Bäcklund transformation

Bäcklund transformations (also written Baecklund or Backlund in English) are transformations of the dependent and independent variables in nonlinear differential equations , which make it possible to combine solutions of one equation or solutions of different equations. They are important in the theory of solitons .

history

Bäcklund transformations are named after Albert Bäcklund , who treated them in several works in the Mathematische Annalen from 1875 to 1882 , and sometimes also after Sophus Lie , who also used them in differential geometry around 1880 . A summary of earlier work was given by Édouard Goursat , while Jean Clairin gave a method for generating Bäcklund transformations at the beginning of the 20th century. After that it became quiet about the method, which experienced a renaissance in the theory of solitons (solutions of nonlinear differential equations) from the 1970s.

Bäcklund transformations for the Sinus-Gordon equation , which was considered in differential geometry (surfaces of negative curvature) as early as the 19th century, had been known for a long time (from Bäcklund himself); Hugo was the first to give the Korteweg-de-Vries equation Wahlquist and Frank Estabrook proposed such a transformation in 1973. A derivation using the method of J. Clairin was given by George Lamb, who in 1967 also obtained multi-soliton solutions of the Sinus-Gordon equation with Bäcklund transformations, the Sinus-Gordon equation this time occurring in the theory of ultra-short laser pulses. The method provided (in addition to the inverse scattering transformation and the direct method of Ryōgo Hirota ) also methods for solving nonlinear evolution equations, whereby it was mostly applied to equations in two independent variables, but transformations for more variables are also known (as for the KP equation ).

definition

In the original definition, Bäcklund and Lie linked

• the two equations , two areas in ${\displaystyle u(x,y)}$ ${\displaystyle v(x_{1},y_{1})}$ ${\displaystyle \mathbb {R} ^{3}}$
• the independent coordinates or as well as ${\displaystyle x,y}$ ${\displaystyle x_{1},y_{1}}$
• the partial derivatives of the two area equations, where z. B. indicates the partial derivative according to , ${\displaystyle u_{x},u_{y},v_{x_{1}},v_{y_{1}}}$ ${\displaystyle u_{x}}$ ${\displaystyle x}$

by four equations or Bäcklund transformations:

${\displaystyle B_{j}(u,u_{x},u_{y},x,y,v,v_{x_{1}},v_{y_{1}},x_{1},y_{1})=0}$, (with j = 1, ..., 4)

In the language of the time, the equations connected two surface elements : one of the two surface equations or was known, the other one was looking for. ${\displaystyle u}$ ${\displaystyle v}$

The integrability condition was also used :

${\displaystyle u_{xy}=u_{yx}}$

or the analog integrability condition for . ${\displaystyle v}$

Goursat and Clairin expanded this so that the second derivatives could also flow into the relations. In modern applications there are mostly solutions of nonlinear partial differential equations (the same or different), which are connected with each other via Bäcklund transforms. ${\displaystyle u,v}$

A modern geometric formulation of Bäcklund transformations takes place via the jet bundle formalism, in which systems of partial differential equations are considered as submanifolds of a jet bundle.

Examples

Cauchy-Riemann differential equations

A simple case of a Bäcklund transformation are the Cauchy-Riemann differential equations between the real part and the imaginary part of a holomorphic function over (the independent complex variable has real part and imaginary part ): ${\displaystyle u}$ ${\displaystyle v}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle z=x+iy}$ ${\displaystyle x}$ ${\displaystyle y}$

${\displaystyle u_{x}=v_{y},\quad u_{y}=-v_{x},\,}$

In this case they are Bäcklund transformations to the Laplace equation , which has both as a solution, so that the integrability conditions (and analogously for ) are satisfied. ${\displaystyle u_{xx}+u_{yy}=0}$ ${\displaystyle u}$ ${\displaystyle v}$ ${\displaystyle u_{xy}=u_{yx}}$ ${\displaystyle v}$

Meets the Laplace equation, it can be reversed via the Bäcklund transformation, a find that also satisfies the Laplace equation. The case here is very simple, since the transformations and the associated invariant differential equation are linear. ${\displaystyle u}$ ${\displaystyle v}$

