Algebraic structures on the flows of dispersionless modified KP equation

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Algebraic structures on the flows of dispersionless modified KP equation

R ILANGOVANE¹^,∗, K KRISHNAKUMAR²and K M TAMIZHMANI³

1Department of Mathematics, Perunthalaivar Kamarajar Institute of Engineering and Technology (PKIET), Karaikal 609 603, India

2Department of Mathematics, Srinivasa Ramanujan Centre, SASTRA Deemed to be University, Kumbakonam 612 001, India

3Department of Mathematics, Ramanujan School of Mathematical Sciences, Pondicherry University, Kalapet 605 014, India

∗Corresponding author. E-mail: ilangovane@gmail.com MS received 30 January 2021; accepted 27 July 2021

Abstract. In this paper, we derive the non-isospectral flows of dispersionless modified Kadomtsev–Petviashvili (dmKP) hierarchies by applying quasiclassical limit in the associated Lax equations of the mKP system. Along with the isospectral flows, we investigate the underlying infinite-dimensional Lie algebraic structure of the dmKP system through the construction of implicit flow representations. In addition to this, we also discuss the correspondence between the non-isospectral flows of dKP and dmKP hierarchies by the dispersionless Miura map.

Keywords. Non-isopectral flows; dispersionless equations; Lie algebraic structures.

PACS No. 02.30.Ik

1. Introduction

In the past three decades, the dispersionless KP (dKP) hierarchy [1–6] and its various reductions have gained lots of interest because they are linked with topological field theory, Whitham hierarchy, its connections to string theory and two-dimensional gravity [1,7–11]. Many properties of this equation, such as twistor construction of dispersionless systems using Orolov functions [5], solutions through hodograph transformations [12,13], tau function theory, dispersionless analogue of Vira- soro constrains [10], etc. were discussed in the recent past. As in the case of dKP hierarchy, the study of dispersionless modified KP (dmKP) hierarchy and its extensions also share many richness related in the fields of mathematical physics [14–19]. However, identifying the Lie algebraic structures by deriving iso and non- isospectral flows of dmKP hierarchy is still in the initial stage. Moreover, the derivation of non-isospectral flows not only plays a crucial in deducting the underlying algebraic structures of a given system, but also quite important from the physical point of view. In the case of soliton systems, the solutions of the isospectral equations explain the behaviour of solitary waves in the

lossless and uniform media whereas the solutions of the non-isospectral equations reveal the nature of solitary waves in a certain type of non-uniform media. In addition, composing the isospectral flows with time- dependant symmetries of an integrable system is also often related with an infinite-dimensional Lie algebraic structure.

Recently, Fu et al studied the algebraic properties of dKP equation such as iso and non-isospectral flows, underlying infinite-dimensional Lie algebraic structures and the Hamiltonian formalisms of the dKP hierarchies using Lax triad approach [20]. On the other hand, Chang and Tu [14] studied the canonical property (preservation of bi-Hamiltonian structure) of the dispersionless Miura map between the dmKP and dKP hierarchies. Using this, they derived the Orlov function of the dmKP hierarchy, which was useful to establish its twistor construction.

Throughout this paper, we follow the field variables u =u(x,y,t)andv =v(x,y,t)for dispersionful hierarchies andU =U(X,Y,T)andV =V(X,Y,T)for dispersionless hierarchies. The dispersionless KP and mKP equations are given as

UT =3U UX+3

4∂⁻¹UY Y(dKP), (1)

0123456789().: V,-vol

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VT = 3

2VX∂⁻¹VY − 3 2VXV² +3

4∂⁻¹VY Y(dmKP), (2)

where U = U(X,Y,T) and V = V(X,Y,T). Equa- tions (1) and (2) are connected by dispersionless Miura transformation [14],

U = 1

2(−V²+∂_X⁻¹VY). (3)

Motivated by these works [14,20], in this paper, we construct the non-isospectral flows of dmKP and obtain the associated infinite-dimensional Lie algebraic structure.

Furthermore, we extend the analysis of dispersionless Miura map between the non-isospectral flows of dKP and dmKP hierarchies.

