Construction of Alternative Modified KdV Equation in (2 + 1) Dimensions
Proceedings of Institute of Mathematics of NAS of Ukraine 00, Vol. 3, Part 1, 377 383 The Construction of Alternative Modified KdV Equation in + 1) Dimensions Kouichi TODA Department of Physics, Keio University,
More informationThe Higher Dimensional Bateman Equation and Painlevé Analysis of Nonintegrable Wave Equations
Symmetry in Nonlinear Mathematical Physics 1997, V.1, 185 192. The Higher Dimensional Bateman Equation and Painlevé Analysis of Nonintegrable Wave Equations Norbert EULER, Ove LINDBLOM, Marianna EULER
More informationCoupled KdV Equations of Hirota-Satsuma Type
Journal of Nonlinear Mathematical Physics 1999, V.6, N 3, 255 262. Letter Coupled KdV Equations of Hirota-Satsuma Type S.Yu. SAKOVICH Institute of Physics, National Academy of Sciences, P.O. 72, Minsk,
More informationComputers and Mathematics with Applications
Computers and Mathematics with Applications 60 (00) 3088 3097 Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa Symmetry
More informationA New Technique in Nonlinear Singularity Analysis
Publ. RIMS, Kyoto Univ. 39 (2003), 435 449 A New Technique in Nonlinear Singularity Analysis By Pilar R. Gordoa, Nalini Joshi and Andrew Pickering Abstract The test for singularity structure proposed by
More informationarxiv:math/ v1 [math.ap] 1 Jan 1992
APPEARED IN BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 26, Number 1, Jan 1992, Pages 119-124 A STEEPEST DESCENT METHOD FOR OSCILLATORY RIEMANN-HILBERT PROBLEMS arxiv:math/9201261v1 [math.ap]
More informationOn the singularities of non-linear ODEs
On the singularities of non-linear ODEs Galina Filipuk Institute of Mathematics University of Warsaw G.Filipuk@mimuw.edu.pl Collaborators: R. Halburd (London), R. Vidunas (Tokyo), R. Kycia (Kraków) 1 Plan
More informationarxiv: v1 [nlin.si] 23 Aug 2007
arxiv:0708.3247v1 [nlin.si] 23 Aug 2007 A new integrable generalization of the Korteweg de Vries equation Ayşe Karasu-Kalkanlı 1), Atalay Karasu 2), Anton Sakovich 3), Sergei Sakovich 4), Refik Turhan
More informationLinearization of Mirror Systems
Journal of Nonlinear Mathematical Physics 00, Volume 9, Supplement 1, 34 4 Proceedings: Hong Kong Linearization of Mirror Systems Tat Leung YEE Department of Mathematics, The Hong Kong University of Science
More informationMACSYMA PROGRAM FOR THE PAINLEVÉ TEST FOR NONLINEAR ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS
. MACSYMA PROGRAM FOR THE PAINLEVÉ TEST FOR NONLINEAR ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS Willy Hereman Mathematics Department and Center for the Mathematical Sciences University of Wisconsin at
More informationIntegrability, Nonintegrability and the Poly-Painlevé Test
Integrability, Nonintegrability and the Poly-Painlevé Test Rodica D. Costin Sept. 2005 Nonlinearity 2003, 1997, preprint, Meth.Appl.An. 1997, Inv.Math. 2001 Talk dedicated to my professor, Martin Kruskal.