Sinus Gordon equation

The solution to the Sinus Gordon equation: ${\displaystyle u}$

${\displaystyle u_{xy}=\sin u}$

can through a Bäcklund transformation

${\displaystyle {\begin{aligned}v_{x}&=u_{x}+2a\sin {\Bigl (}{\frac {u+v}{2}}{\Bigr )}\\v_{y}&=-u_{y}+{\frac {2}{a}}\sin {\Bigl (}{\frac {v-u}{2}}{\Bigr )}\end{aligned}}\,\!}$

(with parameters ) can be linked with another solution of the Sinus-Gordon equation . Since solutions of the same equation are linked here, one speaks of an auto-Bäcklund transformation. ${\displaystyle a}$ ${\displaystyle v}$

Liouville equation

A solution to the nonlinear Liouville equation ${\displaystyle u}$

${\displaystyle u_{xy}=\exp u\,\!}$

can via a Bäcklund transformation from to : ${\displaystyle u}$ ${\displaystyle v}$

${\displaystyle {\begin{aligned}v_{x}&=u_{x}+2a\exp {\Bigl (}{\frac {u+v}{2}}{\Bigr )}\\v_{y}&=-u_{y}-{\frac {1}{a}}\exp {\Bigl (}{\frac {u-v}{2}}{\Bigr )}\end{aligned}}\,\!}$

(with one parameter ) into a solution of the linear equation: ${\displaystyle a}$ ${\displaystyle v}$

${\displaystyle v_{xy}=0}$

transformed and vice versa. Instead of a non-linear differential equation, one only has to solve a much simpler linear differential equation here.

Korteweg-de-Vries equation

A method is considered to obtain new solutions of the Korteweg-de-Vries equation with the help of Bäcklund transformations if a solution u is already known (auto-Bäcklund transformation).

The KdV equation is:

${\displaystyle u_{t}+6uu_{x}+u_{xxx}=0}$

We are looking for the following pair of differential equations, with the solution of the KdV equation. ${\displaystyle u,w}$

${\displaystyle w_{x}=P(u,w,u_{x},u_{t},u_{xx})}$

${\displaystyle w_{t}=Q(u,w,u_{x},u_{t},u_{xx})}$

where the functions only depend on the specified variables, not on the partial derivatives of , and is a constant. It is also required that ${\displaystyle P,Q}$ ${\displaystyle w}$ ${\displaystyle k}$

${\displaystyle w_{xt}=w_{tx}}$ (Integrability condition).

Finding such Bäcklund transformations is not easy. Here the choice delivers

${\displaystyle P=-u_{x}-k^{2}+{(w-u)}^{2}}$

${\displaystyle Q=-u_{t}+4(k^{4}+k^{2}u_{x}-k^{2}{(w-u)}^{2}+u_{x}{(w-u)}^{2}+u_{xx}(w-u))}$

a Bäcklund transformation, because from the integrability condition it follows that the equation: ${\displaystyle w,u}$

${\displaystyle s_{t}-6s_{x}^{2}+s_{xxx}=0}$

for all and it follows from the derivation of this equation that the KdV equation is fulfilled. ${\displaystyle k}$ ${\displaystyle x}$ ${\displaystyle -2s_{x}}$

If you now have a solution, you can use it to construct an infinite number of others. For example, you start with and get: ${\displaystyle u=0}$

${\displaystyle w_{x}=-k_{0}^{2}+w^{2}}$

${\displaystyle w_{t}=-4k_{0}^{2}(w^{2}-k_{0}^{2})}$

The solution is (with ), the 1-soliton solution of the KdV equation. Substituting these, we obtain the 2-soliton solution, etc. Explicit (with the secant hyperbolic ): ${\displaystyle w=-k_{0}\tanh(k_{0}(x-4k_{0}t))=-k_{0}\tanh(\xi _{0})}$ ${\displaystyle \xi _{0}=k_{0}(x-4k_{0}t)}$ ${\displaystyle \operatorname {sech} }$