This paper is organised as follows: in §2, we discuss the basic results such as the iso and non-isospectral flows of the dKP and mKP equations. In §3, we derive the iso and non-isospectral flows of the dmKP equation.

In §4, we investigate the underlying algebraic structure of the dmKP hierarchies. In §5, through dispersionless Miura map we identify the correspondence between the non-isospectral flows of the dKP and dmKP hierarchies.

Section6gives the conclusion.

2. Preliminaries

Throughout this paper, we restrict with following notions to study the iso and non-isospectral flows of evolution equations and obtain their underlying Lie algebraic structures.

Consider an evolution equation

ut = K(u), u ∈M, (4)

where K is a vector field on a linear manifold M of infinite dimension consisting ofC^∞functions u(x,y) defined onR², vanishing rapidly at±∞. Here, one need the solution of (4) to be depending inC^∞ way on the time parametert. This happens becauseMis linear and all the fibres of the tangent bundleTMare replications of the same vector space S, i.e.M=S.

The Gâtaeux derivative of a vector field f(u)∈Min the direction of another vector fieldg(u)∈Mis defined as

f(u)[g(u)] = d

d f(u+g(u))|=0. (5) Using this, S forms a Lie algebra ofC^∞ vector fields onMwith Lie commutator[[·,·]], defined as

[[f,g]](u)= f(u)[g(u)] −g(u)[f(u)]. (6) For convenience, hereafter we use f[g] instead of

f(u)[g(u)]and[[f,g]]instead of[[f,g]](u).

A vector fieldF :M×R→S, i.e.(u,t)→ F(u,t) is said to be a symmetry of (4). Then

Ft + [[F,K]] =0. (7) 2.1 Flows of the dKP equation

In this section, we present the iso and non-isospectral flows of the dKP equation. To derive this, we follow Lax triad [20] approach, a convenient way to obtain the non-isospectral flows. The Lax triad system associated with isospectral flows of dKP is given as

LY = {B2,L}, (8a) LTm = {Bm,L}, (8b) B2,T_m −Bm,Y + {B2,Bm} =0, m=1,2, . . . , (8c) where L is the Laurent series in p, defined as L =

p + _∞

n=1Un+1(X,Y,T)p⁻ⁿ and Bm = (L^m)_≥0, projection onto a polynomial in p, dropping negative powers. Here, T consists of infinite set of time variables,T = (T₁,T₂,T₃, . . .). The bracket {·,·}in (8) is the Poisson bracket in 2D phase space(p,X), defined as

{f,g} = ∂f

∂p

∂g

∂X − ∂f

∂X

∂g

∂p. (9)

From (8), we can generate the isospectral flows of the dKP equation and we list the first four as

UT₁ = K1=UX, (10a)

U_T₂ = K₂=U_Y, (10b)

U_T₃ = K₃=3U U_X+ 3

4∂⁻¹U_{Y Y}, (10c) UT₄ = K4=4U UY +2UX∂⁻¹UY

+1

2∂⁻²UY Y Y. (10d)

Here, one can observe that (10c) is the dKP equation.

Now, we consider the Lax triad system for the non- isospectral flows of the dKP system as follows:

LY = {B2,L}, (11a)

LT_m = {Am,L} +1

2L^m−1, (11b)

B2,Tm −Am,Y + {B2,Am} =0, m =1,2, . . . , (11c) where Am is an m degree polynomial in p, given as Am = _m

i=0gˆ_i⁽^m⁾p^m⁻ⁱ. The unknown functions gˆ⁽_i^m⁾ appearing in Am can be explicitly found by using the following asymptotic condition:

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Am|U=0 =Y p^m+1

2X p^m⁻¹, U=(U2,U3, . . .). (12) From (11), we can deduce the non-isospectral flows of dKP and we present the first four as

U_T₁ =σ1=Y K₁, (13a)

U_T₂ =σ2=Y K₂+1

2XU_X+U, (13b)

U_T₃ =σ3=Y K₃+1

2XU_Y +∂⁻¹U_Y, (13c) U_T₄ =σ4=Y K₄+1

2X K₃+2U²+1

2U_X∂⁻¹U + 3

4∂⁻²UY Y. (13d)

In the above flows,σ3becomes the master symmetry of the dKP equation.

