CONSERVATION LAWS AND LINE SOLITON SOLUTIONS OF A FAMILY OF MODIFIED KP EQUATIONS

. A family of modiﬁed Kadomtsev-Petviashvili equations (mKP) in 2+1 dimensions is studied. This family includes the integrable mKP equa- tion when the coeﬃcients of the nonlinear terms and the transverse dispersion term satisfy an algebraic condition. The explicit line soliton solution and all conservation laws of low order are derived for all equations in the family and compared to their counterparts in the integrable case.

(3) This family (2) can be expected to have wider applications in physical situations where the integrability constraint (3) does not hold.
The main goals will be to determine the line soliton solutions and the low-order conservation laws of the mKP family (2) and to compare the results to the integrable mKP case. First, in section 2, the mKP family (2) is formulated as a local PDE by use of the potential w, with u = w x .
Next, in section 3, all low-order conservation laws of the mKP family in potential form are derived. The admitted conservation laws are found to consist of two topological charges, for arbitrary κ, plus two additional topological charges in the integrable case (3) of the mKP equation. Computational aspects are summarized in an appendix. Unlike the KP equation, the mKP family in potential form does not admit any non-trivial dynamical conserved quantities.
In section 4, all line solitons u = U (x + µy − νt) are derived, where the parameters µ and ν determine the direction and the speed of the line soliton. The basic kinematical properties of these solutions are discussed and compared to the mKP line solitons. In particular, there is a significant qualitative difference between the (extended) mKP case where αγ > 0 and the opposite case where αγ < 0.
Finally, a few concluding remarks are made in section 5.
Hence, we will consider the mKP family in the scaled potential form 0 = w tx + (σ 1 w 2 x + κw y )w xx + w xxxx + σ 2 w yy , σ 1 , σ 2 = ±1, κ > 0, (7) which is a one-parameter family where κ (rescaled) is an arbitrary positive constant. We will refer to σ 1 = 1 as the focussing case, and σ 1 = −1 as the defocussing case; this distinction will be significant when line soliton solutions are considered.
The corresponding scaled mKP family has the form in which the scaled mKP equation is the case namely, 3. Conservation laws. Conservation laws are of basic importance for nonlinear evolution equations because they provide physical, conserved quantities as well as conserved norms. A general treatment of how to find conservation laws is given in Refs. [14,4,6,2]. For the mKP family in potential form (7), a local conservation law is a continuity equation D t T + D x X + D y Y = 0 (11) holding for all solutions w(x, y, t) of equation (7), where T is the conserved density, and (X, Y ) is the spatial flux, which are functions of t, x, y, w, and derivatives of w. When solutions w(x, y, t) are considered in a given spatial domain Ω ⊆ R 2 , every local conservation law yields a corresponding conserved integral satisfying the global balance equation wheren is the unit outward normal vector of the domain boundary curve ∂Ω, and where ds is the arclength on this curve with clockwise orientation. This global equation (13) has the physical meaning that the rate of change of the quantity (12) on the spatial domain is balanced by the net outward flux through the boundary of the domain. A conservation law is locally trivial [14,6,2] if, for all solutions w(x, y, t) in Ω, the conserved density T reduces to a spatial divergence D x Ψ x + D y Ψ y and the spatial flux (X, Y ) reduces to a time derivative −D t (Ψ x , Ψ y ) modulo a spatial curl (D y Θ, −D x Θ), since then the global balance equation (13) becomes an identity. Likewise, two conservation laws are locally equivalent [14,6,2] if they differ by a locally trivial conservation law, for all solutions w(x, y, t) in Ω. We will be interested only in locally non-trivial conservation laws.
Any non-trivial conservation law (11) can be expressed in an equivalent characteristic form [14,6,2] which is given by a divergence identity holding off of the space of solutions w(x, y, t). For the mKP family in potential form (7), conservation laws have the characteristic form whereT ,X,Ỹ , and Q are functions of t, x, y, w, and derivatives of w, and where the conserved densityT and the spatial flux (X,Ỹ ) reduce to T and (X, Y ) when restricted to all solutions w(x, y, t) of equation (7). This divergence identity is called the characteristic equation for the conservation law, and the function Q is called the conservation law multiplier. Note that, when a conservation law is non-trivial, Q will be non-singular when it is evaluated on any solution w(x, y, t).
From the characteristic form (14), all multipliers Q are determined by applying the Euler operator [14,6,2] E w with respect to w, where this operator annihilates a function of t, x, y, w, and derivatives of w iff the function is given by a total divergence. Hence, multipliers Q are the solutions of the determining equation holding off of solutions of equation (7). All multipliers Q up to any specified differential order with respect to w can be found by splitting the determining equation (15) with respect to all variables that do not appear in Q, yielding an overdetermined system to be solved for Q. A variety of methods [17,3,6,2] can be used to derive the conserved densityT and spatial flux (X,Ỹ ) arising from any given multiplier Q.
Here we will explicitly find all low-order conservation laws of the mKP family in potential form (7) by determining all multipliers of differential order at most three where ∂ k w denotes the set of all partial derivatives of order k ≥ 0 of w. Some remarks on the computations are provided in the appendix.
Theorem 3.1. (i) The low-order conservation laws admitted by the mKP family in potential form (7) for arbitrary κ are given by (up to equivalence) where f 1 (t), f 2 (t) are arbitrary functions.
(ii) Additional low-order conservation laws are admitted only when κ 2 = 2, σ 2 σ 1 = −1. These conservation laws consist of (up to equivalence): where f 3 (t), f 4 (t) are arbitrary functions. where the arbitrary function of t appearing in (X, Y ) can be omitted without loss of generality. These charges can be used to introduce corresponding spatial potential systems: where φ is a potential. From conservation laws (21) and (22), we obtain, respectively, and These two spatial potential systems hold for arbitrary κ.
In the case of the mKP equation (9), from conservation law (23), we have while from conservation law (24), we have (30) It is surprising that, in contrast to the situation for the KP equation considered in Ref. [5], there are no dynamical (non-topological) conserved quantities admitted by the mKP family in potential form (7).