${\displaystyle v_{x}=k_{0}\operatorname {sech} ^{2}(\xi _{0})-k_{1}^{2}+{(v+k_{0}\tanh(\xi _{0}))}^{2}}$

If you introduce the function according to Estabrook and Wahlquist with : ${\displaystyle \phi }$ ${\displaystyle v={\frac {k_{1}^{2}-k_{0}^{2}}{\phi -v}}}$

${\displaystyle \phi _{x}=-k_{1}^{2}+\phi ^{2}}$

which is of the same form as the equation of the first approximation. Here you can insert another solution (it was ignored above because it is not restricted, but here it is inserted into the approach for in the denominator) and you get the 2-soliton solution. ${\displaystyle \phi =-k_{1}\coth(\xi _{1})}$ ${\displaystyle v}$

In 1968 Robert Miura used a Bäcklund transformation to obtain the hierarchy of an infinite number of constants of motion in the KdV equation. He also created connections from the KdV equation to the so-called modified KdV equation . ${\displaystyle u_{t}-6u^{2}u_{x}+u_{xxx}=0}$

literature

• C. Rogers, W. Shackwick: Bäcklund transformations and their applications, Elsevier, Academic Press 1982
• C. Rogers, WK Schief: Backlund and Darboux Transformations, Cambridge University Press 2002
• R. Miura (Ed.): Bäcklund Transformations, the Inverse Scattering Method, Solitons, and Their Applications. New York: Springer-Verlag, 1974.
• G. Lamb: Bäcklund transformations at the turn of the century, in RM Miura (ed.), Bäcklund transformations, Springer, Lecture notes in Mathematics 515, 1976, pp. 69-79

Web links

• Baecklund Transformation, Wolfram Mathworld (with literature)
• W.-H. Steeb, Baecklund transformations, Encyclopedia of Mathematics

Individual evidence

1. ^ Bäcklund, On the theory of partial differential equations of the first order, Mathematische Annalen, Volume 17, 1880, pp. 285–328, SUB Göttingen
2. ↑ Bäcklund, on the theory of surface transformations, Mathematische Annalen, Volume 19, 1882, pp. 387-422, SUB Göttingen
3. ↑ Goursat, Lemons sur l'intégration of equations aux derivées partial du ordre second, Volume 2, 1902
4. ^ Clairin, Sur les transformations de Baecklund, Annales scientifiques de l'École Normale Supérieure, Sér. 3, 19 (1902), p. 3-63, numdam ( Memento of the original from August 18, 2016 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. 
5. ^ Clairin, Sur quelques equations aux dérivées partielles du second ordre, Annales de la Faculté des sciences de Toulouse: Mathématiques, Sér. 2, 5 no. 4, 1903, pp. 437–458, numdam ( memento of the original dated August 18, 2016 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. 
6. ↑ Wahlquist, Estabrook, Backlund transformations for solutions of the Korteweg-de-Vries equation, Phys. Rev. Lett., Vol. 31, 1973, p. 1386
7. ↑ GL Lamb, Bäcklund transformations for certain nonlinear evolution equations, J. Math. Phys., Volume 15, 1974, pp. 1257-1265
8. ↑ Lamb, Propagation of ultrashort laser pulses, Phys. Lett. A, Vol. 25, 1967, pp. 181-182
9. ^ For example, C. Rogers, WF Shadwick, Bäcklund transformations and their applications, Academic Press 1982, p. 15
10. ^ C. Rogers, WF Shadwick, 1982, chapter 2
11. ^ Felix Pirani, DC Robinson, WF Shackwick, Local jet bundle formalism of Bäcklund transformations, Reidel 1979
12. ↑ The presentation follows E. Infeld, G. Rowlands, Nonlinear Waves, Solitons and Chaos, Cambridge UP 1990, p. 196ff. Other Bäcklund transformations of the KdV can also be found in the literature.
13. ↑ Miura, Korteweg-de-Vries equation and generalizions 1, 2, J. Math. Phys., Volume 9, 1968, pp. 1202, 1204
14. ^ Infeld, Rowlands, loc. cit., p. 191