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Dispersionless equation

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Dispersionless (or quasi-classical) limits of integrable partial differential equations (PDE) arise in various problems of mathematics and physics and have been intensively studied in recent literature (see e.g. references below). They typically arise when considering slowly modulated long waves of an integrable dispersive PDE system.

Examples

Dispersionless KP equation

The dispersionless Kadomtsev–Petviashvili equation (dKPE), also known (up to an inessential linear change of variables) as the Khokhlov–Zabolotskaya equation, has the form

( u t + u u x ) x + u y y = 0 , ( 1 ) {\displaystyle (u_{t}+uu_{x})_{x}+u_{yy}=0,\qquad (1)}

It arises from the commutation

[ L 1 , L 2 ] = 0. ( 2 ) {\displaystyle =0.\qquad (2)}

of the following pair of 1-parameter families of vector fields

L 1 = ∂ y + λ ∂ x − u x ∂ λ , ( 3 a ) {\displaystyle L_{1}=\partial _{y}+\lambda \partial _{x}-u_{x}\partial _{\lambda },\qquad (3a)}
L 2 = ∂ t + ( λ 2 + u ) ∂ x + ( − λ u x + u y ) ∂ λ , ( 3 b ) {\displaystyle L_{2}=\partial _{t}+(\lambda ^{2}+u)\partial _{x}+(-\lambda u_{x}+u_{y})\partial _{\lambda },\qquad (3b)}

where λ {\displaystyle \lambda } is a spectral parameter. The dKPE is the x {\displaystyle x} -dispersionless limit of the celebrated Kadomtsev–Petviashvili equation, arising when considering long waves of that system. The dKPE, like many other (2+1)-dimensional integrable dispersionless systems, admits a (3+1)-dimensional generalization.

The Benney moment equations

The dispersionless KP system is closely related to the Benney moment hierarchy, each of which is a dispersionless integrable system:

A t 2 n + A x n + 1 + n A n − 1 A x 0 = 0. {\displaystyle A_{t_{2}}^{n}+A_{x}^{n+1}+nA^{n-1}A_{x}^{0}=0.}

These arise as the consistency condition between

λ = p + ∑ n = 0 ∞ A n / p n + 1 , {\displaystyle \lambda =p+\sum _{n=0}^{\infty }A^{n}/p^{n+1},}

and the simplest two evolutions in the hierarchy are:

p t 2 + p p x + A x 0 = 0 , {\displaystyle p_{t_{2}}+pp_{x}+A_{x}^{0}=0,}
p t 3 + p 2 p x + ( p A 0 + A 1 ) x = 0 , {\displaystyle p_{t_{3}}+p^{2}p_{x}+(pA^{0}+A^{1})_{x}=0,}

The dKP is recovered on setting

u = A 0 , {\displaystyle u=A^{0},}

and eliminating the other moments, as well as identifying y = t 2 {\displaystyle y=t_{2}} and t = t 3 {\displaystyle t=t_{3}} .

If one sets A n = h v n {\displaystyle A^{n}=hv^{n}} , so that the countably many moments A n {\displaystyle A^{n}} are expressed in terms of just two functions, the classical shallow water equations result:

h y + ( h v ) x = 0 , {\displaystyle h_{y}+(hv)_{x}=0,}
v y + v v x + h x = 0. {\displaystyle v_{y}+vv_{x}+h_{x}=0.}

These may also be derived from considering slowly modulated wave train solutions of the nonlinear Schrödinger equation. Such 'reductions', expressing the moments in terms of finitely many dependent variables, are described by the Gibbons-Tsarev equation.

Dispersionless Korteweg–de Vries equation

The dispersionless Korteweg–de Vries equation (dKdVE) reads as

u t 3 = u u x . ( 4 ) {\displaystyle u_{t_{3}}=uu_{x}.\qquad (4)}

It is the dispersionless or quasiclassical limit of the Korteweg–de Vries equation. It is satisfied by t 2 {\displaystyle t_{2}} -independent solutions of the dKP system. It is also obtainable from the t 3 {\displaystyle t_{3}} -flow of the Benney hierarchy on setting

λ 2 = p 2 + 2 A 0 . {\displaystyle \lambda ^{2}=p^{2}+2A^{0}.}

Dispersionless Novikov–Veselov equation

The dispersionless Novikov-Veselov equation is most commonly written as the following equation for a real-valued function v = v ( x 1 , x 2 , t ) {\displaystyle v=v(x_{1},x_{2},t)} :

∂ t v = ∂ z ( v w ) + ∂ z ¯ ( v w ¯ ) , ∂ z ¯ w = − 3 ∂ z v , {\displaystyle {\begin{aligned}&\partial _{t}v=\partial _{z}(vw)+\partial _{\bar {z}}(v{\bar {w}}),\\&\partial _{\bar {z}}w=-3\partial _{z}v,\end{aligned}}}

where the following standard notation of complex analysis is used: ∂ z = 1 2 ( ∂ x 1 − i ∂ x 2 ) {\displaystyle \partial _{z}={\frac {1}{2}}(\partial _{x_{1}}-i\partial _{x_{2}})} , ∂ z ¯ = 1 2 ( ∂ x 1 + i ∂ x 2 ) {\displaystyle \partial _{\bar {z}}={\frac {1}{2}}(\partial _{x_{1}}+i\partial _{x_{2}})} . The function w {\displaystyle w} here is an auxiliary function, defined uniquely from v {\displaystyle v} up to a holomorphic summand.

Multidimensional integrable dispersionless systems

See for systems with contact Lax pairs, and e.g., and references therein for other systems.

See also

References

Citations

  1. ^ Sergyeyev, A. (2018). "New integrable (3 + 1)-dimensional systems and contact geometry". Letters in Mathematical Physics. 108 (2): 359–376. arXiv:1401.2122. Bibcode:2018LMaPh.108..359S. doi:10.1007/s11005-017-1013-4. S2CID 119159629.
  2. Calderbank, David M. J.; Kruglikov, Boris (2021). "Integrability via geometry: dispersionless differential equations in three and four dimensions". Communications in Mathematical Physics. 382 (3): 1811–1841. arXiv:1612.02753. doi:10.1007/s00220-020-03913-y. MR 4232780.
  3. Kruglikov, Boris; Morozov, Oleg (2015). "Integrable Dispersionless PDEs in 4D, Their Symmetry Pseudogroups and Deformations". Letters in Mathematical Physics. 105 (12): 1703–1723. arXiv:1410.7104. Bibcode:2015LMaPh.105.1703K. doi:10.1007/s11005-015-0800-z. S2CID 119326497.

Bibliography

External links

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