More informationPainlevé Test for the Certain (2+1)-Dimensional Nonlinear Evolution Equations. Abstract
Painlevé Test for the Certain (2+1)-Dimensional Nonlinear Evolution Equations T. Alagesan and Y. Chung Department of Information and Communications, Kwangju Institute of Science and Technology, 1 Oryong-dong,
More informationA Generalization of the Sine-Gordon Equation to 2+1 Dimensions
Journal of Nonlinear Mathematical Physics Volume 11, Number (004), 164 179 Article A Generalization of the Sine-Gordon Equation to +1 Dimensions PGESTÉVEZ and J PRADA Area de Fisica Teórica, Facultad de
More informationNumerical computation of the finite-genus solutions of the Korteweg-de Vries equation via Riemann Hilbert problems
Numerical computation of the finite-genus solutions of the Korteweg-de Vries equation via Riemann Hilbert problems Thomas Trogdon 1 and Bernard Deconinck Department of Applied Mathematics University of
More informationA note on the G /G - expansion method
A note on the G /G - expansion method Nikolai A. Kudryashov Department of Applied Mathematics, National Research Nuclear University MEPHI, Kashirskoe Shosse, 115409 Moscow, Russian Federation Abstract
More informationSecond Order Lax Pairs of Nonlinear Partial Differential Equations with Schwarzian Forms
Second Order Lax Pairs of Nonlinear Partial Differential Equations with Schwarzian Forms Sen-yue Lou a b c, Xiao-yan Tang a b, Qing-Ping Liu b d, and T. Fukuyama e f a Department of Physics, Shanghai Jiao
More informationLogistic function as solution of many nonlinear differential equations
Logistic function as solution of many nonlinear differential equations arxiv:1409.6896v1 [nlin.si] 24 Sep 2014 Nikolai A. Kudryashov, Mikhail A. Chmykhov Department of Applied Mathematics, National Research
More informationFreedom in the Expansion and Obstacles to Integrability in Multiple-Soliton Solutions of the Perturbed KdV Equation
Freedom in the Expansion and Obstacles to Integrability in Multiple-Soliton Solutions of the Perturbed KdV Equation Alex Veksler 1 and Yair Zarmi 1, Ben-Gurion University of the Negev, Israel 1 Department
More informationA MULTI-COMPONENT LAX INTEGRABLE HIERARCHY WITH HAMILTONIAN STRUCTURE
Pacific Journal of Applied Mathematics Volume 1, Number 2, pp. 69 75 ISSN PJAM c 2008 Nova Science Publishers, Inc. A MULTI-COMPONENT LAX INTEGRABLE HIERARCHY WITH HAMILTONIAN STRUCTURE Wen-Xiu Ma Department
More informationA STEEPEST DESCENT METHOD FOR OSCILLATORY RIEMANN-HILBERT PROBLEMS
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 26, Number 1, January 1992 A STEEPEST DESCENT METHOD FOR OSCILLATORY RIEMANN-HILBERT PROBLEMS P. DEIFT AND X. ZHOU In this announcement
More informationHamiltonian partial differential equations and Painlevé transcendents
Winter School on PDEs St Etienne de Tinée February 2-6, 2015 Hamiltonian partial differential equations and Painlevé transcendents Boris DUBROVIN SISSA, Trieste Lecture 2 Recall: the main goal is to compare
More informationOn bosonic limits of two recent supersymmetric extensions of the Harry Dym hierarchy
arxiv:nlin/3139v2 [nlin.si] 14 Jan 24 On bosonic limits of two recent supersymmetric extensions of the Harry Dym hierarchy S. Yu. Sakovich Mathematical Institute, Silesian University, 7461 Opava, Czech
More informationSymmetry Reductions of (2+1) dimensional Equal Width. Wave Equation
Authors: Symmetry Reductions of (2+1) dimensional Equal Width 1. Dr. S. Padmasekaran Wave Equation Asst. Professor, Department of Mathematics Periyar University, Salem 2. M.G. RANI Periyar University,
More informationLocal and global finite branching of solutions of ODEs in the complex plane
Local and global finite branching of solutions of ODEs in the complex plane Workshop on Singularities and Nonlinear ODEs Thomas Kecker University College London / University of Portsmouth Warszawa, 7 9.11.2014
More informationIntegrability approaches to differential equations
XXIV Fall Workshop on Geometry and Physics Zaragoza, CUD, September 2015 What is integrability? On a fairly imprecise first approximation, we say integrability is the exact solvability or regular behavior
More informationTravelling wave solutions for a CBS equation in dimensions
AMERICAN CONFERENCE ON APPLIED MATHEMATICS (MATH '8), Harvard, Massachusetts, USA, March -6, 8 Travelling wave solutions for a CBS equation in + dimensions MARIA LUZ GANDARIAS University of Cádiz Department
More informationBreaking soliton equations and negative-order breaking soliton equations of typical and higher orders
Pramana J. Phys. (2016) 87: 68 DOI 10.1007/s12043-016-1273-z c Indian Academy of Sciences Breaking soliton equations and negative-order breaking soliton equations of typical and higher orders ABDUL-MAJID
More informationA PERIODICITY PROBLEM FOR THE KORTEWEG DE VRIES AND STURM LIOUVILLE EQUATIONS. THEIR CONNECTION WITH ALGEBRAIC GEOMETRY
A PERIODICITY PROBLEM FOR THE KORTEWEG DE VRIES AND STURM LIOUVILLE EQUATIONS. THEIR CONNECTION WITH ALGEBRAIC GEOMETRY B. A. DUBROVIN AND S. P. NOVIKOV 1. As was shown in the remarkable communication
More informationMeromorphic solutions. of nonlinear ordinary differential equations.
