Lump solutions of the fractional Kadomtsev–Petviashvili equation

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Introduction
The present paper is devoted to the study of fully localized solitary solutions (also known as lump solutions) of the fractional Kadomtsev-Petviashvili (fKP) equation Here the real function u = u(t, x, y) depends on the spatial variable (x, y) ∈ R 2 and the temporal variable t ∈ R + . The linear operator D α x denotes the Riesz potential of order α ∈ R in x-direction, which is defined by multiplication with | · | α on the frequency space, that is where the operator F denotes the extension to the space of tempered distributions S ′ (R n ) of the Fourier transform on the Schwartz space S(R n ) with inverse F −1 (f ) := 1 2π F (f )(−·). We also writef := F (f ). The operator ∂ −1 x is defined as a Fourier multiplier operator on the x-variable as F (∂ −1 x f )(t, ξ 1 , ξ 2 ) = 1 iξ1f (t, ξ 1 , ξ 2 ). In the case α = 2 equation (1.1) becomes the classical Kadomtsev-Petviashvili (KP) equation which was introduced by Kadomtsev & Petviashvili [19] as a weakly two-dimensional extension of the celebrated Korteweg-de Vries (KdV) equation, u t + uu x + u xxx = 0, which is a spatially one-dimensional equation appearing in the context of small-amplitude shallow waterwave model equations. The KP equation comes in two versions: For σ = −1 it is called KP-I and for σ = 1 it is called KP-II. Roughly speaking, the KP-I equation represents the case of strong surface tension, while the KP-II equation appears as a model equation for weak surface tension. Analogously to the classical case, the fKP equation is a two-dimensional extension of the fractional Korteweg-de Vries (fKdV) equation and (1.1) is referred to as the fKP-I equation when σ = −1 and as the fKP-II equation when σ = 1. Notice that for α = 1 in (1.1) we recover the KP-version of the Benjamin-Ono equation. During the last decade there has been a growing interest in fractional regimes such as the fKdV or the fKP equation (see for example [1,4,14,15,16,20,21,25,26,28,29,31] and the references therein). Even though most of these equations are not derived by asymptotic expansions from governing equations in fluid dynamics, they can be thought of as dispersive corrections.
Formally, the fKP equation does not only conserve the L 2 -norm but also the energy Notice that the corresponding energy space includes a zero-mass constraint with respect to x. We refer to [25] for derivation issues and well-posedness results for the Cauchy problem associated with (1.1). The fKP equation is invariant under the scaling u λ (t, x, y) = λ α u(λ α+1 t, λx, λ A lump solution is a traveling-wave solution which decays to 0 as |(x, y)| → ∞.

Main results
Our aim is to study the existence and spatial decay of lump solutions for the fKP equation. Since it is known [9,25] that the fKP-II equation for any α as well as the fKP-I equation for α ≤ 4 5 do not admit any lump solutions in X α 2 ∩ L 3 (R 2 ), the study of this paper is concerned with traveling waves for the fKP-I equation for α > 4 5 . We prove the following two main theorems. Moreover, we study lump solutions and some of their properties numerically. Theorem 1.1 (Existence of lump solutions). For any 4 5 < α there exists a lump solution φ ∈ X α 2 of (1.2) with σ = −1.
The classical KP-I equation possesses an explicit lump solution of the form We would like to point out that de Bouard & Saut studied the existence of of lump solutions for the generalized KP-I equation where p = m/n ≥ 1, m, n relatively prime and n odd in [9]. Furthermore, in their continuation paper [10], de Bouard & Saut investigated the symmetry and decay of lump solutions for (1.4) and showed that for all p ≥ 1 the decay is quadratic. Our studies follow a similar approach as in [9,10]. However, special attention needs to be given to the nonlocal operator D α x . While many proofs can be adapted with a bit more technical effort due to the nonlocal operator, the result on decay of lump solutions in the supercritical case 4 5 < α < 4 3 (which includes the Benjamin-Ono KP version for α = 1) needs a modified approach, since in the supercritical case the symbol of an operator related to the linear dispersion is no longer L 2 -integrable.
