Gauge-invariant description of some (2+1)-dimensional integrable nonlinear evolution equations

Gauge-in v arian t description of some (2+1)-dimensional in tegrable nonlinear ev olution equations V. G. Dubro vsky a nd A. V. Gramolin Nov osibirsk State T echnical Universit y , Karl Marx prosp. 20, Nov osibirsk 630092, R ussia E-mail: dub ro v sky@academ.org an d gramolin@gmail.com Abstract New manifestly gauge-in v ariant forms of tw o-dimensional generalized disp ersiv e long-wa ve and Nizhnik–V eselov–No viko v systems of integrable nonlinear equations are presented. I t is shown how in different gauges from such forms famo u s tw o-dimensional generalization of disp ersiv e long-w av e system of e q uations, Nizhnik–V eselov–No viko v and mod ifi ed Nizhnik–V eselo v –N o v iko v equations and oth er known and new integrable nonlinear equations arise. Miura-type transformations b etw een n on linear equations in different gauges are considered. P ACS num b ers: 02.30.Ik, 02.30.Jr, 02.30.Zz, 05.45.Yv 1 In tro d uction The fundamen tal ideas of gauge inv ar iance and gauge tra ns formations are wide spr e ad and in common use in almost every part of physics. The first a pplications of such ideas in the theory of in teg rable nonlinear equations by Z a kharov and Sha bat [1 ], K uznetsov and Mik ha ilov [2], Zakhar ov and Mikhailov [3], Zakha rov and T akhtadzh yan [4], Konop elchenk o [5], Konop elchenko and Dubrovsky [6 , 7] and others hav e b een made, see als o the bo oks [8–13] and references therein. Now a lo t o f g auge-eq uiv alent to eac h other, in teg rable nonlinear mo dels are well known. In o ne- dimensional ca s e the most famous are the nonlinear Schr¨ odinger and Heisen b erg ferromagnet equatio ns , massive Thirring model and t wo-dimensional relativistic field mo del, KdV a nd mKdV equa tio ns a nd so on; in the tw o -dimensional case the mos t famous are K adomtsev–Petviash vili and mo dified Kadomtsev– Petviash vili nonlinear equa tio ns, Da vey–Stewartson and Ishimor i integrable systems of nonlinear equations and so on. See some references in the b o o ks [8–14 ]. In the present paper, manifestly gauge- inv ariant formulation of t wo-dimensional nonlinear evolution eq ua- tions integrable by the follo wing tw o scalar aux ilia ry linear problems: L 1 ψ =  ∂ 2 ξη + u 1 ∂ ξ + v 1 ∂ η + u 0  ψ = 0 , (1.1) L 2 ψ =  ∂ t + u 3 ∂ 3 ξ + v 3 ∂ 3 η + u 2 ∂ 2 ξ + v 2 ∂ 2 η + ˜ u 1 ∂ ξ + ˜ v 1 ∂ η + v 0  ψ = 0 (1.2) is develope d. Here as usual ξ = x + σy , η = x − σ y , σ 2 = ± 1 and ∂ ξ = ∂ /∂ ξ , ∂ η = ∂ /∂ η , ∂ 2 ξ = ∂ 2 /∂ ξ 2 , etc. Two cases of a ux iliary linear problems (1.1), (1.2 ) with different seco nd auxiliary linear problem (1.2) are studied: • (i) u 3 = κ 1 = const, v 3 = κ 2 = const, third-order pr oblem L 2 ψ = 0, such choice of second a uxiliary problem (1.2) leads to fa mo us Nizhnik–V eselov–No vikov (NVN) [15, 16], mo dified Nizhnik–V eselov– Novik ov (mNVN) [17] and other equations ; • (ii) u 3 = v 3 = 0, u 2 = κ 1 = const, v 2 = κ 2 = co nst, second-o r der pro blem L 2 ψ = 0, such choice of second auxiliar y pr oblem (1.2 ) leads to famous tw o-dimens ional gener alization of disp ersive long- wa ve equation (2DDL W) [18], Dav ey– Stewartson (DS) system of equatio ns [19] and its reductions and o ther equations. 1 All ab ove-men tioned famous in teg rable nonlinear equatio ns via the co mpatibility condition of auxiliary linear problems (1.1) a nd (1.2) in the form of Manak ov’s tr iad repr esentation [20 ] [ L 1 , L 2 ] = B L 1 (1.3) hav e b een previously established [15–18], see also b o oks [12, 13] and references therein. In the pap er, g auge transfo r mations ψ → ψ ′ = g − 1 ψ (1.4) with arbitrary ga uge function g ( ξ , η, t ) of auxiliary linear problems (1.1) and (1.2) are studied. The c onv enient for ga uge-inv aria nt fo r mulation fie ld v a riables, cla s sical gaug e inv aria nt s w 2 , e w 2 , w 1 , w 2 def = u 0 − u 1 ξ − u 1 v 1 = u ′ 0 − u ′ 1 ξ − u ′ 1 v ′ 1 , (1.5) e w 2 def = u 0 − v 1 η − u 1 v 1 = u ′ 0 − v ′ 1 η − u ′ 1 v ′ 1 , (1.6) w 1 def = u 1 ξ − v 1 η = u ′ 1 ξ − v ′ 1 η (1.7) and pure gauge v aria ble ρ c o nnected with field v ariable u 1 ( ξ , η , t ) by the for mula u 1 def = (ln ρ ) η (1.8) are intro duce d. The v a riable ρ co rresp onds to pure gaug e deg rees of freedom and has under (1.4) the following simple law of transformation: ρ → ρ ′ = g ρ. (1.9) Let us ment io n that for the first auxiliar y linear problem (1.1), consider ed as classical partia l differential equation, the inv ar iants w 2 and e w 2 from the early times (see fo r exa mple the classica l b o ok o f F ors yth [21]) as Laplace in v ariants h = w 2 and k = e w 2 are k nown. The main results of the pap er are the following new in tegr able systems of nonlinear equations in terms of field v aria bles ρ, w 1 , w 2 given by (1.5)– (1 .8). In the case (i) of thir d- order linea r auxiliary pr oblem (1.2) the fir st inv aria nt w 1 is equal to zer o w 1 ≡ 0 and the established integrable sys tem of no nlinear eq uations in terms o f ρ , w 2 has the form ρ t = − κ 1 ρ ξξ ξ − κ 2 ρ ηη η − 3 κ 1 ρ ξ ∂ − 1 η w 2 ξ − 3 κ 2 ρ η ∂ − 1 ξ w 2 η + v 0 ρ, (1.10) w 2 t = − κ 1 w 2 ξξ ξ − κ 2 w 2 ηη η − 3 κ 1  w 2 ∂ − 1 η w 2 ξ  ξ − 3 κ 2  w 2 ∂ − 1 ξ w 2 η  η . (1.11) It is remark a ble that the ga uge-inv ariant s ubs ystem of the system (1.10)–(1 .1 1), equation (1.11) fo r the gauge inv ariant w 2 = u 0 − u 1 ξ − u 1 v 1 , co incides in form with the famous NVN equatio n [15, 16 ] u t = − κ 1 u ξξ ξ − κ 2 u ηη η − 3 κ 1  u∂ − 1 η u ξ  ξ − 3 κ 2  u∂ − 1 ξ u η  η . (1.12) Due to the last remark the system (1.10)–(1.1 1) will b e na med b elow a s the Nizhnik–V eselov–Novik ov (NVN) sy stem of equations. In the case (ii) o f second-order linear auxiliary pr oblem (1.2) the established in tegrable system of nonlinear equations in terms of ρ , w 1 and w 2 has the form ρ t = − κ 1 ρ ξξ − κ 2 ρ ηη − 2 κ 1 ρ∂ − 1 η w 2 ξ + 2 κ 2 ρ η ∂ − 1 ξ w 1 + v 0 ρ, (1.13) w 1 t = − κ 1 w 1 ξξ + κ 2 w 1 ηη − 2 κ 1 w 2 ξξ + 2 κ 2 w 2 ηη − 2 κ 1  w 1 ∂ − 1 η w 1  ξ + 2 κ 2  w 1 ∂ − 1 ξ w 1  η , (1.14) w 2 t = κ 1 w 2 ξξ − κ 2 w 2 ηη − 2 κ 1  w 2 ∂ − 1 η w 1  ξ + 2 κ 2  w 2 ∂ − 1 ξ w 1  η . (1.15) 2 The ga uge-inv aria nt subsystem of the system (1 .13)–(1.15), the system of equatio ns (1.14)–(1.15) for inv ariants w 1 = u 1 ξ − v 1 η and w 2 = u 0 − u 1 ξ − u 1 v 1 , for the choice u 1 = 0 , v 1 = v , u 0 = u for w hich w 1 = − v η , w 2 = u , leads to the w ell-known s ystem of e quations [2 2] v t = − κ 1 v ξξ + κ 2 v ηη + 2 κ 1 ∂ − 1 η u ξξ − 2 κ 2 u η + 2 κ 1 v v ξ − 2 κ 2 v η ∂ − 1 ξ v η , (1.16) u t = κ 1 u ξξ − κ 2 u ηη + 2 κ 1  uv  ξ − 2 κ 2  u∂ − 1 ξ v η  η . (1.17) In ter ms of v ar iables v = − q 2 , u = 1 4 (1 + r − q η ) (1.18) the integrable system of nonlinear equations (1.16)–(1.1 7) takes the for m q t = − κ 1 ∂ − 1 η r ξξ + κ 2 r η − κ 1 2  q 2  ξ + κ 2 q η ∂ − 1 ξ q η , (1.19) r t = − κ 1 q ξ + κ 2 ∂ − 1 ξ q ηη − κ 1 q ηξ ξ + κ 2 q ηη η − κ 1  rq  ξ + κ 2  r∂ − 1 ξ q η  η . (1.20) F or the particular v a lue κ 2 = 0 system o f e q uations (1.19)–(1.20) re duces to the famous integrable tw o- dimensional g eneralizatio n of disper sive long-wa ve system of equations [18] q tη = − κ 1 r ξξ − κ 1 2  q 2  ξη , (1.21) r tξ = − κ 1  q r + q + q ξη  ξξ . (1.22) In one-dimensional limit ξ = η bo th s ystems (1.19)–(1.20) with κ 1 − κ 2 = 1 and (1.21)–(1.22) with κ 1 = 1 reduce to the famo us disp ersive lo ng -wa ve equation (s e e , e.g., Br o er [23 ]). It is worth while by this rea son to name the sys tem (1.1 3)–(1.15) as the tw o-dimensional generalized dis pe r sive long- wa ve (2DGDL W) sy stem of equatio ns . In b o th considered cases of the third- and second-o rder auxiliar y linear pro ble m (1.2) the integrable systems of nonlinear equations (1.10)–(1.1 1) and (1.13)–(1.15) hav e common gaug e -transpar ent s tr ucture. They contain corr esp ondingly: • ga uge-inv aria nt subsystems (1.11) and (1.14)–(1.15); • the equatio ns (1.10) a nd (1.13) for the pure gauge v ar iable ρ with some terms containing gauge inv ari- ants. F or the zer o v alues of inv ar iants w 1 = 0, w 2 = 0 both systems (1.10)–(1.11) and (1.13)–(1.1 5) re duce to corres p o nding linea r equations fo r ρ , resp ectively , ρ t = − κ 1 ρ ξξ ξ − κ 2 ρ ηη η + v 0 ρ (1.23) and ρ t = − κ 1 ρ ξξ − κ 2 ρ ηη + v 0 ρ. (1.24) In this pa pe r the NVN (1.1 0)–(1.11) and the 2DGDL W (1.13)–(1.15) systems of integrable nonlinear equations in differen t gauges ar e considered. It is shown that in some different g a uges from (1.1 0)–(1.1 1) famo us Nizhnik–V eselov–Novik ov (NVN) [15, 16] and mo dified Nizhnik–V eselov–Novik ov (mNVN) [1 7] eq uations follow, these equations b y Miura- t yp e transformation ar e connected. It is sho wn a lso that gauge-inv ar iant subsystem (1.1 4)–(1.15) of the 2DGDL W system (1.13)–(1.1 5) contains in par ticula r, the famous case, int eg rable t wo-dimensional g eneralizatio n of disp ers ive long-wa ve system [18] of integrable nonlinea r equations. In some cases the sp ecia l gauge 2DGDL W system (1.1 3)– (1.15) re duce s to the famous Dav ey–Stewartson (DS) system [19] of nonlinear equations and in another 3 sp ecial gaug es to new DS-type systems o f int eg rable nonlinea r equations, these systems by Miura -type transformatio n ar e connected. The plan of our pap er is the following. In sections 2 and 3 via the compatibility condition (1.3) the manifestly gauge- inv a r iant cor resp ondingly integrable NVN sys tem (1.10)–(1.11) and the 2DGDL W system (1.13)–(1.1 5) of nonlinear equations a re derived. Some s p ecia l g auges of NVN (1 .10)–(1.11) and 2DGDL W (1.1 3)–(1.15) int eg rable systems o f no nlinear equations a re considered. Miura-type tra nsforma- tions b etw een solutions of nonlinear equa tions in differen t gauges are es tablished. 