Meromorphic solutions of nonlinear ordinary differential equations Nikolai A. Kudryashov Department of Applied Mathematics, National Research Nuclear University MEPHI, 31 Kashirskoe Shosse, 115409 Moscow,
More informationHamiltonian partial differential equations and Painlevé transcendents
The 6th TIMS-OCAMI-WASEDA Joint International Workshop on Integrable Systems and Mathematical Physics March 22-26, 2014 Hamiltonian partial differential equations and Painlevé transcendents Boris DUBROVIN
More informationBuilding Generalized Lax Integrable KdV and MKdV Equations with Spatiotemporally Varying Coefficients
Journal of Physics: Conference Series OPEN ACCESS Building Generalized Lax Integrable KdV and MKdV Equations with Spatiotemporally Varying Coefficients To cite this article: M Russo and S R Choudhury 2014
More informationOn the Transformations of the Sixth Painlevé Equation
Journal of Nonlinear Mathematical Physics Volume 10, Supplement 2 (2003), 57 68 SIDE V On the Transformations of the Sixth Painlevé Equation Valery I GROMAK andgalinafilipuk Department of differential
More informationarxiv:nlin/ v2 [nlin.si] 14 Sep 2001
Second order Lax pairs of nonlinear partial differential equations with Schwarz variants arxiv:nlin/0108045v2 [nlin.si] 14 Sep 2001 Sen-yue Lou 1,2,3, Xiao-yan Tang 2,3, Qing-Ping Liu 1,4,3 and T. Fukuyama
More informationPainlevé Test and Higher Order Differential Equations
Journal of Nonlinear Mathematical Physics Volume 9, Number 3 2002), 282 30 Article Painlevé Test and Higher Order Differential Equations Uğurhan MUĞAN and Fahd JRAD SUNY at Buffalo, Department of Mathematics,
More informationSimilarity Reductions of (2+1)-Dimensional Multi-component Broer Kaup System
Commun. Theor. Phys. Beijing China 50 008 pp. 803 808 c Chinese Physical Society Vol. 50 No. 4 October 15 008 Similarity Reductions of +1-Dimensional Multi-component Broer Kaup System DONG Zhong-Zhou 1
More informationON THE ALGEBRAIC INDEPENDENCE OF GENERIC PAINLEVÉ TRANSCENDENTS.
ON THE ALGEBRAIC INDEPENDENCE OF GENERIC PAINLEVÉ TRANSCENDENTS. JOEL NAGLOO 1, ANAND PILLAY 2 Abstract. We prove that if y = f(y, y, t, α, β,...) is a generic Painlevé equation from among the classes
More informationThe Modified (G /G)-Expansion Method for Nonlinear Evolution Equations
The Modified ( /-Expansion Method for Nonlinear Evolution Equations Sheng Zhang, Ying-Na Sun, Jin-Mei Ba, and Ling Dong Department of Mathematics, Bohai University, Jinzhou 11000, P. R. China Reprint requests
More informationTowards classification of (2 + 1)-dimensional integrable equations. Integrability conditions I
J. Phys. A: Math. Gen. 31 (1998) 6707 6715. Printed in the UK PII: S0305-4470(98)92996-1 Towards classification of (2 + 1)-dimensional integrable equations. Integrability conditions I A V Mihailov andriyamilov
More informationPoles and α-points of Meromorphic Solutions of the First Painlevé Hierarchy
Publ. RIMS, Kyoto Univ. 40 2004), 471 485 Poles and α-points of Meromorphic Solutions of the First Painlevé Hierarchy By Shun Shimomura Abstract The first Painlevé hierarchy, which is a sequence of higher
More informationNumerical solutions of the small dispersion limit of KdV, Whitham and Painlevé equations
