Additive smoothing: Difference between revisions
en>Helpful Pixie Bot m ISBNs (Build KE) |
en>Trappist the monk m fix CS1 deprecated date parameter errors (test) using AWB |
||
Line 1: | Line 1: | ||
{{Cleanup|date=April 2008}} | |||
==Introduction== | |||
The factorization of a linear partial differential operator (LPDO) is an important issue in the theory of integrability, due to the Laplace-Darboux transformations,<ref>Weiss (1986)</ref> which allow to construct integrable LPDEs. [[Laplace]] solved factorization problem for a '''bivariate hyperbolic operator of the second order''' (see [[Hyperbolic partial differential equation]]), constructing two Laplace invariants. Each [[Laplace invariant]] is an explicit polynomial condition of factorization; coefficients of this polynomial are explicit functions of the coefficients of the initial LPDO. The polynomial conditions of factorization are called '''invariants''' because they have the same form for equivalent (i.e. self-adjoint) operators. | |||
'''Beals-Kartashova-factorization''' (also called BK-factorization) is a constructive procedure to factorize '''a bivariate operator of the arbitrary order and arbitrary form'''. Correspondingly, the factorization conditions in this case also have polynomial form, are invariants and '''coincide with Laplace invariants''' for bivariate hyperbolic operator of the second order. The factorization procedure is purely algebraic, the number of possible factirzations depends on the number of simple roots of the [[Characteristic polynomial]] (also called symbol) of the initial LPDO and reduced LPDOs appearing at each factorization step. Below the factorization procedure is described for a bivariate operator of the arbitrary form, of the order 2 and 3. Explicit factorization formulas for an operator of the order <math> n </math> can be found in<ref>R. Beals, E. Kartashova. Constructively factoring linear partial differential operators in two variables. [http://www.springerlink.com/content/yx664142514k0217/ Theor. Math. Phys. '''145'''(2), pp. 1510-1523 (2005)] </ref> General invariants are defined in<ref> | |||
E. Kartashova. A Hierarchy of Generalized Invariants for Linear Partial Differential Operators. [http://www.springerlink.com/content/lp81238030114354/ Theor. Math. Phys. '''147'''(3), pp. 839-846 (2006)] </ref> and invariant formulation of the Beals-Kartashova factorization is given in<ref> E. Kartashova, O. Rudenko. Invariant Form of BK-factorization and its Applications. Proc. GIFT-2006, pp.225-241, Eds.: J. Calmet, R. W. Tucker, Karlsruhe University Press (2006); [http://arxiv.org/abs/math-ph/0607040/ arXiv]</ref> | |||
==Beals-Kartashova Factorization== | |||
===Operator of order 2=== | |||
Consider an operator | |||
:<math> | |||
\mathcal{A}_2 = a_{20}\partial_x^2 + a_{11}\partial_x\partial_y + a_{02}\partial_y^2+a_{10}\partial_x+a_{01}\partial_y+a_{00}. | |||
</math> | |||
with smooth coefficients and look for a factorization | |||
:<math> | |||
\mathcal{A}_2=(p_1\partial_x+p_2\partial_y+p_3)(p_4\partial_x+p_5\partial_y+p_6). | |||
</math> | |||
Let us write down the equations on <math> p_i</math> explicitly, keeping in | |||