2.2 Flows of the mKP equation

The non-isospectral flows of the mKP equation were investigated by Chen [21] using the Lax pair. Now, we extend this set-up in Lax triad approach for the mKP equation, introduced by Fu et al [20,22] for the KP, semidiscrete KP(∂K P)and dKP equations. The interesting fact behind the construction of Lax triad is to overcome the difficulty in understanding the nature of variablesyandt₂. In the isospectral case,t₂is replaced byywhereas in the case of non-isospectral flows,t2can be thought of as completely independent ofy. Now, the isospectral problem related to mKP equation is given as follows:

L˜ψ=λη, (14a)

ψy = ˜B₂ψ, B˜₂ =∂²+2v0∂, (14b) ψt_m = ˜Bmψ, m=1,2, . . . , (14c) whereψis the eigenfunction andλis the spectral parameter withλt_m =0. Here, the Lax operatorL˜ is given by L˜ =∂+

∞ j=0

vj∂⁻^j, (15)

where∂ =∂x, ∂∂⁻¹ =∂⁻¹∂=1 andvj =vj(x,y,t) with the time variables t = (t₁,t₂, . . .) living in the Schwarz spaceS(R^∞), consisting of all rapidly decreas- ing functions. The operator B˜_m is defined as B˜_m = (L˜^m)≥1, strictly positive differential parts ofL˜^m. Now, by the compatibility conditions of (14), we obtain the Lax triad equations associated with the isospectral flows of the mKP as follows:

L˜y = [ ˜B2,L˜], (16a) L˜t_m = [ ˜Bm,L], (16b) B˜2,t_m − ˜Bm,y+ [ ˜B2,B˜m] =0, m=1,2, . . . , (16c) where[,·,]is defined as[M,N] = M N −N M. Next, we consider the spectral problem related to the non- isospectral flows of the mKP equation [21] withλt_m =

1

2λ^m⁻¹as

Lψ =λψ, (17a)

ψy = ˜B₂ψ, (17b)

ψt_m = ˜Amψ, m =1,2, . . . , (17c) where the operator A˜m is of the form

A˜m =

m−1

i=0

h⁽_i^m⁾∂^m⁻ⁱ, (18)

with the undetermined coefficients h⁽_i^m⁾ thought of functions in vj and its derivatives. The compatibility conditions of (17) read as follows:

L˜y = [ ˜B2,L˜], (19a) L˜t_m = [ ˜Am,L] + 1

2L˜^m⁻¹, (19b)

B˜2,t_m − ˜Am,y+ [ ˜B2,A˜m] =0, m=1,2, . . . . (19c) The functions h^(m)_i in A˜m can be found by substituting (18) in (19b) along with the following asymptotic condition [21]:

A˜m|v=0 =y∂^m+ 1 2x∂^m⁻¹ +m−2

4 ∂^m⁻², v=(v0, v1, . . .). (20)

3. Flows of dispersionless mKP (dmKP) equation

3.1 Isospectral flows dmKP

It is understood that the dispersionless hierarchies are derived by the quasiclassical limit [3,4,10,11,23] of the dispersionful hierarchies. In [5,6], Takasaki and Takebe considered the derivation of dKP hierarchy through Sato’s approach. Following this procedure, we take, x = X, y =Y, t =(t₁, t₂, . . .)=(T₁,T₂, . . .)= T and think of

vj

X ,Y

,T

=Vj(X,Y,T)+O().

