Actual problems of modern science, education and training

Evolution of the scattering data

In this section we derive time evolution of the scattering data which allows us to provide the algorithm for solution of the problem (1)–(3).
We set

 _u 
U  ^{ v },
 
 _^{2 } 

_ 0


L^*  ^
 _x2  4v  2v_{x }

x

d _


^ . (19)

_ 1 4u  2u_{x } d _
 x 
Then the system (1) can be rewritten as follows:

t x
^{U  (L)U  0} . (20)

where
^{(k )  k 2} .
Now we introduce the 'scalar product' notation


V (x), W (x)
 _ [V₁(x)W₁(x)  V₂ (x)W₂ (x)]dx


1 2
^{for V (x) } ^{V (x),V (x)}T ^{, and the vector functions}

   

1
Φ _k, x_  _ f
^{k, x} ^f
^{k, x}^{, 2kf}
^{k, x} ^f
_k, x _T , (21)

2    
Φ _k, x_  _ f _k, x_ f _k, x_, 2kf _k, x_ f _k, x _T , (22)

^Φ3 ^k
_n , x_  _h_n
 ^x ^f
^{kn , x} ^{, 2knhn}
 ^x  ^f
^kn
, x _T . (23)

ELECTRONIC JOURNAL OF ACTUAL PROBLEMS OF MODERN SCIENCE, EDUCATION AND TRAINING. OCTOBER, 2022 10 ISSN 2181-9750

Theorem. If the functions
v  v(x,t),
u  u(x,t)
and
_m  _m (x,t)
are

solutions of the problem (1)-(4), then the scattering data of the operator

T (t, k ) y   y^  v(x,t) y  2ku(x,t) y  k ² y,
depend on t as
^{   х  } ,

^{dr (t, k)}  8ik³r (t,k) ^{, (24)}

k
dt d



dt ⁿ


^t  ^{ 0 , (25)}

d ^ (t) _{ 3 }

Proof. It is easy to show that
ⁿ 8ik_n _n (t) . (26)
dt

t x
^d a_k,t _  (2ik)^1 U dt
 (L^*)U
, Φ₁ ,
_Imk  0,k  0_ , (27)

t x
^d b_k,t _  4ik ²b(k,t)  (2ik)^1 U dt
 (L^*)U
, Φ₂
, _k 
^* _, (28)

dB^ (t) _

2  1 *

ⁿ 4ik_n B_n (t)  (2ik_n )
dt
U_t  (L )U_x , Φ₃
. (29)

If ^{U (x, t)} satisfies (20) then (27), (28) and (29) will take the following form

^d a_k,t _  0,
dt
_Imk  0,k  0_ , (30)

^d b_k,t _  8ik³b(k,t),

dt

_k 
^* _, (31)

^{dBn (t)}  8ik³B^ (t). (32)

It follows that the zeros

dt
k_n  k_n (t),
n n
n 1,2,..., N

of the function

a(k,t)

also do not

n
depend on time, which means (25). From (30), (31) and view of If we use (32) and view of  ^ (t) , we get (26).
The theorem is proved.
r_ (t, k ) , we obtain (24).

Remark 1. The obtained results completely define the time evolution of the scattering data, which allows us to find the solution of the considered problem (1)-(3) via the inverse scattering method.

References:

[1]. Kaup D. J. A Higher-Order Water-Wave Equation and the Method for Solving It, Progress of Theoretical Physics, 1975, vol. 54, issue 2, pp. 396-408.
[2]. Boussinesq J. Theorie de litumescence liquide appelee onde solitarie ou de translation, sepropageant dans un canal rectangulaire, Comptes Rendus Hebdomadaires des Seance de l'Academie des Sciences, 1871, 72, pp.755-759.
[3]. Matveev V.B., Yavor M. I. Solutions Presque Periodiques et a N-solitons de l'Equation Hydrodynamique Nonlineaire de Kaup, Ann.Inst. Henri Poincare, Sect., 1979, A. 31, no. 1, pp. 25-41.

[4]. Smirnov A.O. Real Finite-Gap Regular Solutions of the Kaup-Boussinesq
Equation, Theor. Math. Phys., 1986, vol.66, no.1, pp.19-31.
[6]. Mitropolsky Yu., Bogolyubov N. Jr., Prykarpatsky A., Samoilenko V. Integrable dynamical system: spectral and differential-geometric aspects, Naukova Dunka, Kiev, 1987.
[7]. Maksudov F. G., Guseinov G. Sh. On solution of the inverse scattering problem for a quadratic pencil of one-dimensional Schrodinger operators on the whole axis,
Dokl. Akad. Nauk SSSR, 1986, vol. 289, no. 1, pp. 42--46
[8]. Babadzhanov B.A., Khasanov A.B. Inverse problem for a quadratic pencil of Shturm-Liouville operators with finite-gap pereodic potential on the half-line, Differensial equations , 2007, vol. 43, issue 6, pp. 723-730.
[9]. Babadzhanov B.A., Khasanov A.B., Yakhshimuratov A.B. On the Inverse Problem for a Quadratic Pencil of Sturm-Liouville Operators with Periodic Potential, Diff. Eqs., 2005, vol.41, issue 3, pp. 310-318.