mind the rule of '''left''' composition, i.e. that | |||
:<math> \partial_x (\alpha | |||
\partial_y) = \partial_x (\alpha) \partial_y + | |||
\alpha \partial_{xy}.</math> | |||
Then in all cases | |||
:<math> a_{20} = p_1p_4, </math> | |||
:<math> a_{11} = p_2p_4+p_1p_5, </math> | |||
:<math> a_{02} = p_2p_5, </math> | |||
:<math> a_{10} = \mathcal{L}(p_4) + p_3p_4+p_1p_6, </math> | |||
:<math> a_{01} = \mathcal{L}(p_5) + p_3p_5+p_2p_6, </math> | |||
:<math> a_{00} = \mathcal{L}(p_6) + p_3p_6, </math> | |||
where the notation <math> \mathcal{L} = p_1 \partial_x + p_2 \partial_y </math> is used. | |||
Without loss of generality, <math> | |||
a_{20}\ne 0, | |||
</math> i.e. <math> p_1\ne 0, </math> and it can be taken as 1, <math> | |||
p_1 = 1. </math> Now solution of the system of 6 equations on the variables | |||
:<math> p_2, </math> <math>... </math> <math> p_6 </math> | |||
can be found in '''three steps'''. | |||
'''At the first step''', the roots of a '''''quadratic polynomial''''' have to be found. | |||
'''At the second step''', a linear system of '''''two algebraic equations''''' has to be solved. | |||
'''At the third step''', '''''one algebraic condition''''' has to be checked. | |||
'''Step 1.''' | |||
Variables | |||
:<math> p_2,</math> <math> p_4, </math> <math> p_5 | |||
</math> | |||
can be found from the first three equations, | |||
:<math> a_{20} = p_1p_4, </math> | |||
:<math> a_{11} = p_2p_4+p_1p_5, </math> | |||
:<math> a_{02} = p_2p_5. </math> | |||
The (possible) solutions are then the functions of the roots of a quadratic polynomial: | |||
:<math> | |||
\mathcal{P}_2(-p_2) = a_{20}(- p_2)^2 +a_{11}(- p_2) +a_{02} = 0 | |||
</math> | |||
Let <math> \omega </math> be a root of the polynomial <math> | |||
\mathcal{P}_2, | |||
</math> | |||
then | |||
:<math> p_1=1, </math> | |||
:<math> p_2=-\omega, </math> | |||
:<math> p_4=a_{20},</math> | |||
:<math> p_5=a_{20} \omega +a_{11},</math> | |||
'''Step 2.''' | |||
Substitution of the results obtained at the first step, into the next two equations | |||
:<math> a_{10} = \mathcal{L}(p_4) + p_3p_4+p_1p_6, </math> | |||
:<math> a_{01} = \mathcal{L}(p_5) + p_3p_5+p_2p_6, </math> | |||
yields linear system of two algebraic equations: | |||
:<math> a_{10} = \mathcal{L} a_{20} +p_3 a_{20} +p_6, </math> | |||
:<math> a_{01} = \mathcal{L}(a_{11}+a_{20} \omega)+p_3( a_{11} + a_{20}\omega)- | |||
\omega p_6., </math> | |||
'''In particularly''', if the root <math>\omega</math> is simple, | |||
i.e. | |||
:<math> \mathcal{P}_2'(\omega)=2a_{20}\omega+a_{11}\ne 0,</math> then these | |||
equations have the unique solution: | |||
:<math> p_3 = \frac{\omega a_{10}+a_{01} -\omega\mathcal{L}a_{20}- \mathcal{L}(a_{20} \omega+a_{11})} | |||
{2a_{20}\omega+a_{11}},</math> | |||
:<math> p_6 =\frac{ (a_{20}\omega+a_{11})(a_{10}-\mathcal{L}a_{20})-a_{20}(a_{01} | |||
-\mathcal{L}(a_{20}\omega+a_{11}))}{2a_{20}\omega+a_{11}}.</math> | |||
At this step, for each | |||
root of the polynomial <math> \mathcal{P}_2 </math> a corresponding set of coefficients <math> p_j </math> is computed. | |||
'''Step 3.''' | |||
Check factorization condition (which is the last of the initial 6 equations) | |||