On the existence result: We give a brief outline of the existence proof for lump solutions of (1.4) in [9] by variational methods, since we will be using the same strategy to prove existence of lump solutions of the fKP-I equation (1.1). First consider the constrained minimization problem Via the Lagrange multiplier principle one finds (after rescaling) that solutions of the constrained minimization problem I µ are lump solutions of (1.4). The task is then to prove existence of solutions of I µ and this is achieved using the concentration-compactness theorem (cf. Theorem 2.1). The variational formulation associated with I µ has several good properties. The functional being minimized is just the norm of the space Y . It is therefore immediate that it is coercive, bounded from below and weakly lower semi-continuous; properties which are all advantageous in the context of minimization problems, see [35,Theorem 1.2]. Furthermore, since the norm is homogeneous, it is easily shown that I µ is subadditive as a function of µ and this property is essential in proving that the dichotomy scenario in the concentration-compactness theorem does not occur.
We prove Theorem 1.1 by extending the strategy of [9], outlined above, to the fractional case. Generally speaking, the fractional derivative and the fact that we are allowing for weak dispersion makes the proof of Theorem 1.1 more technical than its classical local counterpart (α = 2). A key ingredient in the proof is the anisotropic Sobolev inequality [25, Lemma 1.1] (see also Proposition 2.2 (ii)), which in particular says that for 4 5 ≤ α, the space X α 2 is continuously embedded in L 3 (R 2 ). This result is what determines the values of α for which we can prove existence of solitary waves. In fact, for α ≤ 4 5 there exist no nontrivial lump solutions of for the fKP-I equation in X α 2 ∩ L 3 (R 2 ) [25, Proposition 1.2]. We would like to mention that there are several existence results on lump solutions using variational approaches for other two-dimensional equations. The full water-wave problem admits lump solutions both for strong [18,5] and weak [6] surface tension. In the strong surface tension case the lump solutions can be approximated by rescalings of KP-I lumps, while in the weak surface tension case the lump solutions can be approximated by rescalings of Davey-Stewartson type solitary waves. The full dispersion KP (FDKP) equation was introduced in [24, chapter 8] as a model for weakly transversal three dimensional water-waves which preserves the dispersion relation of the full water-wave problem. Just as for the classical and fractional KP equation, the FDKP equation can be considered for both strong (FDKP-I) and weak (FDKP-II) weak surface tension. In [11] it was shown that the FDKP-I equation admits lump solutions and later on in [12] it was shown that also the FDKP-II equation possesses lump solutions. This is in contrast to the fKP-II equation, which does not admit any lump solutions [25]. Just like for the full water-wave problem, in the strong surface tension case the lump solutions can be approximated by rescalings of KP-I lumps, while in the weak surface tension case the lump solutions can be approximated by rescalings of Davey-Stewartson type solitary-waves.
On the decay result: The proof of Theorem 1.2 on the decay properties of lump solutions is closely related to that of [3, where the symbol m α is given by 3. An immediate consequence of the discontinuity of the symbol m α at the origin is that any nontrivial, continuous lump solution of (1.2) decays at most quadratically. Let us assume for a contradiction that φ is a nontrivial, continuous lump solution, which decays at infinity as | · | −δ for some δ > 2. Then φ ∈ L 1 (R 2 ), which implies that the Fourier transformation of φ is continuous. Butφ = 1 2 m αφ 2 cannot be continuous at the origin, sinceφ 2 (0, 0) > 0 and m α is discontinuous at the origin. We conclude that the singularity of the symbol m α induced by the transverse direction forces the decay of any nontrivial, continuous lump solution to be at most quadratic. The idea is to study the kernel function K α and to show that it has exactly quadratic decay at infinity (independent of α). Then the decay properties of K α are used to show that also φ decays quadratically at infinity.
On the numerics: We conduct numerical experiments to observe the lump solutions and some of their properties. For this purpose, we generate the solutions numerically by using Petviashvili iteration method. The method was proposed first by Petviashvili [34] to compute the lump solutions of the KP-I equation.
The convergence of the method for the KP-equation was later discussed in [32] and now it is widely used to numerically evaluate traveling wave solutions of evolution equations (see for example [2,30,33] and the references therein).
