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A new integrable 3+1 dimensional generalization of the Burgers equation

A new nonlinear 3+1 dimensional evolution equation admitting the Lax pair is presented. In the case of one spatial dimension, the equation reduces to the Burgers equation. A method of construction of exact solutions, based on a class of discrete symmetries of the former equation is developed. These symmetries reduce to the Cole-Hopf transformation in one-dimensional limit. Some exact solutions are analyzed, in the physical context of spatial dissipative structures and shock wave dressing.


Introduction
The vast majority of known completely integrable nonlinear evolution PDEs is 1+1 dimensional [1], [10]. Dealing with more than a single spatial dimension, one faces fundamental algebraic and geometric obstructions. This fact accounts for the scarcity of integrable d+1 dimensional systems, for d ≥ 2, available today.
from the physical point of view, the equation describes the balance between nonlinearity and dissipation, [2], rather than dispersion, which is more characteristic of integrable evolution equations.
It comes as a model in various problems of continuous media dynamics, condensed matter physics cosmology, etc., see for instance [4], [3], [11]. Mathematically, the Burgers equation is special, as it can be fully linearized via the Cole-Hopf ( [5], [7]) substitution u = −2ν∇ log Θ, AMS subject classification 35Q53, 35Q58 which reduces it to the heat equation for Θ, with diffusivity ν. For the state of the art on the pure mathematical side of the Burgers equation, see for instance [12] and the references contained therein.
The principal content of this note is as follows. Apart from the natural integrable higher dimensional generalization (2) of the Burgers equation (1), the latter can be used as the foundation to construct higher dimensional integrable PDEs, are not fully linearizable, yet their solutions can be found via Lax pairs. In the recent work [13], the 2+1 dimensional BLP (Boiti, Leon and Pempinelli) system was studied and shown to reduce to an integrable two dimensional generalization of the Burgers equation (1). Here we present a 3+1 dimensional nonlinear equation, which contains dissipative additives and has the following properties: 1. It is a scalar second order evolution PDE with quadratic nonlinearity.

General result
Consider the following equation: where u = u(t, x, y, z, t), and all other parameters are constants.
In one dimensional limits equation (4) reduces to the dissipative Burgers equation. Indeed, if we obtain The latter equation boils down to the Burgers equation after the change t → t ′ , such that or simply letting ρ = 0.
In the same fashion, if u = u(z, t) and η(z, t) = u z (z, t), one has the analog of (6).
Finally, the reduction u = u(y, t) results in a linear equation In view of the above, equation (4) can be viewed as a special non-isotropic three dimensional extension of the Burgers equation. To emphasize this, let w = u x , consider µ = ρ = λ = 0 and rewrite (4) as follows: Our main result is the following theorem Theorem 1 Let u(x, y, z, t) be a particular solution of equation (4) and ψ = ψ(x, y, z, t) be a solution of the following linear equation: Then anyũ klm =ũ klm (x, y, z, t), defined by the formulã is also a solution of equation (4).
Theorem 1 rests on the following fact.
Proposition 2 Equation (4) admits the following Lax pair: ψ t = A(u)ψ, cf. (9), and Verification of this proposition a direct calculation. Further in the note we shall refer to equations (11) and (9) as the Lax, or LA-pair for equation (4), and u as a potential. Observe that the spectral equation (11) of the Lax pair can be rewritten in a more compact form: Also observe that if we redefine the operator , then the compatibility condition of the Lax pair equations (11) and (9) will be reduced to a an identity. Namely the operators Theorem 1 implies the following corollary, which follows after successive iteration of (10).
., N is a set of particular solutions of the Lax pair (11), (9), given the potential u, satisfying equation (4), new solutions of (4) are generated by the following rule: As we have indicated earlier, the Burgers equation (6) results from a 1+1 dimensional reduction of equation (4). Conversely, formulae (10), (13) yield a bona fide generalization of the Cole-Hopf substitution (3). Indeed, setting u ≡ 0, k = l = m = 0 in (10), after differentiation in x, we obtain the one dimensional Cole-Hopf transformation, cf. (3). Clearly, the same can be said about the z variable reduction as well.

Proof of Theorem 1
The theorem will follow from the following lemma. (11), (9) with the potential u, which is a solution of equation (4). Then the functioñ