Numerical solutions of the small dispersion limit of KdV, Whitham and Painlevé equations Tamara Grava (SISSA) joint work with Christian Klein (MPI Leipzig) Integrable Systems in Applied Mathematics Colmenarejo,
More informationMultiplication of Generalized Functions: Introduction
Bulg. J. Phys. 42 (2015) 93 98 Multiplication of Generalized Functions: Introduction Ch. Ya. Christov This Introduction was written in 1989 to the book by Ch. Ya. Christov and B. P. Damianov titled Multiplication
More informationStudies on Integrability for Nonlinear Dynamical Systems and its Applications
Studies on Integrability for Nonlinear Dynamical Systems and its Applications Koichi Kondo Division of Mathematical Science Department of Informatics and Mathematical Science Graduate School of Engineering
More informationLinearization of Second-Order Ordinary Dif ferential Equations by Generalized Sundman Transformations
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 6 (2010), 051, 11 pages Linearization of Second-Order Ordinary Dif ferential Equations by Generalized Sundman Transformations Warisa
More informationSymmetry reductions and travelling wave solutions for a new integrable equation
Symmetry reductions and travelling wave solutions for a new integrable equation MARIA LUZ GANDARIAS University of Cádiz Department of Mathematics PO.BOX 0, 50 Puerto Real, Cádiz SPAIN marialuz.gandarias@uca.es
More informationSymmetries and Group Invariant Reductions of Integrable Partial Difference Equations
Proceedings of 0th International Conference in MOdern GRoup ANalysis 2005, 222 230 Symmetries and Group Invariant Reductions of Integrable Partial Difference Equations A. TONGAS, D. TSOUBELIS and V. PAPAGEORGIOU
More informationSecond-order second-degree Painlevé equations related with Painlevé I VI equations and Fuchsian-type transformations
Second-order second-degree Painlevé equations related with Painlevé I VI equations and Fuchsian-type transformations U. Muğan and A. Sakka Citation: J. Math. Phys. 40, 3569 (1999); doi: 10.1063/1.532909
More informationIntroduction to the Hirota bilinear method
Introduction to the Hirota bilinear method arxiv:solv-int/9708006v1 14 Aug 1997 J. Hietarinta Department of Physics, University of Turku FIN-20014 Turku, Finland e-mail: hietarin@utu.fi Abstract We give
More informationExact solutions through symmetry reductions for a new integrable equation
Exact solutions through symmetry reductions for a new integrable equation MARIA LUZ GANDARIAS University of Cádiz Department of Mathematics PO.BOX, 1151 Puerto Real, Cádiz SPAIN marialuz.gandarias@uca.es
More informationA General Formula of Flow Equations for Harry Dym Hierarchy
Commun. Theor. Phys. 55 (211 193 198 Vol. 55, No. 2, February 15, 211 A General Formula of Flow Equations for Harry Dym Hierarchy CHENG Ji-Peng ( Î, 1 HE Jing-Song ( Ø, 2, and WANG Li-Hong ( 2 1 Department
More informationOn a WKB-theoretic approach to. Ablowitz-Segur's connection problem for. the second Painleve equation. Yoshitsugu TAKEI
On a WKB-theoretic approach to Ablowitz-Segur's connection problem for the second Painleve equation Yoshitsugu TAKEI Research Institute for Mathematical Sciences Kyoto University Kyoto, 606-8502, JAPAN
More informationA Short historical review Our goal The hierarchy and Lax... The Hamiltonian... The Dubrovin-type... Algebro-geometric... Home Page.