Applying the Fourier transform to (1.2) with respect to the space variables (x, y) we obtain An iterative algorithm for the equation (1.5) can be proposed as where φ n is the n th iteration of the numerical solution. Since (1.6) is generally divergent the Petviashvili iteration is given as by introducing the stabilizing factor .

Notation and organization of the paper
We first introduce a notation, which is frequently used in the sequel. Let f and g be two positive functions. We write f g (f g) if there exists a constant c > 0 such that f ≤ cg (f ≥ cg). Moreover, we use the notation f g whenever f g and f g.
We conclude the introduction by the organization of the paper: In Section 2 we prove existence of lump solutions for the fKP-I equation (Theorem 1.1) via a variational approach. We also present numerically generated lump solutions and observe the cross-sectional symmetry of the solutions numerically. Section 3 is devoted to the proof of Theorem 1.2, which relies upon a careful study of the decay and regularity of the kernel function K α . The appendix contains some technical results which are needed for the analysis in Section 3.

Existence of solitary solutions
We consider the (rescaled) traveling wave fKP-I equation: Equation (2.1) can be realized as a constrained minimization problem. Indeed, let which we study in the space X α 2 and consider the constrained minimization problem In order to find nontrivial solutions we assume that µ = 0 and without loss of generality we may further assume that µ > 0. Let φ be a solution of (2.2). Then there exists a Lagrange multiplier λ ∈ R such that we find thatφ satisfies the equationφ If λ > 0, this is equation (2.1) and if λ < 0 we apply the transformationφ → −φ and again recover (2.1). Therefore, in order to prove the existence of the solutions of equation (2.1), we will prove existence of solutions of the constrained minimization problem (2.2).
In the sequel, let us fix µ > 0 (this will ensure that I µ > 0, see Corollary 2.3) and let {φ n } n∈N ⊂ X α 2 be a minimizing sequence such that N (φ n ) = µ and lim n→∞ L(φ n ) = I µ . We aim to show that there exists a subsequence (not relabeled) of {φ n } n∈N , which converges to a function φ ∈ X α 2 satisfying L(φ) = I µ and N (φ) = µ. Let us set e n = 1 2 and note that L(φ n ) = R 2 e n d(x, y).
We will use the following version of the concentration-compactness theorem for the sequence {e n } n∈N and show that the concentration scenario occurs. This is then used to construct a convergent subsequence of {φ n } n∈N , converging to a solution of (2.2) admits a subsequence, denoted again by {e n } n∈N , for which one of the following phenomena occurs: • Concentration: There exists a sequence {x n } n∈N ⊂ R d with the property that for each ε > 0, there exists r > 0 with Interpreting I as a mass, Theorem 2.1 says that {e n } n∈N admits a subsequence for which one of the following occur: The the mass spreads out in R n (vanishing), it splits into two parts (dichotomy) or the mass is uniformly concentrated in R n (concentration).

Preliminary results
In this subsection we will gather some of the results we need in order to apply Theorem 2.1.
In particular X α Corollary 2.3. The minimum I µ is positive.
Proof. By Proposition 2.2 we have that

Corollary 2.3 ensures that the minimizer is not given by the trivial solution.
Lemma 2.4. For any α > 0, the space X α 2 is compactly embedded in L 2 loc (R 2 ). Proof. The proof follows essentially the lines in [10, Lemma 3.3]. We include it here for the sake of completeness.
We will show that for any R > 0 there exists a subsequence {φ n k } ∞ n=1 , which converges in L 2 (B R ), where B R is the ball of radius R centered at the origin in R 2 . Let ϕ n = ∂ −1 x φ n . Since we are only interested in convergence in L 2 (B R ), we may assume that ϕ n is supported on B 2R by multiplying ϕ n with a smooth cutoff function ψ such that ψ ≡ 1 in B R and supp(ψ) ⊂ B 2R . It follows then that φ n is supported on B 2R as well. Since {φ n } ∞ n=1 is bounded in X α 2 we can extract a subsequence, which we still denote by {φ n } ∞ n=1 , such that φ n ⇀ φ, for some φ ∈ X α 2 . Moreover, by replacing φ n with φ n − φ, we may assume that φ = 0. Our aim is then to show that We proceed to estimate each integral on the right-hand side of (2.4) separately. For the third integral we can write and for the second one . From these estimates we conclude that, given ε > 0 we can choose R 1 sufficiently large such that In order to deal with the first integral, we first note that since φ n ⇀ 0 in X α 2 , we havê Next we prove that I µ is subadditive as a function of µ, a property which will be crucial when proving that the dichotomy scenario in Theorem 2.1 does not occur.
