Modeling Experimental Nonlinear Dynamics and Chaotic Scenarios

WITH A POLYNOMIAL EIGENVALUE PROBLEM

TAIXI XU, WEIHUA MU, AND ZHIJUN QIAO Received 6 February 2006; Accepted 21 March 2006

M. Antonowicz and A. P. Fordy (1988) introduced the second-order polynomial eigen- value problemLφ=(∂²+ⁿ_i=1viλⁱ)φ=αφ(∂=∂/∂x, α=constant) and discussed its multi-Hamiltonian structures. Forn=1 andn=2, the associated finite-dimensional in- tegrable Hamiltonian systems (FDIHS) have been discussed by Xu and Mu (1990) using the nonlinearization method and Bargmann constraints. In this paper, we consider the general case, that is,nis arbitrary, provide the constrained Hamiltonian systems associ- ated with the above-mentioned second-order polynomial ergenvalue problem, and prove them to be completely integrable.

Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

1. Introduction

In classical mechanics one describes the equation of motion by Hamiltonian systems of the form [2]

dqj

dt ⁼

∂H

∂pj

dpj

dt ^{= −}

∂H

∂qj (j=1,...,n), (1.1)

whereq=(q1,...,q_n)∈Rⁿ, p=(p1,...,p_n)∈Rⁿ, andH=H(q,p) is a smooth function on an open domainΩofR²ⁿ.

It is customary to introduce the “Poisson bracket”{F,G} for two functionsF,G∈ C¹(Ω) by [7]

{F,G} =ⁿ

j=1

∂F

∂qj

∂G

∂pj− ∂F

∂pj

∂G

∂qj

= Fq,Gp

− Fp,Gq

. (1.2)

F,Gare called in involution if{F,G} =0.

Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 13479, Pages1–9

DOI10.1155/IJMMS/2006/13479

Using a notation borrowed from differential geometry we associate with (1.1) the “vec- tor field” or first-order differential operator

XH=ⁿ

j=1

Hpj

∂

∂qj−Hqj

∂

∂pj

. (1.3)

Definition 1.1. A nonconstant functionF∈C(Ω) is called an integral ofXHif

XHF= {F,H} =0. (1.4)

Definition 1.2. A Hamiltonian vector fieldXH inΩ⊂R²ⁿis called “integrable” if it pos- sessesnintegralsFj∈C¹(Ω) satisfying the following conditions

(i){F_j,H} =0, (ii){Fj,Fk} =0,

(iii) the gradientsdFjare linearly independent inΩ.

The first condition expresses that theFjare integrals; the second one means that any two such integrals commute. The third condition is a requirement for nondegeneracy, which we will have to relax frequently.

Since Cao introduced the nonlinearization method to search for finite-dimensional completely integrable Hamiltonian systems [3–6] associated with soliton equations, nu- merous such systems have been obtained by many mathematicians [8–12,14,15].

In this paper, we consider the integrable systems associated with the polynomial eigen- value problem

Lφ=(∂²+ n i=1

v_iλⁱ)φ=αφ(∂=∂/∂x,α=constant). (1.5) Whenn=1, denotev1byv, (1.5) becomes

φ_xx+λvφ=αφ. (1.6)

Equation (1.6) is associated with the Harry-Dym (HD) equation v_t=

1 2^√v

xxx

+ _√2α

v

x

. (1.7)

Remark 1.3. Actually, spectral problem (1.6) generates the Camassa-Holm (CH) equation in its negative-order hierarchy, whereas it produces the HD equation (1.7) in its positive- order hierarchy [9]. Both hierarchies are integrable. Whenn=2, v1=u, v2=v, (1.5) becomes

φ_xx+λu+λ²v φ=αφ. (1.8)

Equation (1.8) is associated with the following coupled Harry-Dym (CHD) equation:

ut= 1

2^√v

xxx

−2α 1

√v

x

, v_t=2u 1

√v+u_x 1

√v.