Page 1 of 46 Department of Mathematics,Shanghai The Hamiltonian Structure and Algebro-geometric Solution of a 1 + 1-Dimensional Coupled Equations Xia Tiecheng and Pan Hongfei Page 2 of 46 Section One A
More informationQUADRATIC EQUATIONS AND MONODROMY EVOLVING DEFORMATIONS
QUADRATIC EQUATIONS AND MONODROMY EVOLVING DEFORMATIONS YOUSUKE OHYAMA 1. Introduction In this paper we study a special class of monodromy evolving deformations (MED). Chakravarty and Ablowitz [4] showed
More informationON THE ASYMPTOTIC REPRESENTATION OF THE SOLUTIONS TO THE FOURTH GENERAL PAINLEVÉ EQUATION
IJMMS 003:13, 45 51 PII. S016117103060 http://ijmms.hindawi.com Hindawi Publishing Corp. ON THE ASYMPTOTIC REPRESENTATION OF THE SOLUTIONS TO THE FOURTH GENERAL PAINLEVÉ EQUATION YOUMIN LU Received 5 June
More informationReductions to Korteweg-de Vries Soliton Hierarchy
Commun. Theor. Phys. (Beijing, China 45 (2006 pp. 23 235 c International Academic Publishers Vol. 45, No. 2, February 5, 2006 Reductions to Korteweg-de Vries Soliton Hierarchy CHEN Jin-Bing,,2, TAN Rui-Mei,
More informationEXACT SOLITARY WAVE AND PERIODIC WAVE SOLUTIONS OF THE KAUP-KUPERSCHMIDT EQUATION
Journal of Applied Analysis and Computation Volume 5, Number 3, August 015, 485 495 Website:http://jaac-online.com/ doi:10.11948/015039 EXACT SOLITARY WAVE AND PERIODIC WAVE SOLUTIONS OF THE KAUP-KUPERSCHMIDT
More informationNew families of non-travelling wave solutions to a new (3+1)-dimensional potential-ytsf equation
MM Research Preprints, 376 381 MMRC, AMSS, Academia Sinica, Beijing No., December 3 New families of non-travelling wave solutions to a new (3+1-dimensional potential-ytsf equation Zhenya Yan Key Laboratory
More informationON THE SYMMETRIES OF INTEGRABLE PARTIAL DIFFERENCE EQUATIONS
Proceedings of the International Conference on Difference Equations, Special Functions and Orthogonal Polynomials, World Scientific (2007 ON THE SYMMETRIES OF INTEGRABLE PARTIAL DIFFERENCE EQUATIONS ANASTASIOS
More informationProlongation structure for nonlinear integrable couplings of a KdV soliton hierarchy
Prolongation structure for nonlinear integrable couplings of a KdV soliton hierarchy Yu Fa-Jun School of Mathematics and Systematic Sciences, Shenyang Normal University, Shenyang 110034, China Received
More informationarxiv: v1 [math-ph] 25 Jul Preliminaries
arxiv:1107.4877v1 [math-ph] 25 Jul 2011 Nonlinear self-adjointness and conservation laws Nail H. Ibragimov Department of Mathematics and Science, Blekinge Institute of Technology, 371 79 Karlskrona, Sweden
More informationFLOQUET THEORY FOR LINEAR DIFFERENTIAL EQUATIONS WITH MEROMORPHIC SOLUTIONS
FLOQUET THEORY FOR LINEAR DIFFERENTIAL EQUATIONS WITH MEROMORPHIC SOLUTIONS R WEIKARD Abstract If A is a ω-periodic matrix Floquet s theorem states that the differential equation y = Ay has a fundamental
More informationSymbolic Computation and New Soliton-Like Solutions of the 1+2D Calogero-Bogoyavlenskii-Schif Equation
MM Research Preprints, 85 93 MMRC, AMSS, Academia Sinica, Beijing No., December 003 85 Symbolic Computation and New Soliton-Like Solutions of the 1+D Calogero-Bogoyavlenskii-Schif Equation Zhenya Yan Key
More informationA Note On Solitary Wave Solutions of the Compound Burgers-Korteweg-de Vries Equation
A Note On Solitary Wave Solutions of the Compound Burgers-Korteweg-de Vries Equation arxiv:math/6768v1 [math.ap] 6 Jul 6 Claire David, Rasika Fernando, and Zhaosheng Feng Université Pierre et Marie Curie-Paris
More informationSome Elliptic Traveling Wave Solutions to the Novikov-Veselov Equation
Progress In Electromagnetics Research Symposium 006, Cambridge, USA, March 6-9 59 Some Elliptic Traveling Wave Solutions to the Novikov-Veselov Equation J. Nickel, V. S. Serov, and H. W. Schürmann University
More informationPainlevé analysis and some solutions of variable coefficient Benny equation
PRAMANA c Indian Academy of Sciences Vol. 85, No. 6 journal of December 015 physics pp. 1111 11 Painlevé analysis and some solutions of variable coefficient Benny equation RAJEEV KUMAR 1,, R K GUPTA and
More informationarxiv:solv-int/ v1 20 Oct 1993
Casorati Determinant Solutions for the Discrete Painlevé-II Equation Kenji Kajiwara, Yasuhiro Ohta, Junkichi Satsuma, Basil Grammaticos and Alfred Ramani arxiv:solv-int/9310002v1 20 Oct 1993 Department
More informationSymmetry and Exact Solutions of (2+1)-Dimensional Generalized Sasa Satsuma Equation via a Modified Direct Method
Commun. Theor. Phys. Beijing, China 51 2009 pp. 97 978 c Chinese Physical Society and IOP Publishing Ltd Vol. 51, No., June 15, 2009 Symmetry and Exact Solutions of 2+1-Dimensional Generalized Sasa Satsuma
More informationEqWorld INDEX.