Proposition 2.5. The infimum I µ is strictly increasing and subadditive as a function of µ, that is from which the statement in the proposition directly follows.
When applying Theorem 2.1 we will be taking integrals over bounded domains. It is therefore useful to consider the norm · α 2 restricted to a bounded domain Ω ⊂ R 2 : We also make the following definition.
When proving that the vanishing scenario does not occur we will make use of the following result.
x φ and let ψ be a smooth cutoff function supported on a bounded Then, The proposition can be generalized to domains, which are given by disjoint unions of type Ω.
Proof. We have that .
We consider each of these terms separately. The first term can be estimated as where we used Poincaré's inequality and the definition ϕ = ∂ −1 x φ. Hence, and in the same way we find , h 2 (y))} and set Ω y := (h 1 (y), h 2 (y)). By the Leibniz' rule for fractional derivatives (see e.g. [17, Theorem 7.6.1]), we can estimate dy.
Using that ψ is a smooth function, we conclude by Poincaré's inequality that Gathering (2.5), (2.6), and (2.7), we have shown that Eventually, when excluding the dichotomy scenario we will make use of the following lemma, which provides a Poincaré-like inequality.
where A 2R,R ⊂ R 2 denotes the annulus centered at the origin of radii 2R > R.

Existence of minimizers
Let {φ n } n∈N ⊂ X α 2 be a minimizing sequence for the constrained minimization problem (2.2), that is, N (φ n ) = µ and lim n→∞ L(φ n ) = I µ . We will apply Theorem 2.1 to the sequence Recall that We will show in Proposition 2.9 and Proposition 2.10 that the vanishing and dichotomy scenarios in Theorem 2.1 does not occur and then use the concentration scenario to construct a convergent subsequence of {φ n } n∈N , converging to a solution φ of (2.2).
Proposition 2.9 (Excluding "vanishing"). No subsequence of {e n } n∈N has the vanishing property in Theorem 2.1.
Proof. Assume for a contradiction that vanishing does occur, that is for each r > 0. Let us cover R 2 with balls B 1,j , j ∈ N, of radius 1 such that each point in R 2 is contained in at most three balls. Let {ψ j } n∈N be a smooth partition of unity such that supp(ψ j ) ⊂ B 1,j . Using Proposition 2.2 (ii) and Proposition 2.7 we find By letting n → ∞ we get N (φ n ) → 0, which contradicts the fact that N (φ n ) = µ > 0.
Proposition 2.10 (Excluding "dichotomy"). No subsequence of {e n } n∈N has the dichotomy property in Theorem 2.1.
Proof. Throughout the proof we will use B R to denote the ball in R 2 centered at the origin of radius R > 0 and A R1,R2 to denote the annulus centered at the origin of radii R 1 > R 2 > 0.