:<math> a_{00} = \mathcal{L}(p_6)+p_3p_6, </math> | |||
written in the known variables <math> p_j </math> and <math> \omega </math>): | |||
:<math> | |||
a_{00} = \mathcal{L} \left\{ | |||
\frac{\omega a_{10}+a_{01} - \mathcal{L}(2a_{20} \omega+a_{11})} | |||
{2a_{20}\omega+a_{11}}\right\}+ \frac{\omega a_{10}+a_{01} - | |||
\mathcal{L}(2a_{20} \omega+a_{11})} | |||
{2a_{20}\omega+a_{11}}\times | |||
\frac{ a_{20}(a_{01}-\mathcal{L}(a_{20}\omega+a_{11}))+ | |||
(a_{20}\omega+a_{11})(a_{10}-\mathcal{L}a_{20})}{2a_{20}\omega+a_{11}} | |||
</math> | |||
If | |||
:<math> | |||
l_2= a_{00} - \mathcal{L} \left\{ | |||
\frac{\omega a_{10}+a_{01} - \mathcal{L}(2a_{20} \omega+a_{11})} | |||
{2a_{20}\omega+a_{11}}\right\}+ \frac{\omega a_{10}+a_{01} - | |||
\mathcal{L}(2a_{20} \omega+a_{11})} | |||
{2a_{20}\omega+a_{11}}\times | |||
\frac{ a_{20}(a_{01}-\mathcal{L}(a_{20}\omega+a_{11}))+ | |||
(a_{20}\omega+a_{11})(a_{10}-\mathcal{L}a_{20})}{2a_{20}\omega+a_{11}} =0, | |||
</math> | |||
the operator <math> \mathcal{A}_2</math> is factorizable and explicit form for the factorization coefficients <math> p_j</math> is given above. | |||
===Operator of order 3=== | |||
Consider an operator | |||
:<math> | |||
\mathcal{A}_3=\sum_{j+k\le3}a_{jk}\partial_x^j\partial_y^k =a_{30}\partial_x^3 + | |||
a_{21}\partial_x^2 \partial_y + a_{12}\partial_x \partial_y^2 +a_{03}\partial_y^3 + | |||
a_{20}\partial_x^2+a_{11}\partial_x\partial_y+a_{02}\partial_y^2+a_{10}\partial_x+a_{01}\partial_y+a_{00}. | |||
</math> | |||
with smooth coefficients and look for a factorization | |||
:<math> | |||
\mathcal{A}_3=(p_1\partial_x+p_2\partial_y+p_3)(p_4 \partial_x^2 +p_5 \partial_x\partial_y + p_6 \partial_y^2 + p_7 | |||
\partial_x + p_8 \partial_y + p_9). | |||
</math> | |||
Similar to the case of the operator <math> \mathcal{A}_2, </math> the conditions of factorization are described by the following system: | |||
:<math> a_{30} = p_1p_4,</math> | |||
:<math> a_{21} = p_2p_4+p_1p_5,</math> | |||
:<math> a_{12} = p_2p_5+p_1p_6,</math> | |||
:<math> a_{03} = p_2p_6,</math> | |||
:<math> a_{20} = \mathcal{L}(p_4)+p_3p_4+p_1p_7,</math> | |||
:<math> a_{11} = \mathcal{L}(p_5)+p_3p_5+p_2p_7+p_1p_8,</math> | |||
:<math> a_{02} = \mathcal{L}(p_6)+p_3p_6+p_2p_8,</math> | |||
:<math> a_{10} = \mathcal{L}(p_7)+p_3p_7+p_1p_9,</math> | |||
:<math> a_{01} = \mathcal{L}(p_8)+p_3p_8+p_2p_9,</math> | |||
:<math> a_{00} = \mathcal{L}(p_9)+p_3p_9, | |||
</math> | |||
with <math>\mathcal{L} = p_1 \partial_x + p_2 \partial_y,</math> and again <math> | |||
a_{30}\ne 0, | |||
</math> i.e. <math> p_1=1, </math> and three-step procedure yields: | |||
'''At the first step''', the roots of a '''''cubic polynomial''''' | |||
:<math> \mathcal{P}_3(-p_2):= a_{30}(-p_2)^3 +a_{21}(- p_2)^2 + | |||
a_{12}(-p_2)+a_{03}=0. | |||
</math> | |||
have to be found. Again <math> \omega </math> denotes a root and first four coefficients are | |||
:<math> p_1=1, </math> | |||
:<math>p_2=-\omega, </math> | |||
:<math>p_4=a_{30}, </math> | |||
:<math>p_5=a_{30} \omega+a_{21}, </math> | |||
:<math>p_6=a_{30}\omega^2+a_{21}\omega+a_{12}. | |||
</math> | |||
'''At the second step''', a linear system of '''''three algebraic equations''''' has to be solved: | |||