EqWorld http://eqworld.ipmnet.ru Exact Solutions > Basic Handbooks > A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC, Boca Raton, 2004 INDEX A
More informationNotes on the Inverse Scattering Transform and Solitons. November 28, 2005 (check for updates/corrections!)
Notes on the Inverse Scattering Transform and Solitons Math 418 November 28, 2005 (check for updates/corrections!) Among the nonlinear wave equations are very special ones called integrable equations.
More informationSymmetry Reductions of Integrable Lattice Equations
Isaac Newton Institute for Mathematical Sciences Discrete Integrable Systems Symmetry Reductions of Integrable Lattice Equations Pavlos Xenitidis University of Patras Greece March 11, 2009 Pavlos Xenitidis
More informationDiophantine integrability
Loughborough University Institutional Repository Diophantine integrability This item was submitted to Loughborough University's Institutional Repository by the/an author. Additional Information: This pre-print
More informationElsayed M. E. Zayed 1 + (Received April 4, 2012, accepted December 2, 2012)
ISSN 746-7659, England, UK Journal of Information and Computing Science Vol. 8, No., 03, pp. 003-0 A modified (G'/G)- expansion method and its application for finding hyperbolic, trigonometric and rational
More informationMultisolitonic solutions from a Bäcklund transformation for a parametric coupled Korteweg-de Vries system
arxiv:407.7743v3 [math-ph] 3 Jan 205 Multisolitonic solutions from a Bäcklund transformation for a parametric coupled Korteweg-de Vries system L. Cortés Vega*, A. Restuccia**, A. Sotomayor* January 5,
More informationA Method for Obtaining Darboux Transformations
Journal of Nonlinear Mathematical Physics 998, V.5, N, 40 48. Letter A Method for Obtaining Darbou Transformations Baoqun LU,YongHE and Guangjiong NI Department of Physics, Fudan University, 00433, Shanghai,
More informationBoussineq-Type Equations and Switching Solitons
Proceedings of Institute of Mathematics of NAS of Ukraine, Vol. 3, Part 1, 3 351 Boussineq-Type Equations and Switching Solitons Allen PARKER and John Michael DYE Department of Engineering Mathematics,
More information2. Examples of Integrable Equations
Integrable equations A.V.Mikhailov and V.V.Sokolov 1. Introduction 2. Examples of Integrable Equations 3. Examples of Lax pairs 4. Structure of Lax pairs 5. Local Symmetries, conservation laws and the
More informationNonlinear Fourier Analysis
Nonlinear Fourier Analysis The Direct & Inverse Scattering Transforms for the Korteweg de Vries Equation Ivan Christov Code 78, Naval Research Laboratory, Stennis Space Center, MS 99, USA Supported by
More informationInverse scattering technique in gravity
1 Inverse scattering technique in gravity The purpose of this chapter is to describe the Inverse Scattering Method ISM for the gravitational field. We begin in section 1.1 with a brief overview of the
More informationSYMMETRIC INTEGRALS DO NOT HAVE THE MARCINKIEWICZ PROPERTY
RESEARCH Real Analysis Exchange Vol. 21(2), 1995 96, pp. 510 520 V. A. Skvortsov, Department of Mathematics, Moscow State University, Moscow 119899, Russia B. S. Thomson, Department of Mathematics, Simon
More informationS.Novikov. Singular Solitons and Spectral Theory
S.Novikov Singular Solitons and Spectral Theory Moscow, August 2014 Collaborators: P.Grinevich References: Novikov s Homepage www.mi.ras.ru/ snovikov click Publications, items 175,176,182, 184. New Results
More informationSYMBOLIC SOFTWARE FOR SOLITON THEORY: INTEGRABILITY, SYMMETRIES CONSERVATION LAWS AND EXACT SOLUTIONS. Willy Hereman
. SYMBOLIC SOFTWARE FOR SOLITON THEORY: INTEGRABILITY, SYMMETRIES CONSERVATION LAWS AND EXACT SOLUTIONS Willy Hereman Dept. of Mathematical and Computer Sciences Colorado School of Mines Golden, Colorado