Assume for a contradiction that the dichotomy scenario in Theorem 2.1 occurs, that is there exist sequences e n dx = I * . (2.8) We will show that this leads to a contradiction, by proving that provided (2.8) holds, there exists two sequences {ω n } n∈N , which have in the limit n → ∞ disjoint support and where ω n = φ n (· + (x n , y n )) is the shift of φ n by (x n , y n ). We shift the function φ n for reasons of convenience in order to work with balls and annuli centered at the origin instead of at (x n , y n ). Notice that if (i) and (ii) hold we obtain a contradiction due to the subadditivity of the I µ stated in Proposition 2.5: Set and µ i := lim n→∞ µ i,n for i = 1, 2. Then (i) implies that µ 1 + µ 2 = µ, since N (ω n ) = µ for all n ∈ N. First we show that µ 1 = 0. If µ 1 = 0, then µ 2 = µ. By setting since lim n→∞ µ µ2,n = 1. But then by using (ii) we obtain which is a contradiction. Hence, µ 1 = 0 and similarly we find µ 2 = 0. Thus, |µ i | > 0 for i = 1, 2 and we can define the rescaled functionsω We are left to show that there exists two sequences {ω n } n∈N , which have in the limit n → ∞ disjoint support and satisfy (i), (ii). To this end, let ϕ n = ∂ −1 x φ n and let χ : R 2 → [0, 1] be a smooth cutoff function such that χ(x, y) = 1 for |(x, y)| ≤ 1 and χ(x, y) = 0 for |(x, y)| ≥ 2. Next let σ n := ϕ n (· + (x n , y n )) and σ (1) n := χ 1n σ n,A2M n ,Mn , σ (2) n := χ 2n σ n,A Nn,Nn /2 , where Eventually, we define ω n := ∂ x σ n , ω (i) n := ∂ x σ (i) n , i = 1, 2. We remark that by definition ω n = φ n (· + (x n , y n )). Furthermore, See Figure 1 for an illustration of the supports for ω In what follows we will prove that the statements (i) and (ii) hold true.
where we used that ω n = ω n on R 2 \ B Nn . The term AN n,Mn ω 3 n d(x, y) tends to zero in view of (2.10) and where we used that ∂ x σ n,A2M n ,Mn = ∂ x σ n = ω n . Using Lemma 2.8, the smoothness of χ 1,n , and (2.9), the first term on the right-hand side above can be estimated by as n → ∞. The second term tends to zero as n → ∞ due to (2.10) and the boundedness of χ 1,n . We conclude as n → ∞, where the second assertion can be shown in the same way. Together with (2.10), equation (2.11) finishes the proof of statement (i).
(ii) We proceed to investigate the limit and show that lim n→∞ L(ω (1) n ) = I * . First consider (2.13) Since ∂ x χ 1,n has support in A 2Mn,Mn a similar argument as in (2.12) shows 1 M n ∂ x χ 1n σ n,A2M n ,Mn L 2 (R 2 ) ω n α 2 ,A2M n ,Mn → 0, as n → ∞, (2.14) by using Lemma 2.8 and (2.9). Hence, we find that both the first and second term on the right-hand side of (2.13) tend to zero as n → ∞. For the third term on the right-hand side of (2.13) we have where we used that supp(χ 1,n ) ⊂ B 2Mn and χ 1,n = 1 on B Mn . Due to (2.9) we find In the same way we can show x ω (1) .

(2.17)
We show first D α 2 x ∂ x χ 1,n σ n,A2M n ,Mn L 2 (R 2 ) M n ω n X α 2 ,A2M n ,Mn , which implies by (2.9), the smoothness of χ 1,n and the boundedness of ω n in X α 2 that the first two terms on the right-hand side of (2.17) tend to zero as n → ∞. As in the proof of Proposition 2.7, an application of Leibniz' rule for fractional derivatives yields D α 2 x ∂ x χ 1,n σ n,A2M n ,Mn L 2 (R 2 ) σ n,A2M n ,Mn L 2 (A2M n ,Mn ) + D α 2 x σ n,A2M n ,Mn L 2 (A2M n ,Mn ) ≤ 2 σ n,A2M n,Mn L 2 (A2M n ,Mn ) + D α 2 x ω n L 2 (A2M n ,Mn ) , where we used interpolation and ∂ x σ n,A2M n ,Mn = ω n in the last inequality. Using Lemma 2.8, the first term on the right-hand side above can by estimated by M n ω n X α 2 ,A2M n,Mn in the same spirit as in (2.12), while the second term is bounded by ω n X α 2 ,A2M n ,Mn . Hence, (2.18) holds true and the first two terms in (2.17) tend to zero as n → ∞.
Again, by applying Leibniz' rule for fractional derivatives and using that χ 1,n is smooth, we find that D α 2 x (χ 1,n ω n ) 2 In the same way we can obtain which proves statement (ii).
This concludes the proof of the proposition.