2 Manifestly gauge-in v arian t form ulation of NVN system of equa- tions It is instructive to der ive integrable nonlinea r equations starting from aux iliary linear problems (1.1) and (1.2) in general p osition, with a ll nonzer o field v ariables . Using the co mpatibility condition (1.3) in the for m of Ma nako v’s triad repr esentation [20] after so me calculations one obtains, equating to zero the coe fficie n ts at different degr ees of partial deriv atives ∂ n ξ ∂ m η of the relation [ L 1 , L 2 ] − B L 1 = 0, the follo wing sy s tem of equations for the field v aria bles u 3 , v 3 , u 2 , v 2 , ˜ u 1 , ˜ v 1 , v 0 and u 1 , v 1 , u 0 : ∂ 4 ξ : u 3 η = 0 , ∂ 4 η : v 3 ξ = 0 , (2.1) ∂ 3 ξ ∂ η : u 3 ξ = 0 , ∂ ξ ∂ 3 η : v 3 η = 0 , (2.2) ∂ 3 ξ : u 3 ξη + u 2 η + u 1 u 3 ξ − 3 u 3 u 1 ξ + v 1 u 3 η = 0 , (2.3) ∂ 3 η : v 3 ξη + v 2 ξ + v 1 v 3 η − 3 v 3 v 1 η + u 1 v 3 ξ = 0 , (2.4) ∂ 2 ξ ∂ η : u 2 ξ − 3 u 3 v 1 ξ = 0 , ∂ ξ ∂ 2 η : v 2 η − 3 v 3 u 1 η = 0 , (2.5) ∂ 2 ξ : u 2 ξη + ˜ u 1 η − 3 u 3 u 1 ξξ − 2 u 2 u 1 ξ + u 1 u 2 ξ + v 1 u 2 η − 3 u 3 u 0 ξ = 0 , (2.6) ∂ 2 η : v 2 ξη + ˜ v 1 ξ − 3 v 3 v 1 ηη − 2 v 2 v 1 η + u 1 v 2 ξ + v 1 v 2 η − 3 v 3 u 0 η = 0 , (2.7) ∂ 2 ξη : ˜ u 1 ξ + ˜ v 1 η − 3 u 3 v 1 ξξ − 3 v 3 u 1 ηη − 2 u 2 v 1 ξ − 2 v 2 u 1 η − B = 0 , (2.8) − ∂ ξ : u 1 t + u 3 u 1 ξξ ξ + v 3 u 1 ηη η + u 2 u 1 ξξ + v 2 u 1 ηη − v 0 η + ˜ u 1 u 1 ξ + ˜ v 1 u 1 η − u 1 ˜ u 1 ξ − v 1 ˜ u 1 η − ˜ u 1 ξη + 3 u 3 u 0 ξξ + 2 u 2 u 0 ξ + B u 1 = 0 , (2.9) − ∂ η : v 1 t + u 3 v 1 ξξ ξ + v 3 v 1 ηη η + u 2 v 1 ξξ + v 2 v 1 ηη − v 0 ξ + ˜ v 1 v 1 η + ˜ u 1 v 1 ξ − u 1 ˜ v 1 ξ − v 1 ˜ v 1 η − ˜ v 1 ξη + 3 v 3 u 0 ηη + 2 v 2 u 0 η + B v 1 = 0 , (2.10) − ∂ 0 : u 0 t + u 3 u 0 ξξ ξ + v 3 u 0 ηη η + u 2 u 0 ξξ + v 2 u 0 ηη + ˜ u 1 u 0 ξ + ˜ v 1 u 0 η − u 1 v 0 ξ − v 1 v 0 η − v 0 ξη + B u 0 = 0 . (2.11) The system o f defining equa tions (2.1)–(2.11) ha s recurrent c har acter and allows us to express via (2 .1 )–(2.7) the field v aria bles u 3 , v 3 , u 2 , v 2 and ˜ u 1 , ˜ v 1 of the second auxilia r y problem (1.2) through the field v ariables u 1 , v 1 , u 0 of the first auxiliary linear problem (1.1). The last three equations (2.9)–(2.11) repre sent the int eg rable system of nonlinear ev olution equations for the field v aria bles u 1 , v 1 and u 0 . In the case of the second auxiliary linea r problem (1 .2) of thir d order fr om relations (2.1 ) and (2.2) it follows that the co efficients u 3 and v 3 are c onstants, u 3 = const = κ 1 , v 3 = const = κ 2 . (2.12) Using (2.1 2) one obtains from the relations (2.3)–(2.5), u 2 ξ = 3 κ 1 v 1 ξ , v 2 η = 3 κ 2 u 1 η , (2.13) u 2 η = 3 κ 1 u 1 ξ , v 2 ξ = 3 κ 2 v 1 η . (2.14) F r o m (2.13)–(2.1 4) the imp o r tant relation b etw een field v aria bles u 1 , v 1 , u 1 ξ = v 1 η (2.15) 4 and expre ssions for v aria bles u 2 and v 2 , u 2 = 3 κ 1 v 1 + const 1 , v 2 = 3 κ 2 u 1 + const 2 , (2.16) follow. A r ising in (2.1 6), fo r simplicity the constants b eing equal to zer o are chosen b elow. By the us e of (2.6) and (2.7) taking into acco unt (2 .12), (2.1 5) and (2.16) one deriv es the expre s sions for ˜ u 1 and ˜ v 1 , ˜ u 1 = 3 κ 1 ∂ − 1 η u 0 ξ − 3 κ 1 ∂ − 1 η ( u 1 v 1 ξ ) + 3 κ 1 2 v 2 1 + f 1 ( ξ , t ) , (2.17) ˜ v 1 = 3 κ 2 ∂ − 1 ξ u 0 η − 3 κ 2 ∂ − 1 ξ ( v 1 u 1 η ) + 3 κ 2 2 u 2 1 + g 1 ( η , t ) , (2.18) including as ‘constants’ of in teg ration the arbitra r y functions f 1 ( ξ , t ) and g 1 ( η , t ) whic h for simplicit y are chosen be low as equal to zero v alues. Inserting ˜ u 1 and ˜ v 1 from (2.17), (2.18) into (2.8) and taking into account (1.5), (2.12), (2 .1 5)–(2.18) one deriv es the expr e s sion for the co efficient B , B = − 3 κ 1 v 1 ξξ − 3 κ 2 u 1 ηη − 3 κ 1 v 1 v 1 ξ − 3 κ 2 u 1 u 1 η + 3 κ 1 ∂ − 1 η u 0 ξξ + 3 κ 2 ∂ − 1 ξ u 0 ηη − 3 κ 1 ∂ − 1 η  u 1 v 1 ξ  ξ − 3 κ 2 ∂ − 1 ξ  v 1 u 1 η  η = 3 κ 1 ∂ − 1 η w 2 ξξ + 3 κ 2 ∂ − 1 ξ w 2 ηη . (2.19) The last three equa tio ns (2.9)–(2.11) of the system (2.1)–(2.11) are the evolution e quations for the field v a riables u 1 , v 1 and u 0 . By the use of (1.5), (2.12), (2.15)–(2.1 9) after s o me calcula tions (by sing ling out in some terms the combination of field v ariables w 2 = u 0 − u 1 ξ − u 1 v 1 coinciding with gauge inv aria nt (1.5)) these equatio ns can be represented in the following conv enient fo r m: u 1 t = − κ 1 u 1 ξξ ξ − κ 2 u 1 ηη η − κ 1  v 3 1 + 3 v 1 v 1 ξ  η − κ 2  u 3 1 + 3 u 1 u 1 η  η − 3 κ 1  v 1 ∂ − 1 η w 2 ξ  η − 3 κ 2  u 1 ∂ − 1 ξ w 2 η  η + v 0 η , (2.20) v 1 t = − κ 1 v 1 ξξ ξ − κ 2 v 1 ηη η − κ 1  v 3 1 + 3 v 1 v 1 ξ  ξ − κ 2  u 3 1 + 3 u 1 u 1 η  ξ − 3 κ 1  v 1 ∂ − 1 η w 2 ξ  ξ − 3 κ 2  u 1 ∂ − 1 ξ w 2 η  ξ + v 0 ξ , (2.21) u 0 t = − κ 1 u 0 ξξ ξ − κ 2 u 0 ηη η − 3 κ 1 v 1 u 0 ξξ − 3 κ 2 u 1 