(1.9)

We have already obtained the integrable Hamiltonian systems associated with Harry- Dym and coupled Harry-Dym equations [13]. For general positive integern, the associ- ated integrable system is given in this paper. In the next section, we give the Hamiltonian system associated with the polynomial eigenvalue problem, and inSection 3, we obtain the involutive integrals and prove they are linearly independent.

2. The Hamiltonian system Consider the evolution equation

φtm= −1

2B^(m)_x φ+B^(m)φx, (2.1)

where

B^(m)=^m

−1

j=0

bjλ^m−j, b−j=0 (j=1, 2,...). (2.2) From the solvability condition of (1.5) and (2.1), the hierarchy of evolution equations of potentialsv=(v1,...,vn)^T can be written as

v_i=

⎡

⎢⎢

⎢⎢

⎢⎢

⎣

0 ··· 0 J0

... · · J1

0 · · ...

J0 J1 ··· J_n−1

⎤

⎥⎥

⎥⎥

⎥⎥

⎦

⎡

⎢⎢

⎢⎢

⎢⎣ b_m−n

bm−n−1

... bm−1

⎤

⎥⎥

⎥⎥

⎥⎦. (2.3)

Also, from the solvability condition, it is found thatbksatisfies

J0bj+J1bj+1+···+Jnbj+n=0, (2.4) or

Kb_j,b_j+1,...,b_j+n−1 T=Jb_j+1,b_j+2,...,b_j+n T

, (2.5)

or

KG_j=JG_j+1, (2.6)

where

K=

⎡

⎢⎢

⎢⎢

⎢⎢

⎣

0 ··· 0 J0

... · · J1

0 · · ...

J0 J1 ··· J_n−1

⎤

⎥⎥

⎥⎥

⎥⎥

⎦

, J=

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎣

0 ··· 0 J0 0

... · · J1 0

0 · · ... ...

J0 J1 ··· J_n−2 0

0 0 ··· 0 −Jn

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎦

(2.7)

are the Lenard pair of operators Gj=

bj,bj+1,...,bj+n−1 , J0=1

2∂³−2α∂, Ji=vi∂+∂vi (i=1, 2,...,n).

(2.8) It is evident that ifφis a solution of (1.5), then

n i=0

λⁱJ_iφ²=0. (2.9)

Rewrite it as

n i=0

λⁱJiP=0. (2.10)

LettingP=_∞

j=0Pjλ^−j; we find thatPj satisfies the same relationship (2.4) asbj does.

Multiplying both sides of (2.10) byPand integrating it once, we get P_xxP−1

2P_x²−2αP²+ 2 n i=1

λⁱv_iP²=C(λ). (2.11) SetP0=V_n^−1/2,C(λ)=λⁿ. By substitutingP=_∞

j=0Pjλ^−j into (2.11), we find thatPj= b_j,

bk+nb⁻0¹= −1 4

k j=0

bjxxbk−j+1 8

k j=0

bjxbk−j,x+α 2

k j=0

bjbk−j

−1 2

n−1 i=0

k+i

j=0

bjbk+i−j−1 2vn

k+n−1 j=1

bjbk+n−j (k=1, 2,...).

(2.12)

Proposition 2.1. Letλjbe an eigenvalue of (1.5) andφj an eigenfunction corresponding toλ_j. Then

gradλj= δλj

δv1,...,δλj

δv1

T

=

λjφ²_j,λ²_jφ²_j,...,λⁿ_jφ²_j ^T, Kgradλ_j=λ_jJgradλ_j.

(2.13)

Define the Lenard sequence recursively: G0=(b0,...,bn−1)^T, KGj−1=JGj (j=1, 2,...),Xj=JGj(j=0, 1, 2,...) are the CHD vector fields.

Let

G0=^N

j=1

gradλ_j. (2.14)

Then

b_j=

Λ^j+1φ,φ (j=0, 1, 2,...,n−1), (2.15) where·,·is the standard inner-product inR^N,Λ=diag(λ1,...,λ_N).