By Proposition 2.9 and Proposition 2.10 the scenarios of "vanishing" and "dichotomy" in Theorem 2.1 are ruled out and the only possibility left is the concentration scenario. Hence, there exists {(x n , y n )} n∈N ⊂ R 2 such that for each ε > 0 there exists r > 0 with Br(xn,yn) e n d(x, y) ≥ I µ − ε, for all n ∈ N.
This implies that, for r sufficiently large, R 2 \Br (xn,yn) e n < ε. (2.20) By taking a subsequence we may assume that (2.20) holds for all n ∈ N. This implies in particular that φ n (· − (x n , y n )) L 2 (|(x,y)|>r) < ε. (2.21) Since {φ n } n∈N is a bounded sequence in X α 2 we may assume φ n ⇀ φ ∈ X α 2 . From Proposition 2.4 we know that X α 2 is compactly embedded in L 2 loc (R 2 ), and therefore φ n → φ strongly in L 2 loc (R 2 ). By combining this with (2.21) we can use Cantors diagonal extraction process to extract a subsequence, still denoted by φ n , converging strongly in L 2 (R 2 ). Proposition 2.2 (ii) implies that for α > 4 5 , φ n → φ in L 3 (R 2 ) as well, which yields that N (φ) = µ. Finally, since L(φ) = 1 2 φ α 2 and the norm is weakly lower semicontinuous, we find Hence, φ is a solution of the minimization problem (2.2).  In Figure 2, we represent the numerically generated lump solution of the KP-I equation and the cross sections φ(x, 0) and φ(0, y) of both numerical and analytical solutions. Choosing the number of grid points as N x = N y = 2 13 for both x and y coordinates we see that the L ∞ -norm of the difference of numerical and exact solutions is approximately of order 10 −5 after 50 iterations. In Figure 2 we also present the variation of three different errors with the number of iterations in semi-log scale. In the next experiment we consider the fractional case. Figure 3 depicts the profiles of the numerical solutions for α = 1.7 and α = 1.35, respectively. It can be seen from the numerical results that the lump solutions become more peaked for smaller values of α. Therefore, to ensure the required numerical accuracy we need to increase the number of grid points to 2 14 for both x and y directions when α = 1.35. In this case the Fourier coefficients go down to 10 −5 . To obtain the same numerical accuracy for smaller values of α, we need to increase the number of Fourier modes more, which is not accessible due to the limits of computation.
We also observe the cross-sectional symmetry of the lump solutions of the fKP-I equation numerically. We present several x and y-cross sections of the solutions for various α. We consider the cases α = 2, α = 1.7 and α = 1.35 in Figure 4, Figure 5, and Figure 6, respectively. The numerical results indicate symmetry in both x and y directions.

Decay of lump solutions
Throughout this section, unless specifically stated otherwise, we assume that α > 4 5 . The existence of lump solutions u(t, x, y) = φ(x − ct, y) for the fKP-I equation, where φ ∈ X α 2 , was proved in the previous section. The function φ satisfies the (rescaled) traveling wave fKP-I equation which can be written in convolution form as where the symbol m α is given by Let us recall from Remark 1.3 that any nontrivial, continuous solution φ of (3.2) decays at most quadratically.
In this section, we show that any nontrivial solution φ ∈ X α 2 of (3.2) decays indeed exactly quadratically, that is we prove Theorem 1.2.
The idea is to study the kernel function K α and to show that it has quadratic decay at infinity (independent of α). Then the decay properties of K α are used to show that also φ decays quadratically at infinity. In the sequel we denote by r : R 2 → R the function It will be useful to note that for all a ≥ 1 we have that r a is convex, so that Notice that by (3.3) and Young's inequality for some 1 ≤ q, q ′ ≤ ∞ with 1 = 1 q + 1 q ′ , so that the statement is proved provided that Before studying the properties of the kernel function K α , we state the following two lemmata, which yield some a priori regularity of lump solutions in the energy space.
Lemma 3.1. Any solution φ of (3.2) in the energy space X α 2 satisfies φ ∈ L r (R 2 ) for all 2 ≤ r < ∞ and In particular, φ is uniformly continuous and decays to zero at infinity.