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When→0 in the Lax operator (15), we get L˜ =∂+

∞ j=1

vj

X ,Y

,T

(∂)⁻^j

=∂+ ∞

j=1

Vj(X,Y,T)+O()

(∂)⁻^j, ∂ =∂X. (21) By considering the Wentzel–Kramers–Brillouin (WKB) asymptotic expansion of the wave function with action ofS[6] as

ψ =exp 1

S(X,Y,T, λ) , →0, (22) and defining p =∂XS, called momentum function, we observe that ⁱ∂ⁱψ → pⁱψ, → 0. Using this limit procedure, we arrive at

L˜ = p+ ∞

j=1

Vj(X,Y,T)p⁻^j. (23) Now, by taking the quasiclassical limit in (16), we get (L˜)Y = [ ˜B2,L˜], (24a) (L˜)T_m = [ ˜Bm,L˜], (24b) B˜2,T_m −B˜m,Y + [ ˜B2,B˜m] =0, m=1,2, . . . .

(24c) As approaches zero, we obtain the following system associated for the isospectral flows of dmKP:

L˜Y = { ˜B2,L},˜ (25a) L˜Tm = { ˜Bm,L},˜ (25b) B˜2,Tm − ˜Bm,Y + { ˜B2,B˜m} =0, (25c) andB˜m = (L˜^m)≥1, projection onto a polynomial in p with strictly positive powers and the Poisson bracket is as defined in (9). Now, we list the first fewB˜m’s:

B˜1 = p, (26a)

B˜2 = p²+2V0p, (26b)

B˜3 = p³+3V0p²+3(V₀²+V1)p, (26c) B˜4 = p⁴+4V0p³+2(3V₀²+2V1)p²

+4(V₀³+3V₁V₀+V₂)p. (26d) From (25c), we consider the equation for the isospectral flows of the dmKP equation as follows:

V0,T_m = 1

2p(B˜m,Y − { ˜B2,B˜m}), m =1,2, . . . . (27)

For various values ofm, we get

V_0,T₁ =V_0,X, (28a)

V₀_,_T₂ =V₀_,_Y, (28b)

V₀_,_T₃ = 3

2(−2V₁_,_XV₀+V₁_,_Y +2V₀_,_XV₁−2V₀_,_XV₀²

+2V0,YV0), (28c)

V0,T₄ =2(−2V2,XV0+V2,Y −6V1,XV₀²+3V1,YV0

+2V₀_,_XV₂−4V₀_,_XV₀³+3V₀_,_YV₁+3V₀_,_YV₀²).

(28d) From (25a), expressingV₁,V₂,V₃, . . .in terms ofV₀in the above expression, we get

VT₁ = ˜K1 =VX, (29a)

V_T₂ = ˜K₂ =V_Y, (29b)

V_T₃ = ˜K₃ = 3

2V_X∂⁻¹V_Y

−3

2V_XV²+ 3

4∂⁻¹V_{Y Y}, (29c)

V_T₄ = ˜K₄ =V_X∂⁻²V_{Y Y} −2V_X∂⁻¹(V V_Y) +2VY∂⁻¹VY −2VYV²+V∂⁻¹VY Y

−∂⁻¹(V_Y²)+1

2∂⁻²VY Y Y −∂⁻¹(V VY Y). (29d) Equation (29c) is the dmKP equation, which together with (29a), (29b), (29d),. . .form the isospectral flows of the dmKP equation.

3.2 Non-isospectral flows of the dmKP equation In (17), we considered the spectral parameterλ, which is a function of infinite set of variables corresponding to the non-isospectral problem of mKP equation. Using the averaging procedure in the variables, we getλTm = (/2)λ^m⁻¹. Along with this fact, we look into system (19) as follows:

L˜Y = [ ˜B2,L˜], (30a) L˜_T_m = [A˜_m,L] +

2L˜^m⁻¹, (30b) B˜_2,T_m −²A˜_m,Y + [ ˜B₂, A˜_m] =0, m =1,2, . . .