From (2.4), (2.14), and (2.15), we have

k−1 j=0

Jn−j

Λ^k−jφ,φ=0 (k=1, 2,...,n), (2.16) which yields

vn−k=

Λφ,φ⁻¹∂⁻¹Λφ,φJn

Λ^kφ,φ+···+Jn−k+2

Λ²φ,φ. (2.17) By making use of the recursion formula ofvk, we have the following proposition.

Proposition 2.2. The constraint between the potentials and the eigenfunctions (1.5) is of the form

vn=

Λφ,φ⁻², (2.18)

v_n−k=^k

j=1

a_jΛφ,φ^−(j+2)

l1+_···+lj=k−j

Λ^l1+2φ,φ···

Λ^lj+2φ,φ (k=1, 2,...,n−1), (2.19) whereaj=(−1)^j(j+ 1),a0=1, andφj,λjsatisfy (1.5).

We now consider the following system instead of (1.5):

φ_jxx+ n i=1

v_iλⁱ_jφ_j=αφ_j (j=1, 2,...,N), (2.20) whereλ_j =λ_kwhenj =k. Let

q=

q1,q2,...,q_N ^T=

φ1,φ2,...,φ_N ^T; (2.21) then (2.20) can be condensed as

q_xx+ n i=1

v_iΛⁱq=αq. (2.22)

By using the identity k i=1

Λ^k−jq,q i j=1

ajγi,j= k i=1

aj k−j i=0

Λⁱq,qγk−i,j (2.23)

and substituting (2.18), (2.19) into (2.22), we get px=αq−

n−1 i=0

aj

Λq,qⁱ⁺²·

l1+_···+li+1

Λ^l1+2q,q···

Λ^li+2q,qΛ^li+1+2q, qx=p,

(2.24)

which can be written in canonical Hamiltonian system q_x=∂H0

∂p , p_x= −∂H0

∂q , (2.25)

wherep=(p1,...,pN)^T=(q1x,...,qNx)^T, H0=1

2p,p −α

2q,q+1 2

n−2 i=0

biΛq,q⁻⁽ⁱ⁺²⁾·

l1+_···+li+1=n−2₋i

Λ^l1+2q,q···

Λ^li+1+2q,q bi= a_i

i+ 1⁼(−1)ⁱ,i=0, 1,....

(2.26) 3. Involutivity and integrability

Consider the constraint of (2.12) F_n+k=

bk+n

b0 +1 4

k j=0

b_jxxb^k−j−1 8

k j=0

b_jxb_k−j,x

+1 2

n−1 i=1

v_i

k+i

j=0

b_jb_k+i−j−α 2

k j=0

b^k_j^−j+1 2v_n

k+n−1 j=1

b_jb_k+n−j

A

,

(3.1)

where subscriptAmeans to substitutebj= Λ^j+1q,qinto (2.12). So Fn+k=1

2

k−1 j=0

Λ^j+1q,px

Λ^k−j−1q,q+1 2

k j=0

Λ^j+1p,pΛ^k−j+1q,q

+1 2

Λ^k+1q,px

Λq,q −1 2

Λ^j+1q,pΛ^k−j+1p,q

−α 2

k j=0

Λ^j+1q,qΛ^k−j+1q,q+1 2

n−1 i=0

vi k+i

j=0

Λ^j+1q,qΛ^k+i−j+1q,q

+1 2vn

k+n−1 j=1

Λ^j+1q,pΛ^k+i−j+1q,q+Λq,q⁻¹

Λ^k+n+1q,q.

(3.2)

Through direct calculations from (2.18), (2.19), and (2.24), we have Fn+k=1

2

n−1 i=0

biΛq,q⁻⁽ⁱ⁺¹⁾·

l¹+···+li+1=n−1−i

Λ^l1+2q,q···

Λ^li+2q,qΛ^ii+1+k+2q,q +1

2 k j=0

Λ^j+1p,pΛ^k−j+1q,q−

Λ^j+1q,qΛ^k−j+1p,q, k=1, 2,....