Next, we aim to bootstrap the smoothness. By Hölder's inequality it is clear also that (φ 2 ) y ∈ L r (R 2 ) for all 2 ≤ r < ∞. Due to the Leibniz rule for fractional derivatives (see e.g. [17, Theorem 7.6.1]) and Hölder's inequality we can estimate which yields that also D α x φ 2 ∈ L r (R 2 ) for all 2 ≤ r < ∞. This can be used to bootstrap the smoothness of φ, by using Reiterating the argument yields and eventually D k x φ ∈ L r (R 2 ) for all 2 ≤ r < ∞ and k ∈ N, which implies that φ ∈ H ∞ (R 2 ). Eventually, since H ∞ (R 2 ) is embedded into the space of uniformly continuous functions on R 2 and φ is L 2 (R 2 )-integrable, we deduce that φ decays to zero at infinity.
Proof. The proof follows essentially the lines in [10, Lemma 3.1]. Here, we proceed formally by omitting the truncation function at infinity. First, let us multiply (3.1) by x 2 φ and integrate over R 2 . Then Using integration by parts we find that In view of Lemma A.1, the nonlocal part can be written as Adding the above equalities we obtain that Multiplying (3.1) by y 2 φ instead yields Again, using integration by parts, we find that Adding (3.6) and (3.7), while keeping in mind that φ ∈ X α 2 , we can estimate Using that φ is continuous and tends to zero at infinity (see Lemma 3.1), there exists R > 0 such that φ(x, y) ≤ 1 2 for |(x, y)| ≥ R and we conclude Properties of the kernel function K α .
We will first concentrate on the regularity properties of K α .
where we used the change of variables z = ξ2 |ξ1|(1+|ξ1| α ) 1 2 . Since the second integral above is clearly convergent for any p ≥ 1, we find that m α ∈ L p (R 2 ) if and only if Remark 3.4. The above lemma implies that m α ∈ L 2 (R 2 ) if and only if α > 4 3 , which is the L 2 -critical exponent. In this case it follows immediately, that also K α ∈ L 2 (R 2 ) and the proof of Theorem 1.2 can be done essentially by following the lines in [10]. In the supercritical case 4 5 < α ≤ 4 3 , which in particular includes the Benjamin-Ono KP equation for α = 1, the symbol m α belongs to an L p -space with p > 2 so that the integrability properties of the kernel K α are a priori not clear. Proof. Let χ : R 2 → R be a compactly supported, radial, smooth function with χ(0, 0) = 1. Setm α := χm α . Thenm α has compact support and F −1 (m α ) is real analytic. Now, setm α := (1 − χ)m α . Thenm α is smooth. Let us fix (x 0 , y 0 ) = (0, 0) and let ψ : R 2 → R be a compactly supported, smooth function with ψ(x, y) = 1 in an arbitrarily small neighborhood of (x 0 , y 0 ) and ψ(0, 0) = 0. Then also Ψ k (x, y) = |(x, y)| −2k ψ is smooth and compactly supported. Notice thatΨ k = −∆ −kψ and Since Ψ k is smooth with compact support, we know thatΨ k ∈ S(R 2 ). Furthermore ∆ km α is smooth with ∆ km α (ξ) at least as 1 |·| α+2k at infinity for an arbitrary choice of k ∈ N. In particular, F −1 (m α * ψ) = F −1 (m α )ψ is smooth, which yields that F −1 (m α ) is smooth outside the origin. We conclude that is smooth outside the origin.
Let us now investigate the behavior of K α at infinity. We show that the decay is quadratic, independently of the value of α > 0. Proposition 3.6. For any α > 0, we have that r 2 K α belongs to L ∞ (R 2 ).
where we used that and a 2 = ξ 2 1 + |ξ 1 | α+2 . Let us consider the case where ξ ≥ 0 (the proof works similarly for ξ < 0). Assume for the moment that y = 0. Setting we can write where G(ξ) := ixξ − |y|ξ(1 + ξ α ) 1 2 . Using integration by parts, we obtain Applying again integration by parts, we find In order to lighten the notation, we set Using Lemma A.2, we find We are left to consider the case when y = 0, that is Notice that where δ 0 denotes the delta distribution centered at zero and g ∈ L 1 (R). Thus x → x 2 K α (x, 0) belongs to L ∞ (R). This concludes the proof.