u 0 ηη − 3 κ 1  v 1 ξ + v 2 1  u 0 ξ − 3 κ 2  u 1 η + u 2 1  u 0 η − 3 κ 1  u 0 ∂ − 1 η w 2 ξ  ξ − 3 κ 2  u 0 ∂ − 1 ξ w 2 η  η + v 0 ξη + u 1 v 0 ξ + v 1 v 0 η . (2.22) Remem b er that in the cons idered case due to (2 .15) the first in v ariant w 1 = u 1 ξ − v 1 η = 0 is equal to zero. Due to the equality u 1 ξ = v 1 η one ca n reduce the set of dep endent v a riables u 1 , v 1 and u 0 in the system (2.20)–(2.22) to t wo v ariables ρ , w 2 (or equiv alently to v a riables φ = ln ρ , w 2 ) defined by the relations u 1 def = φ η = ρ η ρ , v 1 def = φ ξ = ρ ξ ρ , (2.23) w 2 = u 0 − u 1 ξ − u 1 v 1 = u 0 − φ ξη − φ ξ φ η = u 0 − ρ ξη ρ . (2.24) Indeed the insertion o f u 1 = φ η and v 1 = φ ξ int o (2.2 0) and (2.21) reduce s b oth these equations to the s ing le one equa tion φ t = − κ 1 φ ξξ ξ − κ 2 φ ηη η − κ 1 ( φ ξ ) 3 − κ 2 ( φ η ) 3 − 3 κ 1 φ ξ φ ξξ − 3 κ 2 φ η φ ηη − 3 κ 1 φ ξ ∂ − 1 η w 2 ξ − 3 κ 2 φ η ∂ − 1 ξ w 2 η + v 0 , (2.25) or in terms of v aria bles ρ , w 2 to the equation ρ t = − κ 1 ρ ξξ ξ − κ 2 ρ ηη η − 3 κ 1 ρ ξ ∂ − 1 η w 2 ξ − 3 κ 2 ρ η ∂ − 1 ξ w 2 η + v 0 ρ. (2.26) One can show a lso that the exclusio n of field v aria ble v 0 from the last equa tio n (2.22) by the use of deriv atives v 0 ξ , v 0 η and v 0 ξη (calculated from the first tw o equatio ns (2.20) and (2.21)) leads to the following nonlinear evolution equatio n for the second in v ariant w 2 : w 2 t = − κ 1 w 2 ξξ ξ − κ 2 w 2 ηη η − 3 κ 1  w 2 ∂ − 1 η w 2 ξ  ξ − 3 κ 2  w 2 ∂ − 1 ξ w 2 η  η . (2.27) 5 So by the c ha ng e of v ariables (2.23), (2.24) the in tegr able sy stem of nonlinear equations (2.20)–(2.22) is reduced to the following equiv alent integrable system of nonlinea r equa tions: ρ t = − κ 1 ρ ξξ ξ − κ 2 ρ ηη η − 3 κ 1 ρ ξ ∂ − 1 η w 2 ξ − 3 κ 2 ρ η ∂ − 1 ξ w 2 η + v 0 ρ, (2.28) w 2 t = − κ 1 w 2 ξξ ξ − κ 2 w 2 ηη η − 3 κ 1  w 2 ∂ − 1 η w 2 ξ  ξ − 3 κ 2  w 2 ∂ − 1 ξ w 2 η  η . (2.29) Equiv alently , in ter ms of v ar iables φ = ln ρ a nd w 2 , the system of equations (2.28)–(2.29) takes the for m φ t = − κ 1 φ ξξ ξ − κ 2 φ ηη η − κ 1 ( φ ξ ) 3 − κ 2 ( φ η ) 3 − 3 κ 1 φ ξ φ ξξ − 3 κ 2 φ η φ ηη − 3 κ 1 φ ξ ∂ − 1 η w 2 ξ − 3 κ 2 φ η ∂ − 1 ξ w 2 η + v 0 , (2.30) w 2 t = − κ 1 w 2 ξξ ξ − κ 2 w 2 ηη η − 3 κ 1  w 2 ∂ − 1 η w 2 ξ  ξ − 3 κ 2  w 2 ∂ − 1 ξ w 2 η  η . (2.31) Note that equation (2.29) (or (2.31)) for the g auge in v aria nt w 2 exactly coincides in form with the fa mo us NVN equatio n [1 5, 16 ]. Due to this r eason it is w o rthwhile to na me the integrable systems (2.28)–(2.29) (or (2.30)–(2.31)) as the NVN system of equations. The NVN system o f equations (2.28)–(2.29) (or (2.3 0)–(2.31)) has gaug e-transpa rent structure. It co n- tains: • explicitly gauge- inv a riant subsystem — equation (2 .29) (or (2.31)) for in v aria nt w 2 ; • equa tion (2.28) (or (2.30)) for pure gauge v ar iable ρ (or φ ) with some terms cont a ining gauge in v ari- ant w 2 and field v ar iable v 0 from the second linear auxiliary pro blem (1.2). Manako v’s tria d representation (1.3) for the NVN system o f equatio ns (2.28)–(2.29) (or (2.30)–(2.31)), due to formulae (2.12)–(2.19) and (2.23)–(2.24), includes the following ope rators L 1 , L 2 of auxiliar y linea r problems and co efficient B ( w 2 ): L 1 = ∂ 2 ξη + ρ η ρ ∂ ξ + ρ ξ ρ ∂ η + w 2 + ρ ξη ρ , (2.32) L 2 = ∂ t + κ 1 ∂ 3 ξ + κ 2 ∂ 3 η + 3 κ 1 ρ ξ ρ ∂ 2 ξ + 3 κ 2 ρ η ρ ∂ 2 η + 3 κ 1  ρ ξξ ρ +  ∂ − 1 η w 2 ξ   ∂ ξ + 3 κ 2  ρ ηη ρ +  ∂ − 1 ξ w 2 η   ∂ η + v 0 , (2.33) B ( w 2 ) = 3 κ 1 ∂ − 1 η w 2 ξξ + 3 κ 2 ∂ − 1 ξ w 2 ηη . (2.34) In the case w 2 = 0 of zero in v aria nt the NVN system o f equatio ns (2.28)–(2.29) (or (2.30)–(2.31)) reduces to linear equation ρ t = − κ 1 ρ ξξ ξ − κ 2 ρ ηη η + v 0 ρ, (2.35) which is int eg rable by auxilia ry linea r pr oblems (1.1) and (1.2) with L 1 and L 2 from (2 .32), (2.33) un- der w 2 = 0. The compatibility condition in this ca se, due to B ( w 2 ) = 0, has Lax form [ L 1 , L 2 ] = 0 . In ter ms of v ariable φ = ln ρ linear equation (2.35) loo ks like Burger s-type equation of the thir d o rder φ t = − κ 1 φ ξξ ξ − κ 2 φ ηη η − κ 1 ( φ ξ ) 3 − κ 2 ( φ η ) 3 − 3 κ 1 φ ξ φ ξξ − 3 κ 2 φ η φ ηη + v 0 , (2.36) which linear izes by the substitution φ = ln ρ and cons equently is C-in tegr able. Let us denote b y C ( φ, u 0 , v 0 ) the gauge whic h cor resp onds to nonzero field v ariables u 1 = φ η , v 1 = φ ξ , u 0 and v 0 of linear pro blems (1.1) and (1.2) and co nsequently to NVN system (2 .3 0)–(2.31) in gene r al po sition. Under different gauges from NVN system differe nt integrable nonlinea r equations follow, which are gauge- equiv a lent to each other. The solutions of these eq uations by some Miura-type transformatio n ar e connected. F or example in the gauge C (0 , u 0 , 0) the NVN system of equations (2.30)–(2.31) reduces to the famous NVN equatio n [15, 1 6] for the field v ariable u 0 , u 0 t = − κ 1 u 0 ξξ ξ − κ 2 u 0 ηη η − 3 κ 1  u 0 ∂ − 1 η u 0 ξ  ξ − 3 κ 2  u 0 ∂ − 1 ξ u 0 η  η . (2.37) 6 In another gauge C ( φ, 0 , v 0 ) the NVN system (2.30)–(2.31) takes the form φ t = − κ 1 φ ξξ ξ − κ 2 φ ηη η − κ 1 ( φ ξ ) 3 − κ 2 ( φ η ) 3 + 3 κ 1 φ ξ ∂ − 1 η  φ ξ φ η  ξ + 3 κ 2 φ η ∂ − 1 ξ  φ ξ φ η  η + v 0 , (2.38)  ∂ 2 ξη + φ η ∂ ξ + φ ξ ∂ η  φ t =  ∂ 2 ξη + φ η ∂ ξ + φ ξ ∂ η  h − κ 1 φ ξξ ξ − κ 2 φ ηη η − κ 1 ( φ ξ ) 3 − κ 2 ( φ η ) 3 + 3 κ 1 φ ξ ∂ − 1 η  φ ξ φ η  ξ + 3 κ 2 φ η ∂ − 1 ξ  φ ξ φ η  η i , (2.39) and conse quently to the following system of equations: φ t = − κ 1 φ ξξ ξ − κ 2 φ ηη η − κ 1 ( φ ξ ) 3 − κ 2 ( φ η ) 3 + 3 κ 1 φ ξ ∂ − 1 η  φ ξ φ η  ξ + 3 κ 2 φ η ∂ − 1 ξ  φ ξ φ η  η + v 0 , (2.40)  ∂ 2 ξη + φ η ∂ ξ + φ ξ ∂ η  v 0 = 0 (2.41) is equiv ale nt. F or v 0 = 0 system of equations (2.40)–(2.41) reduces to the famous modified Nizhnik–V es e lov– Novik ov equation φ t = − κ 1 φ ξξ ξ − κ 2 φ ηη η − κ 1 ( φ ξ ) 3 − κ 2 ( φ η ) 3 + 3 κ 1 φ ξ ∂ − 1 η  φ ξ φ η  ξ + 3 κ 2 φ η ∂ − 1 ξ  φ ξ φ η  η , (2.42) which at first in the pap er [17] o f K onop elchenk o in a differ e nt context was discov er ed. Let us mention that the considered version (2.42) of mNVN equation derived in the pr esent pap er in the framework of ma nifestly gauge-inv ar iant description is differen t from the mNVN equation disco vered in the pap er [2 4]. The new system of eq uations (2 .40)–(2.41) can be named as modified NVN (mNVN) system of equa- tions. This s y stem due to (2.32)–(2.3 4) and to the choice of the gaug e C ( φ, 0 , v 0 ) has the following triad representation (1.3) with tr ia d ( L 1 , L 2 , B ): L 1 = ∂ 2 ξη + φ η ∂ ξ + φ ξ ∂ η , (2.43) L 2 = ∂ t + κ 1 ∂ 3 ξ + κ 2 ∂ 3 η + 3 κ 1 φ ξ ∂ 2 ξ + 3 κ 2 φ η ∂ 2 η + 3 κ 1  φ 2 ξ − ∂ − 1 η  φ ξ φ η  ξ  ∂ ξ + 3 κ 2  φ 2 η − ∂ − 1 ξ  φ ξ φ η  η  ∂ η + v 0 , (2.44) B ( w 2 ) = − 3 κ 1 φ ξξ ξ − 3 κ 2 φ ηη η − 3 κ 1 ∂ − 1 η  φ ξ φ η  ξξ − 3 κ 2 ∂ − 1 ξ  φ ξ φ η  ηη . (2.45) The mNVN equation (2.42) has triad representation (2.4 3)–(2.45) with v 0 = 0. It is evident that the so lutio ns u 0 and φ o f NVN (2.37) and mNVN (2.4 2) equations v ia inv ariant w 2 = u 0 = − φ ξη − φ ξ φ η (calculated in different gauges C (0 , u 0 , 0) and C ( φ, 0 , 0 )) b y Miur a -type transfor- mation u 0 = − φ ξη − φ ξ φ η (2.46) are connected. In one-dimensional limit, under ∂ ξ = ∂ η , the mNVN equation (2.42) reduces to the mKdV equation in potential for m φ t = − κ φ ξξ ξ + 2 κ ( φ ξ ) 3 , (2.47) where κ = κ 1 + κ 2 . In terms of v ariable v 1 = φ ξ this is mKdV equa tio n v 1 t = − κ v 1 ξξ ξ + 6 κ v 2 1 v 1 ξ . (2.48) 3 Manifestly gauge-in v arian t form ulation of t w o-dimensional gen- eralization of the disp ersiv e long-w a v e equations system In the case of second-order linear aux ilia ry problem (1.2) the co efficients u 3 , v 3 in the s ystem of r elations (2.1) – (2.11) have zero v alues u 3 = v 3 = 0. The relations (2.3)–(2.5) lea d to cons tant v alues for the coe fficient s u 2 and v 2 , u 2 = const = κ 1 , v 2 = const = κ 2 . (3.1) 7 By integration of relations (2.6) a nd (2.7) one immediately obtains the expressions for the c o efficients ˜ u 1 and ˜ v 1 , ˜ u 1 = 2 κ 1 ∂ − 1 η u 1 ξ + f 2 ( ξ , t ) , ˜ v 1 = 2 κ 2 ∂ − 1 ξ v 1 η + g 2 ( η , t ) , (3.2) where f 2 ( ξ , t ) and g 2 ( η , t ) are arbitrar y functions which below, for s implicit y , chosen equal to zer o v alues. Inserting (3.1)–(3 .2) into (2.8) one obtains taking into account (1 .7) the expr ession for coefficient B , B = − 2 κ 1 v 1 ξ − 2 κ 2 u 1 η + 2 κ 1 ∂ − 1 η u 1 ξξ + 2 κ 2 ∂ − 1 ξ v 1 ηη = 2 κ 1 ∂ − 1 η w 1 ξ − 2 κ 2 ∂ − 1 ξ w 1 η . (3.3) The las t three r elations (2.9)–(2.1 1) o f the s ystem (2.1)–(2.1 1) are nonlinea r evolution equations for the field v a riables u 1 , v 1 and u 0 . By the use of (3.1)–(3.3) after some calcula tions thes e equations can be represented (by s ingling out in some ter ms the combinations of field v ar iables w 1 = u 1 ξ − v 1 η and w 2 = u 0 − u 1 ξ − u 1 v 1 coinciding with gauge inv aria nts (1.5)–(1.7)) in the follo wing con venien t form: u 1 t = − κ 1 v 1 ξη − κ 2 u 1 ηη − 2 κ 2 u 1 u 1 η − κ 1 w 1 ξ − 2 κ 1 w 2 ξ − 2 κ 1 u 1 ξ ∂ − 1 η u 1 ξ + 2 κ 2  u 1 ∂ − 1 ξ w 1  η + v 0 η , (3.4) v 1 t = − κ 1 v 1 ξξ − κ 2 u 1 ξη − 2 κ 1 v 1 v 1 ξ − κ 2 w 1 η − 2 κ 2 w 2 η − 2 κ 2 v 1 η ∂ − 1 ξ v 1 η − 2 κ 1  v 1 ∂ − 1 η w 1  ξ + v 0 ξ , (3.5) u 0 t = − κ 1 u 0 ξξ − κ 2 u 0 ηη − 2 κ 1 u 0 ξ v 1 − 2 κ 2 u 0 η u 1 − 2 κ 1  u 0 ∂ − 1 η w 1  ξ + 2 κ 2  u 0 ∂ − 1 ξ w 1  η + v 0 ξη + u 1 v 0 ξ + v 1 v 0 η . (3.6) Let us emphasize that the integrable sys tem of nonlinear eq uations (3.4)–(3.6) a rises as a c o mpatibility condition of a uxiliary linear pr oblems ( 1 .1) and (1 .2) in the form (1.3) of Ma nako v’s tr iad representation in the general p osition. The sy s tem contains thr ee evolution equations for the field v ariables u 1 , v 1 