(3.3)

Set

Gk=1 2

k j=0

Λ^j+1p,pΛ^j+1q,q−

Λ^j+1q,pΛ^k−j+1p,q, Qk=Fk+n−Gk.

(3.4)

It is known (see Cao [3]) thatGkare in involution. Using the identity l

i=0

Λ^l+k+j−ip,pΛⁱq,q+ k i=0

Λⁱp,pΛ^l+k+j−iq,q

=

l+k+j i=0

Λ^l+k+j−ip,pΛⁱq,q−

l+j−1 i=l+1

Λ^l+k+j−ip,pΛⁱq,q,

(3.5)

we can show by straightforward calculations that Qk,Gl

+Gk,Ql

+Qk,Ql

=0. (3.6)

So

F_k+n,F_l+n

=0. (3.7)

Since allλkare distinct, the Vandermonde determinant ofλ1,λ2,...,λN is not zero. Then it is easy to see that

gradF_n+k= ∂Fn+k

∂q1 ,...,∂Fn+k

∂q_N ,∂Fn+k

∂p1 ,...,∂Fn+k

∂p_N

, k=1, 2,..., (3.8) are functionally independent. So we have the following proposition.

Proposition 3.1. The Hamiltonian system (R^2N,dp∧dq,H0) is completely integrable in the sense of Liouville.

Consider the systems obtained from (2.1) φjtm= −1

2R⁽_x^m)φj+R^(m)φjx (j=1, 2,...,N). (3.9)

Substitutingb−j= Λ^j+1q,q(j=1, 2,...) into it, we have qtm=^m

−1

j=0

Λ^j+1q,qΛ^m−jp−

Λ^j+1q,pΛ^m−jq,

ptm= ∂

∂xqtm=^m

−1

j=0

Λ^j+1q,pΛ^m−jp−

Λ^j+1p,pΛ^m−jq +

m−1 j=0

Λ^j+1q,qΛ^m−jpx−

Λ^j+1q,px Λ^m−jq.

(3.10)

Through direct calculation from (2.24), (3.10) can be written in canonical Hamilton- ian system

q_tm=∂Fn+m−1

∂p , p_tm= −∂Fn+m−1

∂q . (3.11)

Proposition 3.2. The Hamiltonian systems in the last equation are completely integrable in the sense of Liouville, and if (p,q) satisfies (2.25) and (3.11), thenvgiven by (2.18) and (2.19) is a solution of CHD equation.

Proof. SinceF_k are in involution, the systems (3.11) (m=1, 2,...) are completely inte- grable. Observe that (2.3) is deduced from the solvability condition of (2.22) and (3.9);

(2.25) and (3.11) are obtained by substituting (2.18) and (2.19) into (2.22) and (3.9), respectively. It is easy to see that if (q,p) satisfies both (2.25) and (3.11), thenvgiven by

(2.18) and (2.19) is a solution of CHD equation.

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Taixi Xu: Department of Mathematics, Southern Polytechnic State University, 1100 South Marietta Parkway, Marietta, GA 30060, USA

E-mail address:[email protected]

Weihua Mu: Department of Mathematics, Shijiazhuang Railway Institute, Hebei 050043, China E-mail address:[email protected]

Zhijun Qiao: Department of Mathematics, University of Texas – Pan American, 1201 W. University Drive Edinburg, TX 78541, USA

E-mail address:[email protected]

Special Issue on

Modeling Experimental Nonlinear Dynamics and Chaotic Scenarios

Call for Papers

Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system. Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision. In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.

Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from “Qualitative Theory of Differential Equations,”

allowing more precise analysis and synthesis, in order to produce new vital products and services. Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.

This proposed special edition of the Mathematical Prob- lems in Engineering aims to provide a picture of the impor- tance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.

Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophis- ticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.

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Guest Editors

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