(30c) and the undetermined operator (18) with boundary condition (20) becomes

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A˜_m =

m−1

i=0

h_i⁽^m⁾(∂)^m⁻ⁱ, (31a) A˜m|_V=0 =

Y (∂)^m +X

2(∂)^m⁻¹+m−2

4 (∂^m⁻²) , (31b) with V = (V0,V1, . . .)+ O(). Now, by taking the dispersionless limit, i.e. as →0 in (30) and (31) L˜Y = { ˜B2,L},˜ (32a) L˜T_m = { ˜Am,L} +1

2L˜^m⁻¹, (32b)

B˜2,T_m − ˜Am,Y + { ˜B2,A˜m} =0, (32c)

where A˜m =

m−1

i=0

H⁽_i^m⁾p^m⁻ⁱ, (33)

with the condition onA˜matV=0 A˜m|V=0 =Y p^m+1

2X p^m⁻¹, V=(V₀,V₁, . . .). (34) Here, we observe that the undetermined coefficient func- tionH⁽_i^m⁾ becomes a function of Vj and its derivative with respect to X andY. Now, by substituting (33) in (32b) and using (34) for various values ofm, we present the first fewA˜mas

A˜1 =YB˜1, (35a)

A˜2 =YB˜2+ X

2B˜1, (35b)

A˜3 =YB˜3+ X

2B˜2, (35c)

A˜4 =YB˜4+ X

2B˜3+1

2∂⁻¹(V₁)p. (35d) One can observe that from (32c), we obtain the non- isospectral flow generating equation for dmKP as

V0,T_m = 1

2p(A˜m,Y − { ˜B2,A˜m}), m=1,2, . . . . (36)

By consideringm=1,2,3,4, we get V0,T₁ =Y V0,X +1

2, (37a)

V_0,T₂ =Y V_0,Y + X

2V_0,X+ 1

2V₀, (37b)

V_0,T₃ = 3

2Y(−2V_1,_XV₀+V_1,Y +2V_0,XV₁−2V_0,XV₀² +2V₀_,_YV₀)+1

2X V₀_,_Y +3 2V₁+1

2V₀², (37c) V0,T₄ =2Y(−2V2,XV0+V2,Y −6V1,XV₀²+3V1,YV0

+2V0,XV2−4V0,XV₀³+3V0,YV1+3V0,YV₀²) +3

4X(−2V1,XV0+V1,Y +2V0,XV1

−2V0,XV₀²+2V0,YV0)+ 1

4∂⁻¹V1,Y

+1

2V0,X∂⁻¹V1+2V2+4V1V0+1

2V₀³. (37d) Replacing V1,V2,V3, . . . in (37a), (37b), (37c) and (37d) with V0 = V, we arrive at the first four non- isospectral flows of dmKP as follows:

VT₁ = ˜σ1=YK˜1+1

2, (38a)

V_T₂ = ˜σ2=YK˜₂+ X

2V_X+ V

2, (38b)

VT₃ = ˜σ3=YK˜3+ X

2 K˜2− 1

4V²+ 3

4∂⁻¹VY, (38c) VT₄ = ˜σ4=YK˜4+ X

2 K˜3+ 1

4VX∂⁻²VY −1

4VX∂⁻¹V² +V∂⁻¹VY +5

8∂⁻²VY Y − 5

4∂⁻¹(V VY)− 1 2V³.

(38d) Equation (38c) is the non-isospectral dmKP equation, which together with (38a), (38b), (38d),. . .form the non- isospectral flows of the dmKP equation.

4. Algebraic structure of the dmKP equation To study the Lie algebraic structure, we make use of the implicit flow representations on the iso and non- isospectral flows of the dmKP equation withV₀ =V as follows:

Isospectral case:

B˜2[ ˜Kn] = ˜Bn,Y − { ˜B2,B˜n}, (39a)

B˜n|V=0= pⁿ. (39b)

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Non-isospectral case:

B˜2[ ˜σm] = ˜Am,Y − { ˜B2,A˜m}, (40a) A˜m|V=0 =Y p^m+1

2X p^m⁻¹, m≥2. (40b) Lemma 1. If a functionRtakes the form_m₋₁

i=0 aip^m⁻ⁱ, a_i ∈MandZ = Z(V)∈M, then the equation B˜2[Z] =RY − { ˜B2,R}, R|V=0 =0, (41) only admits zero solution Z =0, R =0. Here,B˜2 =

p²+2V p, where we have takenV₀ =V. Proof. The proof is straightforward.