More informationKorteweg-de Vries flow equations from Manakov equation by multiple scale method
NTMSCI 3, No. 2, 126-132 2015) 126 New Trends in Mathematical Sciences http://www.ntmsci.com Korteweg-de Vries flow equations from Manakov equation by multiple scale method Mehmet Naci Ozer 1 and Murat
More informationSTRUCTURE OF BINARY DARBOUX-TYPE TRANSFORMATIONS FOR HERMITIAN ADJOINT DIFFERENTIAL OPERATORS
Ukrainian Mathematical Journal, Vol. 56, No. 2, 2004 SRUCURE OF BINARY DARBOUX-YPE RANSFORMAIONS FOR HERMIIAN ADJOIN DIFFERENIAL OPERAORS A. K. Prykarpats kyi 1 and V. H. Samoilenko 2 UDC 517.9 For Hermitian
More informationSpectral difference equations satisfied by KP soliton wavefunctions
Inverse Problems 14 (1998) 1481 1487. Printed in the UK PII: S0266-5611(98)92842-8 Spectral difference equations satisfied by KP soliton wavefunctions Alex Kasman Mathematical Sciences Research Institute,
More informationOWELL WEEKLY JOURNAL
Y \»< - } Y Y Y & #»»» q ] q»»»>) & - - - } ) x ( - { Y» & ( x - (» & )< - Y X - & Q Q» 3 - x Q Y 6 \Y > Y Y X 3 3-9 33 x - - / - -»- --
More informationarxiv:nlin/ v1 [nlin.si] 3 Sep 2002
On the non-integrability of a fifth order equation with integrable two-body dynamics D.D. Holm & A.N.W. Hone arxiv:nlin/0209007v1 [nlin.si] 3 Sep 2002 June 21, 2014 Abstract We consider the fifth order
More informationarxiv:nlin/ v1 [nlin.si] 13 Apr 2005
Diophantine Integrability R.G. Halburd Department of Mathematical Sciences, Loughborough University Loughborough, Leicestershire, LE11 3TU, UK (Dated: received January 1, revised March 7, 005) The heights
More informationOn the classification of certain curves up to projective tranformations
On the classification of certain curves up to projective tranformations Mehdi Nadjafikhah Abstract The purpose of this paper is to classify the curves in the form y 3 = c 3 x 3 +c 2 x 2 + c 1x + c 0, with
More informationFrom the Kadomtsev-Petviashvili equation halfway to Ward s chiral model 1
Journal of Generalized Lie Theory and Applications Vol. 2 (2008, No. 3, 141 146 From the Kadomtsev-Petviashvili equation halfway to Ward s chiral model 1 Aristophanes IMAKIS a and Folkert MÜLLER-HOISSEN
More informationSupersymmetric Sawada-Kotera Equation: Bäcklund-Darboux Transformations and. Applications
Supersymmetric Sawada-Kotera Equation: Bäcklund-Darboux Transformations and arxiv:1802.04922v1 [nlin.si] 14 Feb 2018 Applications Hui Mao, Q. P. Liu and Lingling Xue Department of Mathematics, China University
More informationarxiv:nlin/ v1 [nlin.si] 7 Jul 2005
Completeness of the cubic and quartic Hénon-Heiles Hamiltonians Robert CONTE Micheline MUSETTE and Caroline VERHOEVEN Service de physique de l état condensé URA 464 CEA Saclay F 99 Gif-sur-Yvette Cedex
More informationHints on the Hirota Bilinear Method
Vol. 112 (2007) ACTA PHYSICA POLONICA A No. 6 Hints on the Hirota Bilinear Method P.P. Goldstein The Andrzej Soltan Institute for Nuclear Studies Hoża 69, 00-681 Warsaw, Poland (Received October 11, 2007)
More informationExact Travelling Wave Solutions of the Coupled Klein-Gordon Equation by the Infinite Series Method
Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 93-9466 Vol. 6, Issue (June 0) pp. 3 3 (Previously, Vol. 6, Issue, pp. 964 97) Applications and Applied Mathematics: An International Journal (AAM)
More informationAlgebraic Spectral Relations forelliptic Quantum Calogero-MoserProblems
Journal of Nonlinear Mathematical Physics 1999, V.6, N 3, 263{268. Letter Algebraic Spectral Relations forelliptic Quantum Calogero-MoserProblems L.A. KHODARINOVA yz and I.A. PRIKHODSKY z y Wessex Institute
More information