:<math> a_{20}-\mathcal{L} a_{30} = p_3 a_{30} +p_7,</math> | |||
:<math> a_{11}-\mathcal{L}(a_{30} \omega + a_{21}) = p_3(a_{30}\omega+a_{21})- \omega p_7+p_8,</math> | |||
:<math> a_{02}-\mathcal{L}(a_{30}\omega^2+a_{21}\omega+a_{12})= p_3 (a_{30}\omega^2+a_{21}\omega+a_{12})-\omega p_8.</math> | |||
'''At the third step''', '''''two algebraic conditions''''' have to be checked. | |||
===Operator of order <math>n</math>=== | |||
==Invariant Formulation== | |||
'''Definition''' The operators <math> \mathcal{A} </math>, <math> \tilde{\mathcal{A}} </math> are called | |||
equivalent if there is a gauge transformation that takes one to the | |||
other: | |||
:<math> | |||
\tilde{\mathcal{A}} g= e^{-\varphi}\mathcal{A} (e^{\varphi}g). | |||
</math> | |||
BK-factorization is then pure algebraic procedure which allows to | |||
construct explicitly a factorization of an arbitrary order LPDO <math> \tilde{\mathcal{A}} </math> | |||
in the form | |||
:<math> | |||
\mathcal{A}=\sum_{j+k\le n}a_{jk}\partial_x^j\partial_y^k=\mathcal{L}\circ | |||
\sum_{j+k\le (n-1)}p_{jk}\partial_x^j\partial_y^k | |||
</math> | |||
with first-order operator <math> \mathcal{L}=\partial_x-\omega\partial_y+p</math> where <math> \omega</math> is '''an arbitrary simple root''' of the characteristic polynomial | |||
:<math> | |||
\mathcal{P}(t)=\sum^n_{k=0}a_{n-k,k}t^{n-k}, \quad | |||
\mathcal{P}(\omega)=0.</math> | |||
Factorization is possible then for each simple root <math>\tilde{\omega}</math> '''iff''' | |||
for <math>n=2 \ \ \rightarrow l_2=0,</math> | |||
for <math>n=3 \ \ \rightarrow l_3=0, l_{31}=0,</math> | |||
for <math>n=4 \ \ \rightarrow l_4=0, l_{41}=0, l_{42}=0,</math> | |||
and so on. All functions <math>l_2, l_3, l_{31}, l_4, | |||
l_{41}, \ \ l_{42}, ...</math> are known functions, for instance, | |||
:<math> l_2= a_{00} - \mathcal{L}(p_6)+p_3p_6, </math> | |||
:<math> l_3= a_{00} - \mathcal{L}(p_9)+p_3p_9, </math> | |||
:<math> l_{31} = a_{01} - \mathcal{L}(p_8)+p_3p_8+p_2p_9,</math> | |||
and so on. | |||
'''Theorem''' All functions | |||
:<math>l_2= a_{00} - \mathcal{L}(p_6)+p_3p_6, | |||
l_3= a_{00} - \mathcal{L}(p_9)+p_3p_9, | |||
l_{31}, ....</math> | |||
are '''invariants''' under gauge transformations. | |||
'''Definition''' Invariants <math>l_2= a_{00} - \mathcal{L}(p_6)+p_3p_6, | |||
l_3= a_{00} - \mathcal{L}(p_9)+p_3p_9, | |||
l_{31}, .... .</math> are | |||
called '''generalized invariants''' of a bivariate operator of arbitrary | |||
order. | |||
In particular case of the bivariate hyperbolic operator its generalized | |||
invariants '''coincide with Laplace invariants''' (see [[Laplace invariant]]). | |||
'''Corollary''' If an operator <math> \tilde{\mathcal{A}} </math> is factorizable, then all | |||
operators equivalent to it, are also factorizable. | |||
Equivalent operators are easy to compute: | |||
:<math> e^{-\varphi} \partial_x e^{\varphi}= \partial_x+\varphi_x, \quad e^{-\varphi}\partial_y e^{\varphi}= | |||
\partial_y+\varphi_y,</math> | |||
:<math> e^{-\varphi} \partial_x \partial_y e^{\varphi}= e^{-\varphi} \partial_x e^{\varphi} | |||
e^{-\varphi} \partial_y e^{\varphi}=(\partial_x+\varphi_x) \circ (\partial_y+\varphi_y)</math> | |||