Theorem 1. Suppose that

˜Bm,Bñ = ˜Bm [ ˜Kn] − ˜Bn[ ˜Km] + { ˜Bm,Bñ}, (42a) ˜Bm,Añ = ˜Bm [ ˜σn] − Ãn[ ˜Km] + { ˜Bm,Añ}, (42b) Ãm,Añ = Ãm[ ˜σn] − Ãn[ ˜σm] + { Ãm,Añ}. (42c) Then we have

B˜2[[[ ˜Km,Kñ]]] = ˜Bm,BñY − { ˜B2, ˜Bm,Bñ}, (43a) B˜2[[[ ˜Km,σñ]]] = ˜Bm,AñY − { ˜B2, ˜Bm,Añ}, (43b) B˜2[[[ ˜σm,σñ]]] = Ãm,AñY − { ˜B2, Ãm,Añ}, (43c) and

˜Bm,B˜n|V=0 =0, (44a)

˜Bm,A˜n|V=0= 1

2mp^m+n−2, (44b)

˜Am,A˜n|V=0=1

2(m−n)

Y p^m⁺ⁿ⁻²+1

2X p^m⁺ⁿ⁻³

,

∀m,n ≥2. (44c)

Proof. We only prove eqs (43c) and (44c), and the other equations can be proved in the same manner. Consider the LHS of (43c) and expand it as

B˜2[[[ ˜σm,σ˜n]]] =(B˜2[σm])[σn] −(B˜2[σn])[σm]. (45) First, we take

(B˜2[σm])[σn] =(A˜m,Y − { ˜B2,A˜m})[σn]

= ˜A_m,Y[σn] − { ˜B2[σn],A˜m} − { ˜B2,A˜m[σn]}

= Ãm,Y[σn] − { Ãn,Y,A˜m} + {{ ˜B2,Añ},A˜m}

− { ˜B2,A˜m[σn]}. (46) Similarly

(B˜2[σn])[σm] = Ãn,Y[σm] − { Ãm,Y,Añ}

+ {{ ˜B2,A˜m},A˜n} − { ˜B2,A˜_n[σm]}.

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Subtracting (47) from (46) and using the Jacobi identity, we arrive

B˜2[[[ ˜σm,σñ]]] = Ãm,AñY − { ˜B2, Ãm,Añ}.

Next, observe thatσ˜n|V=0 =0,∀n ≥2. Using this in (42c) with (40b), we can easily prove (44c). Hence, the theorem is proved.

Theorem 2. The isospectral flows { ˜Ki} and non- isospectral flows{ ˜σi}of dmKP equation constitute the following infinite-dimensional Lie algebra of Virasoro type:

[[ ˜Kn,K˜m]] =0, (48a)

[[ ˜Km, σ˜n]] = 1

2mK˜n+m−2, (48b)

[[ ˜σm, σ˜n]] = 1

2(m−n)σ˜n+m−2, ∀m,n≥2. (48c) Proof. We only prove (48c). In the same way (48a) and (48b) could be proved. Now, we take

2ω1 =2[[ ˜σm,σñ]] −(m−n)σ˜m+n−2, (49a) ω2= Ãm,Añ −1

2(m−n)A˜m+n−2. (49b) Using (43c) and (44c) together with implicit flow representations (40a) and (40b), we have

2ω1 =ω2,Y − { ˜B2, ω2}, ω2|V=0 =0. (50) From Lemma1, it is immediate thatω1=0 andω2 =0.

This shows that (48c) is true. Hence, the theorem is proved.

5. Dispersionless Miura map

Let us consider the more generalised form ofLas L= p^m+

m

i=1

am−ip^m⁻ⁱ + ∞

j=1

a₋jp⁻^j, (51) where am−1,am−2, . . . ,a0,a₋1, . . . are functions of X,YandT. Now, takeφ(X,Y,T)as any arbitrary function, independent of pand define [14]

L˜ =e⁻^ad^φL

=L− {φ,L} + 1

2!{φ,{φ,L}}

− 1

3!{φ,{φ,{φ,L}}} + · · ·

= ∞ n=0

1

n!(φX)ⁿ∂ⁿpL, φX = ∂φ

∂X, (52)

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where {·,·}is the Poisson bracket defined in (9). The map defined in (52) is called the dispersionless Miura map [14], which relates the isospectral flows between the dKP and dmKP equations. Now, we review some lemmas discussed in [14].

