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There are many results on kink-shaped structures in mathematical literature.
For example, these are the structures generated by the oscillating solutions
of the Van der Pol oscillator type rapid-slow systems (A.N. Dorodnitsyn, E.F. Mishchenko, N.H. Rozov),
the stair type contrast structures (see the papers of A.B. Vasilieva, V.F. Butuzov and their colleagues).
Such structures were studied earlier by P. Fife and W. Greenley without the description of the
neighbourhood of front set.
However, only the case of at most two independent variables was considered earlier.
And the mathematical rigorous investigation of process of generating such kink-shaped structures is missing as yet.
Our work would fill these lacunas.
The front sets of generating kink-shaped structures will be described by the matching asymptotis expansions method.
Verification of expansions obtained will be realized by means of differential inequalities.
Namely it will be verification for the singularly perturbed solutions of ordinary differential equations
with the independent variable running through whole real axis.
Till now the asymptotics verification was performed only for the finite domains of independent variables running.
We will use the results of catastrophe theory concerning to the typical nature of butterfly-shaped
catastrophe in analysis of generating four-dimensional kink-shaped structures.
This approach admits to detect the specific character of multi-dimensional case unknown earlier.
Problems
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Describe and verify rigorously the asymptotics of special solutions of ordinary differential equations
$u_x=u^3-tu+x$ and $u_{xx}=u^3-tu+x$, while $x^2+t^2\to\infty$.
In principal member in $\epsilon$ they describe the processes of generating kink-shaped structures
for the solutions of partial differential equations with a small parameter $L(\epsilon Y,u,D_Y u, D^2_{YY)u)=0$
in the neighbourhood of the assembling points of the solutions of equation $L(\epsilon Y,u,0,0)=0$.
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Describe the behavior of special solutions of ordinary differential equations
$u_x=u^5-tu^3+yu^2+zu+x$ and $u_{xx}=u^5-tu^3+yu^2+zu+x$ for the large values of $x,y,z,t$.
These special solutions describe generation of the kink-shaped structures in the neighbourhood of the butterfly type
singularities.
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Results
We began to study the above posed problems in 2003.
Results of 2003:
- In the papers [1, 2] we show that the special solutions of the equations
$u_x=u^3-tu+x$ and $u_{xx}=u^3-tu+x$ describe the processes of genesis of kink-shaped structures for the
wide class of the solutions of singularly perturbed partial differential equations.
- In [2] the rigorous mathematical results are proved concerning to the complete asymptotic expansions
of these special solutions while $x\to \pm \infty$ with $t$ fixed.
We hope this result help us to describe their uniform asymptotics while $x^2+t^2 \to \infty$
with arbitrary order of precision.
The reasons stated in [1, 2] allow us to suppose the special solutions of ordinary equations
$u_{x}=u^5-tu^3+zu^2+yu^+x$ and $u_{xx}=u^5-tu^3+zu^2+yu^+x$ became kink-shaped for sufficiently large $t$.
And we hope to obtain the analytic description for asymptotics of these special solutions for the large values of
$x,t,z$ and $y$.
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Publications
- Suleimanov B.I.,
Cusp catastrophe in slowly varying equilibriums. JETP, 2002, V. 95, Is. 5. P. 944-956.
- Il'in A.M., Suleimanov B.I.,
On two special functions, concerned with the cusp singularity. Doklady RAN, 2002. V. 387, N 2. P. 156-158. (in Russian)
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