and so on. Some example are given below: | |||
:<math> A_1=\partial_x \partial_y + x\partial_x + 1= \partial_x(\partial_y+x), \quad | |||
l_2(A_1)=1-1-0=0;</math> | |||
:<math>A_2=\partial_x \partial_y + x\partial_x + \partial_y +x + 1, \quad | |||
A_2=e^{-x}A_1e^{x};\quad l_2(A_2)=(x+1)-1-x=0;</math> | |||
:<math>A_3=\partial_x \partial_y + 2x\partial_x + (y+1)\partial_y +2(xy +x+1), \quad | |||
A_3=e^{-xy}A_2e^{xy}; \quad l_2(A_3)=2(x+1+xy)-2-2x(y+1)=0;</math> | |||
:<math>A_4=\partial_x \partial_y +x\partial_x + (\cos x +1) \partial_y + x \cos x +x +1, \quad | |||
A_4=e^{-\sin x}A_2e^{\sin x}; \quad l_2(A_4)=0.</math> | |||
==Transpose== | |||
Factorization of an operator is the first step on the way of solving corresponding equation. But for solution we need '''right''' factors and BK-factorization constructs '''left''' factors which are easy to construct. On the other hand, the existence of a certain right factor of a LPDO is equivalent to the existence of a corresponding left factor of the '''transpose''' of that operator. | |||
'''Definition''' | |||
The transpose <math> \mathcal{A}^t</math> of an operator | |||
<math> | |||
\mathcal{A}=\sum a_{\alpha}\partial^{\alpha},\qquad \partial^{\alpha}=\partial_1^{\alpha_1}\cdots\partial_n^{\alpha_n}. | |||
</math> | |||
is defined as | |||
<math> | |||
\mathcal{A}^t u = \sum (-1)^{|\alpha|}\partial^\alpha(a_\alpha u). | |||
</math> | |||
and the identity | |||
<math> | |||
\partial^\gamma(uv)=\sum \binom\gamma\alpha \partial^\alpha u,\partial^{\gamma-\alpha}v | |||
</math> | |||
implies that | |||
<math> | |||
\mathcal{A}^t=\sum (-1)^{|\alpha+\beta|}\binom{\alpha+\beta}\alpha (\partial^\beta a_{\alpha+\beta})\partial^\alpha. | |||
</math> | |||
Now the coefficients are | |||
<math> \mathcal{A}^t=\sum \tilde{a}_{\alpha} \partial^{\alpha},</math> | |||
<math> \tilde{a}_{\alpha}=\sum (-1)^{|\alpha+\beta|} | |||
\binom{\alpha+\beta}{\alpha}\partial^\beta(a_{\alpha+\beta}). | |||
</math> | |||
with a standard convention for binomial coefficients in several | |||
variables (see [[Binomial coefficient]]), e.g. in two variables | |||
:<math> | |||
\binom\alpha\beta=\binom{(\alpha_1,\alpha_2)}{(\beta_1,\beta_2)}=\binom{\alpha_1}{\beta_1}\,\binom{\alpha_2}{\beta_2}. | |||
</math> | |||
In particular, for the operator <math> \mathcal{A}_2 </math> the coefficients are | |||
<math> \tilde{a}_{jk}=a_{jk},\quad j+k=2; \tilde{a}_{10}=-a_{10}+2\partial_x a_{20}+\partial_y | |||
a_{11}, \tilde{a}_{01}=-a_{01}+\partial_x a_{11}+2\partial_y a_{02},</math> | |||
:<math> | |||
\tilde{a}_{00}=a_{00}-\partial_x a_{10}-\partial_y a_{01}+\partial_x^2 a_{20}+\partial_x \partial_x | |||
a_{11}+\partial_y^2 a_{02}. | |||
</math> | |||
For instance, the operator | |||
:<math> \partial_{xx}-\partial_{yy}+y\partial_x+x\partial_y+\frac{1}{4}(y^2-x^2)-1 </math> | |||
is factorizable as | |||
:<math> \big[\partial_x+\partial_y+\tfrac12(y-x)\big]\,\big[...\big]</math> | |||
and its transpose <math> \mathcal{A}_1^t </math> is factorizable then as | |||
<math> \big[...\big]\,\big[\partial_x-\partial_y+\tfrac12(y+x)\big].</math> | |||
==See also== | |||
* [[Partial derivative]] | |||