Line soliton solutions. A line soliton is a solitary wave in two dimensions,
with U, U , U , etc. → 0 as |x|, |y| → ∞, (32) where the parameters µ and ν determine the direction and the speed of the wave.
A more geometrical form for a line soliton is given by writing x + µy = (x, y) · k with k = (1, µ) being a constant vector in the (x, y)-plane. The travelling wave variable can then be expressed as where the unit vectork = (cos θ, sin θ), tan θ = µ (34) gives the direction of propagation of the line soliton, and the constant c = ν/|k|, |k| 2 = 1 + µ 2 (35) gives the speed of the line soliton. Since the direction of propagation stays the same under changing the direction angle by ±π while simultaneously changing the sign of the speed, we will take the domain of θ to be − 1 2 π < θ ≤ 1 2 π. We will now derive the explicit line soliton solutions (31) for the scaled mKP family (8). It will be convenient to use the coordinate form of the travelling wave variable ξ = x + µy − νt for this derivation. Thus, we have u x = U , u y = µU , u t = −νU , and so on, while ∂ −1 x u y = µ∂ −1 ξ U = µU by the solitary wave conditions (32). Substitution of the line soliton expression (31) into equation (8) yields a nonlinear fourth-order ODE We can straightforwardly integrate this ODE twice to obtain a second-order ODE, and then we can use an integrating factor U to obtain a separable first-order ODE after use of conditions (32).
(38b) With respect to the x axis, the angle θ of the direction of motion of the line soliton is given by arctan(µ), while the speed of the line soliton is given by ν/ 1 + µ 2 . These two parameters obey the kinematic condition (38b) which depends crucially on the signs of σ 1 and σ 2 .

4.1.
Subfamily containing the mKP equation. To begin, we examine the case σ 1 σ 2 = −1, where the mKP family constitutes a one-parameter (κ) extension of the mKP equation. The line soliton (38) in this case is given by with the kinematic conditions In the focussing case, σ 1 = 1, there is a minimum negative speed c > −µ 2 / 1 + µ 2 which is a function of the angle θ = arctan(µ), while there is no maximum speed. These minimum and maximum speeds are independent of κ. In the defocussing case, σ 1 = −1, the speed has both a positive minimum and maximum, µ 2 / 1 + µ 2 < c < (1 + 1 6 κ 2 )µ 2 / 1 + µ 2 , which depends on κ where κ 2 = 2 recovers the mKP equation. The kinematically allowed region in (c, θ) is plotted in Fig. 1a for the focussing case and in Figs. 1b and 2 for the defocussing case.
In the focussing case, the line soliton has height h = 6(ν + µ 2 )/( (κ 2 + 6)µ 2 + 6ν + κµ) and width (proportional to) w = 2/ ν + µ 2 . These expressions can be inverted and substituted into the line soliton (40), yielding for the profile of the line soliton in terms of its height and width. Notice that it does not depend on κ and hence it is the same as for the ordinary mKP line soliton. Plots of this profile are shown in Fig. 3a.
In the focussing case, the line soliton has height h = 6(ν + µ 2 )/( (κ 2 − 6)µ 2 − 6ν + κµ) and width (proportional to) w = 2/ ν + µ 2 . The profile of this line soliton in terms of its height and width is the same as the profile (43) in the (extended) mKP case. Plots are shown in Fig. 3a.  Similarly in the defocussing case, the height and width of the line soliton are h = 6(ν − µ 2 )/( (κ 2 − 6)µ 2 + 6ν + κµ) and w = 2/ ν − µ 2 . The profile in terms of its height and width is the same as the profile (44) in the (extended) mKP case. Plots are shown in Fig. 3b. 5. Concluding remarks. We have obtained in explicit form all of the line soliton solutions and all of the low-order conservation laws for the family (2) of mKP equations.
When αγ > 0, the family includes the well-known integrable mKP equation (1). The line solitons in this case have qualitatively similar kinematic properties to the mKP line soliton.
In contrast, when αγ < 0, the family is a strict generalization of the mKP equation (1). The kinematic properties of the line solitons in this case are qualitatively different compared to the mKP line soliton. In particular, in the focussing case α < 0, the speed of the line soliton is strictly positive, and in the defocussing case α > 0, the speed will be strictly negative if κ 2 < 6α|γ|.
Our results can be used as a starting point to investigate the stability of the line soliton solutions and to determine whether their stability depends on integrability condition (3).
Acknowledgments. S.C.A. is supported by an NSERC research grant and thanks the University of Cádiz for additional support during the period when this work was initiated.
Appendix. The determining equation (15) for multipliers (16) with differential order less than four splits with respect to the set of variables {∂ 4 w, ∂ 5 w, ∂ 6 w}. We have carried out the setting up and splitting of the determining equation by using Maple. This yields an overdetermined system consisting of 3356 equations to be solved for Q as well as for κ = 0, with σ 2 1 = σ 2 2 = 1. Solving the system is a nonlinear problem because Q appears linearly in products with κ. We use the Maple package 'rifsimp' to find the complete case tree of solutions. For each solution case in the tree, we solve the system of equations by using Maple 'pdsolve' and 'dsolve', and we check that the solution has the correct number of free constants/functions and satisfies the original overdetermined system. Finally, we merge overlapping cases by following the method explained in Ref. [15].