Lemma 2 [14]. LetL˜be defined as above, then L˜≥1=e⁻^ad^φL≥0−L≥0|p=φX. (53) Lemma 3 [14]. If f(X,Y,T,p)andg(X,Y,T,p)are any two functions, then

e⁻âd^φ{f,g} = {e⁻âd^φf,e⁻âd^φg}.

Lemma 4 [14]. IfL˜is defined as above, then L˜Tq =e⁻^ad^φLTq− {φTq,e⁻^ad^φL}.

Lemma 5 [14]. IfF(X,Y,T,p)andG(X,Y,T,p)are any two functions, then

e⁻âd^φ(F G)=(e⁻âd^φF)(e⁻âd^φG).

Using the above lemmas, we construct the following lemmas and theorems to prove the correspondence between the non-isospectral flows of the dKP and dmKP equations. From Lemma5, we getL˜^k =e⁻^ad^φL^k. Now, we defineB˜k=(L˜^k)_≥1andBk =(L^k)_≥0.

Lemma 6. IfB˜kandBkare defined as above, then B˜k,Tq =e⁻^ad^φBk,Tq − {φTq,e⁻^ad^φBk} −(Bk|p=φX)Tq. Proof. Using Lemma3, we obtain

B˜k,T_q =(e⁻^ad^φBk−Bk|p=φX)T_q

=e⁻^ad^φBk,T_q +φX,T_q

∞ n=0

1

n!φⁿX∂ⁿp⁺¹Bk

−(Bk|p=φX)T_q

=e^−adφBk,T_q − {φT_q,e^−adφBk} −(Bk|p=φX)T_q. Hence, the lemma is proved.

Lemma 7. If f(X,Y,T) and G(X,Y,T,p) are any two functions, then

e^−adφ(f G)= fe^−adφG.

Proof. From (52), the proof is straightforward.

Lemma 8. Suppose, A˜k =YB˜k+1

2XB˜k−1+ ˜hk, Ak =YBk+1

2XBk−1+hk,

where h˜k(X,Y,T,p) andhk(X,Y,T,p) are any two functions withh˜k=e⁻^ad^φhk−hk|p=φX, then

A˜k =e⁻^ad^φAk−Ak|p=φX.

Proof. Using Lemma7, we consider LHS=Y(e⁻^ad^φBk−Bk|p=φX)

+ 1

2X(e⁻^ad^φBk−1−Bk−1|p=φX)+ ˜hk

and from Lemma8, we get LHS=e⁻^ad^φ

YBk+ 1 2XBk−1

−

YBk|p=φX +1

2XBk−1|p=φX

+ ˜hk

=e⁻^ad^φAk−e⁻^ad^φhk−Ak|p=φX+hk|p=φX+ ˜hk

=e⁻^ad^φAk−Ak|p=φX. Hence, the lemma is proved.

Lemma 9. IfA˜kandAkare defined as above, then A˜k,T_q =e⁻^ad^φAk,T_q − {φT_q,e⁻^ad^φAk}

−(Ak|p=φX)T_q.

Proof. Using Lemma9and taking the proof similar with Lemma7, it is easy to prove this lemma.

Theorem 3. IfL˜,A˜kandAkare defined as above, then L˜Tk − { ˜Ak,L} −˜ 1

2L˜^k⁻¹

=e⁻^ad^φ

LTk− {Ak,L} −1 2L^k⁻¹

+ {Ak|p=φX −φTk,e⁻^ad^φL}.

Proof. Using Lemmas5and9, LHS=e⁻^ad^φLTk − {φTk,e⁻^ad^φL}

− {e⁻^ad^φAk−Ak|p=φX,e⁻^ad^φL}

− 1

2e⁻^ad^φL^k⁻¹

=e⁻^ad^φ

LT_k − {Ak,L} −1 2L^k⁻¹

+ {Ak|p=φX −φTk,e⁻^ad^φL}

=RHS.