* [[Invariant (mathematics)]] | |||
* [[Invariant theory]] | |||
* [[Characteristic polynomial]] | |||
== Notes == | |||
<references/> | |||
== References == | |||
* J. Weiss. Bäcklund transformation and the Painlevé property. [http://www2.appmath.com:8080/site/few/few.html] J. Math. Phys. '''27''', 1293-1305 (1986). | |||
* R. Beals, E. Kartashova. Constructively factoring linear partial differential operators in two variables. [http://www.springerlink.com/content/yx664142514k0217/ Theor. Math. Phys. '''145'''(2), pp. 1510-1523 (2005)] | |||
* E. Kartashova. A Hierarchy of Generalized Invariants for Linear Partial Differential Operators. [http://www.springerlink.com/content/lp81238030114354/ Theor. Math. Phys. '''147'''(3), pp. 839-846 (2006)] | |||
* E. Kartashova, O. Rudenko. Invariant Form of BK-factorization and its Applications. Proc. GIFT-2006, pp.225-241, Eds.: J. Calmet, R. W. Tucker, Karlsruhe University Press (2006); [http://arxiv.org/abs/math-ph/0607040/ arXiv] | |||
[[Category:Multivariable calculus]] | |||
[[Category:Differential operators]] | |||
[[Category:Partial differential equations]] |
Revision as of 12:40, 4 January 2014
Introduction
The factorization of a linear partial differential operator (LPDO) is an important issue in the theory of integrability, due to the Laplace-Darboux transformations,[1] which allow to construct integrable LPDEs. Laplace solved factorization problem for a bivariate hyperbolic operator of the second order (see Hyperbolic partial differential equation), constructing two Laplace invariants. Each Laplace invariant is an explicit polynomial condition of factorization; coefficients of this polynomial are explicit functions of the coefficients of the initial LPDO. The polynomial conditions of factorization are called invariants because they have the same form for equivalent (i.e. self-adjoint) operators.
Beals-Kartashova-factorization (also called BK-factorization) is a constructive procedure to factorize a bivariate operator of the arbitrary order and arbitrary form. Correspondingly, the factorization conditions in this case also have polynomial form, are invariants and coincide with Laplace invariants for bivariate hyperbolic operator of the second order. The factorization procedure is purely algebraic, the number of possible factirzations depends on the number of simple roots of the Characteristic polynomial (also called symbol) of the initial LPDO and reduced LPDOs appearing at each factorization step. Below the factorization procedure is described for a bivariate operator of the arbitrary form, of the order 2 and 3. Explicit factorization formulas for an operator of the order can be found in[2] General invariants are defined in[3] and invariant formulation of the Beals-Kartashova factorization is given in[4]
Beals-Kartashova Factorization
Operator of order 2
Consider an operator
with smooth coefficients and look for a factorization
Let us write down the equations on explicitly, keeping in mind the rule of left composition, i.e. that
Then in all cases
Without loss of generality, i.e. and it can be taken as 1, Now solution of the system of 6 equations on the variables
can be found in three steps.