Lemma 4 Let ψ be a solution of the Lax pair equaitons
also satisfies (11), (9), with the same potential u.
To prove the lemma, observe the validity of the following commutator relations, for i, j = 1, 2, 3: Lemma 4 then follows by induction. Now, to prove Theorem 1, let us introduced three intertwining operators with the quantities f i , g i to be found (naturally, ∂ 1,2,3 = ∂ x,y,z , respectively), such that the operators D i have the following property: for some u i = u i (x, y, z, t), The commutation relations (16) determine the maps u → u i , which come from substitution of (15) into (16). The explicit form of the operators D i is found as follows. Substituting (15) into (16) and equating the components at the same partial derivatives results in a system of nonlinear equations (which is not quoted because of its bulk) whence it follows: where c i -are constants, which will be further assigned zero values. The quantity v = v(x, y, z, t) is a solution of the following nonlinear equation: Therefore (17) or explicitly (18) indicate that for u ≡ 0, the function "−v" satisfies equation (4). Then automatically the quantity ψ = e −u will satisfy the Lax pair equations (11) and (9). In fact, the L-equation, cf. (11), is satisfied as the identity. The A-equation (9) however is satisfied only if u is a solution of (4). On the other hand, by Lemma 4, the functions of the formψ klm defined via relation (14) are also solutions of (11) and (9), with the same potential u. Rewriting them as ψ klm = exp(v klm ) and substituting into (9) one verifies that the quantities v klm are indeed solutions of equation (18). Theorem 1 and formula (10) now follow from the second relation from (17).
Remark. Formula (10) has a countenance similar to the Darboux transformation, which is a standard tool for construction of exact solutions of nonlinear PDEs (usually 1+1, more rarely 2+1 dimensional) which admit Lax pairs, see e.g. [9] for the general theory, applications and references. However (10) does not represent a bona fide Darboux transform for two following reasons.
1. Darboux transforms, representing discrete symmetries of a Lax pair, possess a non-trivial kernel on the solution space of the pair. In other words, there always exists some Lax pair solution which zeroes the transform. This is the property which enabled one Crum, [6], to write down the determinant formulae for successive Darboux transforms. Transformation (10) however does not have this property.
It is known that in addition to the Darboux transform, fairly rich spectral problems, such as the Zakharov-Shabat problem for the Nonlinear Schrödinger equation or its two-dimensional generalization for the Davey-Stewartson equations, admit another discrete symmetry, namely the Schlesinger transform, [8]. The difference between (10) and the latter transformations lies in the fact that for the Schlesinger transform, the potential transformation rules can be locally defined without using the Lax pair solution ψ, while (10) certainly does so. This feature is shared by (10) and the standard Darboux transformation.
2. To construct exact solutions of nonlinear PDEs via the Darboux transform, one has to take advantage of the solution of the full Lax pair as a system of equations. In order to get (10) however, we have used the solution of the A-equation (9) only. The L-equation (11) has been used only as a tool to prove Theorem 1. To this effect, transformation (10) combines essential features of the Darboux and Cole-Hopf transformations.

Some exact solutions
Let us use the above formalism in order to construct some exact solutions of equation (8) (which is the equation for u x , where u is a solution of equation (4) with µ = λ = ρ = 0). We consider equation (8), because it appears to be a closer relation of the Burgers equation and is likely to be interesting from the physics point of view. Example 4.1. As the first example let us consider dressing on the vacuum background u ≡ 0. In this case the function ψ(x, y, z, t) = a 2 x 2 + b 2 y 2 + c 2 z 2 + 2 a 1 a 2 + a 2 c 2 t + s 2 , where a, b, c, s are some real constants, is clearly a solution of the Lax pair equations (11) and (9). Substituting (19) in (10) we derive theũ klm . After differentiating it with respect to x and choosing k = l = m = 0, we obtain a solution w of equation (8) as follows: Physically, solution (20) describes a rationally localized impulse, vanishing as t → +∞. To ensure that (20) is non-singular for t ≥ 0, one should impose the inequality a 2 ≥ −a 2 a 1 /c 2 on the coefficients. Moreover, if a 2 = −a 2 a 1 /c 2 , the solution in question is rationally localized and stationary.
Remark. The fact that there exists a localized stationary solution in an equation containing dissipative terms may appear somewhat paradoxical from the point of view of physics. However solution (20) is stationary only if a 1 a 2 < 0. One can see that these constants appear in the dissipative terms of equation (8), a 1 characterizing the dissipation along the x and a 2 along the z axes. The fact that a 1 and a 2 should have different signs implies that dissipation in the direction of one axis is compensated by instability in the direction of the other. The balance of these two effects results in the stationary solution, which can be regarded as a three-dimensional dissipative structure. A similar situation occurs with two-dimensional stationary solutions of the BLP equation, cf. [13].
Clearly the quantity U (x, t) = 2νξ(x, t) solves the one dimensional Burgers equation As a starting point let us take a shock wave solution of (24), e.g.
where a and v are constants. Seek a solution of (9) as a superposition where η = x − vt, while the 2N quantities β k and γ k can in general be functions of η. For simplicity however let us render them constants to be determined. Substitution of (26) into (9) yields N linear equations for A k (η): In equation (27) the indices for the quantities A k , β k , γ k and σ k ≡ b 1 β k + b 2 γ k have been omitted, while the quantity ξ came from (25), dot standing for differentiation with respect to η. Equation (27) can be simplified further. To do so, let us introduce a new independent variable: In terms of q equation (27) becomes ν q 2 − aq 2 A ′′ (q) + σ q 2 − aq A ′ (q) + (δ − σq) A(q) = 0, where δ = a 2 γ 2 − σξ 0 , while prime stands for differentiation in q. The latter equation can be simplified further via a substitution reducing (28) to the following equation for W (q): In a particular case of the dependence one can solve (9) explicitly as where W k and V k are arbitrary constants. Substitution of (31) into (10) yields exact solutions of equations (4), (8). The described procedure can be regarded as the shock wave dressing.