and u 0 . These equations include als o the field v aria ble v 0 from the second auxiliary linear problem. T he pres ence of these four dependent v aria bles u 1 , v 1 , u 0 and v 0 in system (3.4 )–(3.6) of three nonlinear equations reflects gaug e freedom of auxiliar y linear problems (1.1) and (1.2) a nd the corresp onding int eg rable systems of nonlinea r equations. In contrast to the cas e consider ed in the previous section, the first in v ariant w 1 = u 1 ξ − v 1 η 6 = 0 is not equal to zero. One can show that the first t wo equations (3.4) and (3.5) of last system under change o f v ar ia bles u 1 = φ η = ρ η ρ , v 1 = − ∂ − 1 η w 1 + φ ξ = − ∂ − 1 η w 1 + ρ ξ ρ , (3.7) w 2 = u 0 − φ ξη − φ ξ φ η + φ η ∂ − 1 η w 1 = u 0 − ρ ξη ρ + ρ η ρ ∂ − 1 η w 1 , (3.8) reduce to the single one equation of the form ρ t = − κ 1 ρ ξξ − κ 2 ρ ηη − 2 κ 1 ρ∂ − 1 η w 2 ξ + 2 κ 2 ρ η ∂ − 1 ξ w 1 + v 0 ρ, (3.9) or in terms of v aria ble φ = ln ρ to the equation φ t = − κ 1 φ ξξ − κ 2 φ ηη − κ 1 ( φ ξ ) 2 − κ 2 ( φ η ) 2 − 2 κ 1 ∂ − 1 η w 2 ξ + 2 κ 2 φ η ∂ − 1 ξ w 1 + v 0 . (3.10) The condition of equalit y of mixture deriv atives v 0 ξη and v 0 ηξ , calculated from (3.4) and (3.5), leads to the following nonlinear ev o lution equation in terms of gauge inv ar iants w 1 and w 2 , w 1 t = − κ 1 w 1 ξξ + κ 2 w 2 ηη − 2 κ 1 w 2 ξξ + 2 κ 2 w 2 ηη − 2 κ 1  w 1 ∂ − 1 η w 1  ξ + 2 κ 2  w 1 ∂ − 1 ξ w 1  η . (3.11) One can show also that the exclusion of free field v ariable v 0 from the last equa tio n (3.6) by the us e of deriv atives v 0 ξ , v 0 η and v 0 ξη , calculated from the first tw o equatio ns (3.4) a nd (3.5), leads to ano ther evolution equatio n in terms of in v ariants w 1 and w 2 , w 2 t = κ 1 w 2 ξξ − κ 2 w 2 ηη − 2 κ 1  w 2 ∂ − 1 η w 1  ξ + 2 κ 2  w 2 ∂ − 1 ξ w 1  η . (3.12) 8 So by the change of v a riables (3.7), (3.8) the integrable system (3.4)–(3.6) of nonlinea r equations of second o rder is reduced to the following equiv alent integrable system of nonlinear equations: ρ t = − κ 1 ρ ξξ − κ 2 ρ ηη − 2 κ 1 ρ∂ − 1 η w 2 ξ + 2 κ 2 ρ η ∂ − 1 ξ w 1 + v 0 ρ, (3.13) w 1 t = − κ 1 w 1 ξξ + κ 2 w 1 ηη − 2 κ 1 w 2 ξξ + 2 κ 2 w 2 ηη − 2 κ 1  w 1 ∂ − 1 η w 1  ξ + 2 κ 2  w 1 ∂ − 1 ξ w 1  η , (3.14) w 2 t = κ 1 w 2 ξξ − κ 2 w 2 ηη − 2 κ 1  w 2 ∂ − 1 η w 1  ξ + 2 κ 2  w 2 ∂ − 1 ξ w 1  η . (3.15) In terms of v ariables φ = ln ρ , w 1 and w 2 the integrable s ystem (3.1 3)–(3.1 5) takes the form φ t = − κ 1 φ ξξ − κ 2 φ ηη − κ 1 ( φ ξ ) 2 − κ 2 ( φ η ) 2 − 2 κ 1 ∂ − 1 η w 2 ξ + 2 κ 2 φ η ∂ − 1 ξ w 1 + v 0 , (3.16) w 1 t = − κ 1 w 1 ξξ + κ 2 w 1 ηη − 2 κ 1 w 2 ξξ + 2 κ 2 w 2 ηη − 2 κ 1  w 1 ∂ − 1 η w 1  ξ + 2 κ 2  w 1 ∂ − 1 ξ w 1  η , (3.17) w 2 t = κ 1 w 2 ξξ − κ 2 w 2 ηη − 2 κ 1  w 2 ∂ − 1 η w 1  ξ + 2 κ 2  w 2 ∂ − 1 ξ w 1  η . (3.18) In terms of v ariables φ = ln ρ , w 2 and e w 2 = w 2 + w 1 the integrable system (3.13)–(3.15) converts into more symmetrical form φ t = − κ 1 φ ξξ − κ 2 φ ηη − κ 1 ( φ ξ ) 2 − κ 2 ( φ η ) 2 − 2 κ 1 ∂ − 1 η w 2 ξ + 2 κ 2 φ η ∂ − 1 ξ w 1 + v 0 , (3.19) w 2 t = κ 1 w 2 ξξ − κ 2 w 2 ηη − 2 κ 1  w 2 ∂ − 1 η ( e w 2 − w 2 )  ξ + 2 κ 2  w 2 ∂ − 1 ξ ( e w 2 − w 2 )  η , (3.20) e w 2 t = − κ 1 e w 2 ξξ + κ 2 e w 2 ηη − 2 κ 1  e w 2 ∂ − 1 η ( e w 2 − w 2 )  ξ + 2 κ 2  e w 2 ∂ − 1 ξ ( e w 2 − w 2 )  η . (3.21) Remem b er for conv enience that due to (1.5)–(1.8) in equiv alent to each other sy stems of nonlinear equations (3.13)–(3.15), (3.16)–(3.18) and (3.19)–(3 .2 1) the v aria bles φ = ln ρ , w 1 , w 2 and e w 2 are connected with the field v aria bles u 1 , v 1 , u 0 of the linear pro ble m (1.1) by the form ulae u 1 = ρ η ρ = φ η , v 1 = ρ ξ ρ − ∂ − 1 η w 1 = φ ξ − ∂ − 1 η w 1 , w 1 = u 1 ξ − v 1 η , (3.22) w 2 = u 0 − u 1 ξ − u 1 v 1 = u 0 − φ ξη − φ η φ ξ + φ η ∂ − 1 η w 1 = u 0 − ρ ξη ρ + ρ η ρ ∂ − 1 η w 1 , (3.23) e w 2 = w 2 + w 1 . (3.24) Int eg rable system of nonlinear equations (3.13)–(3.15) (and analogo usly equiv alent systems (3.16)–(3.18) or (3.19)–(3.2 1)) for the choice of v aria bles ρ = 1; u 1 = 0 , v 1 = v, u 0 = u ; v 0 = 2 κ 1 ∂ − 1 η w 2 ξ (3.25) for which w 1 = − v η , w 2 = u , reduces to known system of equations v t = − κ 1 v ξξ + κ 2 v ηη − 2 κ 2 u η + 2 κ 1 v v ξ + 2 κ 1 ∂ − 1 η u ξξ − 2 κ 2 v η ∂ − 1 ξ v η , (3.26) u t = κ 1 u ξξ − κ 2 u ηη + 2 κ 1  uv  ξ − 2 κ 2  u∂ − 1 ξ v η  η , (3.27) derived in differen t context by Konop elchenko [22]. F or the particular v alues κ 1 = 1 a nd κ 2 = 0, system of equations (3 .26)–(3.27) reduces to famous int eg rable tw o-dimensio nal genera lization of dis p er sive long-wa ve sys tem of equations v tη = − v ξξ η + 2 u ξξ +  v 2  ξη , (3 .28) u t = u ξξ + 2  uv  ξ , (3.29) discov ered by Boiti, Leon and Pempinelli [1 8 ]. It is interesting to note tha t in a different context the system of equations (3 .20)–(3.21) for Laplace in v aria nts h = w 2 and k = e w 2 in the case κ 1 = 1, κ 2 = 0 in the