At the first step, the roots of a quadratic polynomial have to be found.
At the second step, a linear system of two algebraic equations has to be solved.
At the third step, one algebraic condition has to be checked.
Step 1. Variables
can be found from the first three equations,
The (possible) solutions are then the functions of the roots of a quadratic polynomial:
Let be a root of the polynomial then
Step 2. Substitution of the results obtained at the first step, into the next two equations
yields linear system of two algebraic equations:
In particularly, if the root is simple, i.e.
equations have the unique solution:
At this step, for each root of the polynomial a corresponding set of coefficients is computed.
Step 3. Check factorization condition (which is the last of the initial 6 equations)
written in the known variables and ):
If
the operator is factorizable and explicit form for the factorization coefficients is given above.
Operator of order 3
Consider an operator
with smooth coefficients and look for a factorization
Similar to the case of the operator the conditions of factorization are described by the following system:
with and again i.e. and three-step procedure yields:
At the first step, the roots of a cubic polynomial
have to be found. Again denotes a root and first four coefficients are
At the second step, a linear system of three algebraic equations has to be solved:
At the third step, two algebraic conditions have to be checked.
Operator of order
Invariant Formulation
Definition The operators , are called equivalent if there is a gauge transformation that takes one to the other:
BK-factorization is then pure algebraic procedure which allows to construct explicitly a factorization of an arbitrary order LPDO in the form
with first-order operator where is an arbitrary simple root of the characteristic polynomial
Factorization is possible then for each simple root iff
and so on. All functions are known functions, for instance,
and so on.
Theorem All functions
are invariants under gauge transformations.
Definition Invariants are called generalized invariants of a bivariate operator of arbitrary order.
In particular case of the bivariate hyperbolic operator its generalized invariants coincide with Laplace invariants (see Laplace invariant).
Corollary If an operator is factorizable, then all operators equivalent to it, are also factorizable.
Equivalent operators are easy to compute:
and so on. Some example are given below:
Transpose
Factorization of an operator is the first step on the way of solving corresponding equation. But for solution we need right factors and BK-factorization constructs left factors which are easy to construct. On the other hand, the existence of a certain right factor of a LPDO is equivalent to the existence of a corresponding left factor of the transpose of that operator.
Definition The transpose of an operator is defined as and the identity implies that
Now the coefficients are
with a standard convention for binomial coefficients in several variables (see Binomial coefficient), e.g. in two variables
In particular, for the operator the coefficients are
For instance, the operator
is factorizable as
and its transpose is factorizable then as
See also
Notes
- ↑ Weiss (1986)
- ↑ R. Beals, E. Kartashova. Constructively factoring linear partial differential operators in two variables. Theor. Math. Phys. 145(2), pp. 1510-1523 (2005)
- ↑ E. Kartashova. A Hierarchy of Generalized Invariants for Linear Partial Differential Operators. Theor. Math. Phys. 147(3), pp. 839-846 (2006)
- ↑ E. Kartashova, O. Rudenko. Invariant Form of BK-factorization and its Applications. Proc. GIFT-2006, pp.225-241, Eds.: J. Calmet, R. W. Tucker, Karlsruhe University Press (2006); arXiv
References
- J. Weiss. Bäcklund transformation and the Painlevé property. [1] J. Math. Phys. 27, 1293-1305 (1986).
- R. Beals, E. Kartashova. Constructively factoring linear partial differential operators in two variables. Theor. Math. Phys. 145(2), pp. 1510-1523 (2005)
- E. Kartashova. A Hierarchy of Generalized Invariants for Linear Partial Differential Operators. Theor. Math. Phys. 147(3), pp. 839-846 (2006)
- E. Kartashova, O. Rudenko. Invariant Form of BK-factorization and its Applications. Proc. GIFT-2006, pp.225-241, Eds.: J. Calmet, R. W. Tucker, Karlsruhe University Press (2006); arXiv