This completes the theorem.

Theorem 4. IfB˜k,Bk,A˜kandAkare defined as above, then

(8)

B˜2,T_k − ˜Ak,Y + { ˜B2,A˜k}

=e⁻^ad^φ

B2,T_k −Ak,Y + {B2,Ak} + {φY −B2|p=φX,e⁻^ad^φAk}

− {φT_k −Ak|p=φX,e⁻^ad^φB2} +(Ak|p=φX)Y −(B2|p=φX)T_k.

Proof. From Lemmas7,9and 10, we consider LHS=e⁻^ad^φB2,T_k − {φT_k,e⁻^ad^φB2} −(B2|p=φX)T_k

−e^−adφAk,Y + {φY,e^−adφAk} +(Ak|p=φX)Y

+{e⁻^ad^φB2−B2|p=φX,e⁻^ad^φAk−Ak|p=φX}

=e⁻^ad^φ

B2,T_k−Ak,Y+{B2,Ak}

+{φY,e⁻^ad^φAk}

− {φT_k,e⁻^ad^φB2}+(Ak|p=φX)Y−(B2|p=φX)T_k

− {e⁻^ad^φB2,Ak|p=φX} − {B2|p=φX,e⁻^ad^φAk} + {B2|p=φX,Ak|p=φX}.

Observe that {B2|p=φX,Ak|p=φX} = 0 since B2|p=φX

andAk|p=φX are independent of p. Therefore, LHS=e⁻^ad^φ

B2,T_k −Ak,Y + {B2,Ak} + {φY −B2|p=φX,e^−adφAk}

− {φTk −Ak|p=φX,e⁻^ad^φB2} +(Ak|p=φX)Y −(B2|p=φX)T_k

=R.H.S.

Hence, the theorem is proved.

The given theorems and lemmas are discussed for the general form ofLin (51). Now, we consider that the par- ticular form ofLdefined for the dKP equation, becomes subcase of (51). It is easy to identify that the functionsBk

andAk for dmKP also become subcases of the general form ofBk andAk which are discussed in the lemmas and theorems in this section.

Theorem 5. IfL= p+_∞

i=1Ui+1p⁻ⁱ, the associated B2 = p²+2U2andAksatisfy non-isospectral flows of dKP equation(11)with

φT_m =Am|p=φX, φY =B2|p=φX,

thenL˜ =e⁻^ad^φLsatisfies the non-isospectral flows of dmKP equation(32).

Proof. Equation (32a) is common to both iso and non- isospectral flows of the dKP equation and the proof of (32a) was already given in [14]. Hence, we omit the proof of (32a).

Now, using the given condition φT_m = Am|p=φX in Theorem5, the proof of (32b) is obvious. Together with the given conditionφY = B2|p=φX, consider Theorem 4, and we obtain

B˜2,Tm − ˜Am,Y + { ˜B2,A˜m} =(Am|p=φX)Y −(B2|p=φX)Tm. Using the compatibility condition betweenφY andφTm, we conclude that (32c) also holds good. This completes the theorem.

6. Conclusion

In this article, we have constructed Lax triad structures (14) for the mKP equation and subsequently imple- mented quasiclassical limit in (14) to obtain the associated Lax triad equations (25) for the dmKP equation.

In [20,22], Fu et al clearly investigated that generating the non-isospectral flows for (2+1)-dimensional equations using triad formalisms overcome the difficulty in understanding the nature of time variableT₂and space variableY. Making use of this fact, from the triad equations, we derived the isospectral flows {Kl(U)}

and non-isospectral flows{σr(U)}of the dmKP equation. By composing these flows, we have also obtained infinite-dimensional Lie algebras with centreless Kac–

Moody–Virasoro structure. Finally, from Theorems3–5 we demonstrated that the correspondence between the non-isospectral flows of the dKP and dmKP hierarchies through dispersionless Miura map is preserved as in the case of isospectral flows.

Acknowledgements

The authors would like to thank Prof. D J Zhang and Prof. Y Ohta for their valuable suggestions to make this manuscript in complete form.

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