pap er of W eis s [2 5 ] was consider ed. By this reason and due to the remarks in section 1 (see (1 .13)–(1.22) and discussion therein) it is worth while to name the in tegrable system of nonlinear e quations (3.13)–(3.15) (and analogously equiv alent sy s tems (3.1 6)–(3.18) or (3.1 9)–(3.2 1)) a s a tw o-dimensio nal gener alization o f disp e rsive long -wa ve (2DGDL W) system of equations. All considered equiv alent to each other, 2DGDL W in tegr able systems of nonlinear equations (3.13)–(3.15), (3.16)–(3.18) and (3.19)–(3.21) have a common gaug e-transpar ent structure: 9 • they co ntain explicitly gauge-inv ariant subsystems (3.14)–(3.1 5), (3.17)–(3.18) of nonlinear equa tions for gaug e inv ariants w 1 and w 2 (or equiv alently subsystem (3 .2 0)–(3.21) for gauge inv a riants w 2 and e w 2 ); • they include equation (3.13) for pure gauge v ariable ρ (o r equation (3.16) for v ariable φ = ln ρ ) (with simple rule of gauge transformation ρ → ρ ′ = g ρ ) with additional terms containing gauge in v ariants and field v ar ia ble v 0 . Such structure of 2DGDL W systems reflects existing gauge free do m in auxilia ry linear problems (1 .1) and (1.2). Due to formulae (1 .5 ), (1.7) a nd (3.1)–(3.3) 2DGDL W sys tem (3.13)–(3.1 5) has tria d re pr esentation [ L 1 , L 2 ] = B ( w 1 ) L 1 with o p er ators L 1 , L 2 and coefficient B ( w 1 ) o f the following forms: L 1 = ∂ 2 ξη + ρ η ρ ∂ ξ +  ρ ξ ρ −  ∂ − 1 η w 1   ∂ η + w 2 + ρ ξη ρ − ρ η ρ ∂ − 1 η w 1 , (3.30) L 2 = ∂ t + κ 1 ∂ 2 ξ + κ 2 ∂ 2 η + 2 κ 1 ρ ξ ρ ∂ ξ + 2 κ 2  ρ η ρ −  ∂ − 1 ξ w 1   ∂ η + v 0 , (3 .31) B ( w 1 ) = 2 κ 1 ∂ − 1 η w 1 ξ − 2 κ 2 ∂ − 1 ξ w 1 η . (3.32) Let us consider some particular gaug e s of established 2DGDL W systems of equations equations (3.13)– (3.15), (3.16)–(3.18) and (3.19)–(3.21). It is co nvenien t to denote the g auge in general p ositio n by the symbo l C ( u 1 , v 1 , u 0 ). In the gauge C ( u 1 = φ η , v 1 = φ ξ , u 0 = φ ξη + φ ξ φ η ) which due to (1.5)–(1.7) cor r esp onds to zero v alues of inv aria n ts w 1 and w 2 w 1 = u 1 ξ − v 1 η = 0 , w 2 = u 0 − u 1 ξ − u 1 v 1 = 0 , e w 2 = 0 , (3.33) the 2DGDL W system of equations (3.19)–(3.21) r educes to tw o- dimens ional Burg ers equa tion in p otential form φ t = − κ 1 φ ξξ − κ 2 φ ηη − κ 1 ( φ ξ ) 2 − κ 2 ( φ η ) 2 + v 0 , (3.34) or in terms of v a riable ρ co nnected with φ b y Hopf-Cole transforma tio n φ = ln ρ , to linear diffusion eq ua tion ρ t = − κ 1 ρ ξξ − κ 2 ρ ηη + v 0 ρ. (3.35) Equation (3.34) (or (3.3 5)) due to our constructio n is a compatibilit y condition in Lax form [ L 1 , L 2 ] = B ( w 1 ) L 1 ≡ 0 (3.36) of linear problems (1.1) a nd (1.2) with op era tors L 1 , L 2 given b y (3 .30), (3.31) under substitution w 1 = w 2 = 0. In another s imple gaug e C ( u 1 = φ η , v 1 = 0 , u 0 = 0) corresp o nding due to (3.22)–(3.2 4) to the in v ariants w 1 = φ ξη , w 2 = − φ ξη , e w 2 = 0 , (3.37) the 2DGDL W system of equations (3.1 9)–(3.21) for the c hoice v 0 = 0 again reduces to the single equa tio n of Burg ers type in p otential form φ t = κ 1 φ ξξ − κ 2 φ ηη − κ 1 ( φ ξ ) 2 + κ 2 ( φ η ) 2 . (3.38) This equatio n linearize s by Hopf-Cole tr ansformatio n φ = − ln ρ to cor resp onding linear equation ρ t = κ 1 ρ ξξ − κ 2 ρ ηη . (3.39) In the les s trivial ga uge C ( u 1 = 0 , v 1 = − q ξ /q , u 0 = p q ) the in v ariants w 1 , w 2 and e w 2 due to (3.2 2)–(3.2 4) are given by the following expr essions: w 1 =  ln q  ξη , w 2 = u 0 = p q , e w 2 = p q +  ln q  ξη , (3.40) 10 the v aria ble ρ due to (3.2 2) has constant v alue, consequen tly the v ariable φ = 0. In this case due to (3.1 9) v 0 = 2 κ 1 ∂ − 1 η w 2 ξ = 2 κ 1 ∂ − 1 η  p q  ξ (3.41) and from the 2DGDL W system of equations (3.19)–(3.2 1) one obtains after some calculations the famous DS sys tem of equations [19] for the field v a riables p and q , p t = κ 1 p ξξ − κ 2 p ηη + 2 κ 1 p ∂ − 1 η  p q  ξ − 2 κ 2 p ∂ − 1 ξ  p q  η , (3.42) q t = − κ 1 q ξξ + κ 2 q ηη − 2 κ 1 q ∂ − 1 η  p q  ξ + 2 κ 2 q ∂ − 1 ξ  p q  η . (3.43) One can consider also the gauge C ( u 1 = p η , v 1 = q ξ , u 0 = p η q ξ ) in which due to (3.2 2)–(3.24) the in v ari- ants have the follo wing expressions through q and p : w 1 = p ξη − q ξη , w 2 = − p ξη , e w 2 = − q ξη . (3.44) Substitution o f w 1 , w 2 and e w 2 from (3.4 4) into the sy stem (3.19)–(3.2 1) leads to the following three equations for p and q . F rom equation (3.19) for φ ≡ p one o btains p t = κ 1 p ξξ − κ 2 p ηη − κ 1 ( p ξ ) 2 + κ 2 ( p η ) 2 − 2 κ 2 p η q η + v 0 . (3.45) Equations (3.2 0) and (3.21) for w 2 and e w 2 in ter ms of v ar iables p , q take the forms p t = κ 1 p ξξ − κ 2 p ηη − κ 1 ( p ξ ) 2 + κ 2 ( p η ) 2 + 2 κ 1 ∂ − 1 η  p ξη q ξ  − 2 κ 2 ∂ − 1 ξ  p ξη q η  , (3.46) q t = − κ 1 q ξξ + κ 2 q ηη + κ 1 ( q ξ ) 2 − κ 2 ( q η ) 2 − 2 κ 1 ∂ − 1 η  q ξη p ξ  + 2 κ 2 ∂ − 1 ξ  q ξη p η  . (3.47) Equations (3.4 5) and (3.46) are compatible for the choice of v 0 given by the formula v 0 = 2 κ 1 ∂ − 1 η  p ξη q ξ  + 2 κ 2 ∂ − 1 ξ  q ξη p η  , (3.48) and the system of three equations (3 .45)–(3.47) reduces to system of tw o equa tio ns (3.4 6)–(3.47) containing in nonlo cal terms der iv a tives p ξη q ξ , p ξη q η , etc. Analogously in the g auge C ( u 1 = p η , v 1 = q ξ , u 0 = 0) it follows for w 1 , w 2 and e w 2 due to (3.22)–(3.24) w 1 = p ξη − q ξη , w 2 = − p ξη − p η q ξ , e w 2 = − q ξη − p η q ξ . (3.49) Equation (3.16) for φ ≡ p v ia (3.49) takes the form p t = κ 1 p ξξ − κ 2 p ηη − κ 1 ( p ξ ) 2 + κ 2 ( p η ) 2 − 2 κ 2 p η q η + 2 κ 1 ∂ − 1 η  p η q ξ  ξ + v 0 . (3.50) Equation (3.17) via substitutions from (3.4 9) transforms to the for m p t − q t = κ 1  p + q  ξξ − κ 2  p + q  ηη − κ 1 ( p ξ − q ξ ) 2 + κ 2 ( p η − q η ) 2 + 2 κ 1 ∂ − 1 η  p η q ξ  ξ − 2 κ 2 ∂ − 1 ξ  p η q ξ  η . (3.51) By substr a ction of equation (3.51) from equation (3.50) one obtains the ev olution equation for q : q t = − κ 1 q ξξ + κ 2 q ηη + κ 1 ( q ξ ) 2 − κ 2 ( q η ) 2 − 2 κ 1 p ξ q ξ + 2 κ 2 ∂ − 1 ξ  p η q ξ  η + v 0 . (3.52) Equation (3.18) for the in v aria nt w 2 due to (3.49) in terms of v ar iables p , q is  p ξη + p η q ξ  t = κ 1  p ξη + p η q ξ  ξξ − κ 2  p ξη + p η q ξ  ηη − 2 κ 1  ( p ξη + p η q ξ )( p ξ − q ξ )  ξ + 2 κ 2  ( p ξη + p η q ξ )( p η − q η )  η . (3.53) Equations (3.50), (3 .5 2) and (3.53) are compatible with each other if the field v aria ble v 0 satisfies the equatio n v 0 ξη + p η v 0 ξ + q ξ v 0 η = 0 . (3.54) 11 F or the simple c hoice v 0 ≡ 0 one obtains from the s ystem of the three equations (3.50), (3.52) a nd (3.53) the following equiv alent s ystem of tw o equations: p t = κ 1 p ξξ − κ 2 p ηη − κ 1 ( p ξ ) 2 + κ 2 ( p η ) 2 − 2 κ 2 p η q η + 2 κ 1 ∂ − 1 η  p η q ξ  ξ , (3.55) q t = − κ 1 q ξξ + κ 2 q ηη + κ 1 ( q ξ ) 2 − κ 2 ( q η ) 2 − 2 κ 1 p ξ q ξ + 2 κ 2 ∂ − 1 ξ  p η q ξ  η . (3.56) A t first this system of equations has been derived in another co nt ex t in the pap er of Kono pelchenko [22]. In conclusion, let us derive Miura -type transformatio ns b etw een different systems of DS-type equations of seco nd order o btained in this section in differ ent g auges. F or conv enience le t us denote by capital letters P ≡ p , Q ≡ q the solutions of the DS famous sy stem (3.42)–(3.43) of equa tions. By the use of in v ariants w 1 and w 2 one o btains the following relations b etw een v ar ia bles ( P ≡ p , Q ≡ q ) of DS system (3.4 2)–(3.43) and v a riables p , q of the system (3.4 6)–(3.47), w 1 =  ln Q  ξη = p ξη − q ξη , w 2 = P Q = − p ξη . ( 3 .57) One derives from (3.57), Q = e p − q , P = − p ξη e q − p . (3.58) Quite analogously for the pair o f DS systems (3.4 2)–(3.43) and (3.55)–(3.56) one has w 1 =  ln Q  ξη = p ξη − q ξη , w 2 = P Q = − p ξη − p η q ξ . (3.59) One obta ins from (3.59), Q = e p − q , P = −  p ξη + p η q ξ  e q − p . (3.60) T r ansformatio ns (3.58) a nd (3.60) a llow us to obtain solutions of the famous DS system of e qua- tions (3.4 2)–(3.43) from the systems o f equations (3.4 6)–(3.47) and (3.5 5)–(3.56), thes e tra ns formations are Miura -type transforma tio ns b eing ga uge-equiv alent to other DS-type s ystems of equations of seco nd order. 4 Conclusion In conclusion let us underline once a gain that ideas of ga uge inv aria nc e now ar e in c ommon us e in the theory o f integrable nonlinear evolution equations. There a re k nown attempts to develop inv ar iant descr ip- tion o f s ome nonlinear integrable equa tio ns co nsidered in the pre sent pape r by the use of matrix linear auxiliary pro ble ms . This was done for example in the pap er [26] for the Nizhnik–V eselov–Novik ov and Dav ey–Stewartson equations in the framework of the classica l inv ariant theory of second-or der linear par tia l differential eq ua tions. Matrix linear auxiliary problems hav e a bigge r num b er of degree s of freedom than the scala r, the p erfor- mance of reductio ns from gener al p o sition to in tegr able nonlinear equations is more difficult; all this leads to the need of considera tion gauge transformations under some restrictions, manifestly the ga uge-inv aria nt description of integrable nonlinear equations in this case is far from completion and requires additiona l resear ch work. Ac kno wledgmen t This research was suppo rted b y Nov o sibirsk State T echnical University scient ific gr a nt of fundamental re- searches in 2007–2 008 years. 12 References [1] Za kharov V E and Shabat A B 1974 F u nct. Anal . A ppl. 8 226 [2] Kuz nets ov E A and Mikhailo v A V 1 977 The or. Math. Phys. 30 19 3 [3] Za kharov V E and Mikhailov A V 1978 Sov. Phys.—JETP 47 1017 [4] Za kharov V E and T akhtadzhy an L A 1 979 The or. Math. Phys. 38 17 [5] Ko nop elchenk o B G 198 2 Phys. L ett. 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