In order to determine the L p -regularity of K α , it is left to investigate the behaviour of the kernel function close to the origin. To do so, we will use that |∇m α | h α , where (3.8) Notice also that ∂ −1 x K α (ξ 1 , ξ 2 ) = − i h(ξ 1 , ξ 2 ) and (3.2) can be written as Lemma 3.7 (The symbol h α ). We have that a) h α ∈ L p (R 2 ) if and only if 1 2 + 3 2(1+α) < p < 2 and Proof. Similar as in the proof of Lemma 3.3 we compute where we used the change of variables z = ξ2 |ξ1|(1+|ξ1| α ) 1 2 . Since the last integral above is bounded for all p > 1, we find that h α ∈ L p (R 2 ) if and only if Since the Fourier transform is a bounded function from L p (R 2 ) to L p ′ (R 2 ) for p ∈ [1, 2] and p ′ being the dual conjugate to p, we obtain immediately that Thereby, part (a) is proved. In order to prove part (b) we proceed as in the proof of Proposition 3.6. We have that 1 G ′ (ξ) , we obtain after integration by parts where G(ξ) = ixξ − |ξ|(1 + ξ α ) 1 2 |y|. In view of Lemma A.3 we find that (3.10) If y = 0, we have where δ 0 denotes the delta distribution centered at zero and g ∈ L 1 (R). We deduce that x → |x|H α (x, 0) is a bounded function. Together with (3.10) this proves the claim that rH α ∈ L ∞ (R 2 ).
Proposition 3.8. The kernel function K α satisfies the regularity Proof. We know already from Lemma 3.5 and Proposition 3.6 that K α is smooth outside the origin and r 2 K α ∈ L ∞ (R 2 ). Introducing a smooth truncation function ρ : R + → R + , which is compactly supported in a neighborhood of zero, denoted by B ⊂ R 2 , with ρ(0) = 1, we find that (1 − ρ(r))K α ∈ L s (R 2 ) for all s > 1, (3.11) where r(x, y) = |(x, y)|. In order to determine the regularity of K α close to zero, recall that |∇m α | |h α |, where h α defined in (3.8) so that |∇m α | ∈ L q (R 2 ) for 1 2 + 3 2(1 + α) < q < 2, due to Lemma 3.7 (a). Now, we use that the Fourier transformation is a bounded operator from L p (R 2 ) to L p ′ (R 2 ) when p ∈ [1, 2] and p ′ is the dual conjugate of p and obtain that and in fact rK α ∈ L s (B) for 1 ≤ s < 4+α 2−α , by Hölder's inequality and the boundedness of B. Then, we estimate In view of (3.11) we deduce that K α ∈ L r (R 2 ) for 1 < r < 8 + 2α 8 − α .
A non-optimal decay rate.
First notice that φ inherits the integrability properties of K α , since by the L 2 -integrability of φ. Interpolating between the boundedness of φ, which is due to Lemma 3.1, and the L r -integrability of φ for 1 < r < 8+2α 8−α , we actually find that φ ∈ L p (R 2 ) for all p > 1.
Proof. We use the regularity in Proposition 3.9 to improve the decay rate by estimating where we also used the convexity of r 1+δ . The first norm on the right-hand side above is clearly bounded by Young's inequality, Proposition 3.6 and the L 2 -integrability of φ. For the second norm, let ε > 0 be a small constant so that 0 < δ < 1 1+ε < 1. Using that K α ∈ L 1+ε (R 2 ) for ε > 0 small enough, we estimate .
Notice that By our choice of ε > 0, we have (1 − δ) 1+ε ε > 1, so the above norm is bounded by (3.12), which concludes the proof of the statement.

Proof of Theorem 1.2
In view of the discussion at the beginning of this section and Lemma 3.1, we obtain our main result provided that (A) and (B) at the beginning of the section are satisfied. The statement in (A) is proved in Proposition 3.6, while the first part of statement (B) follows from Proposition 3.8, where it is shown that Now, we make use of the non-optimal decay estimate in Proposition 3.10 to show that indeed r 2 φ 2 ∈ L r ′ (R 2 ), where r ′ is the dual conjugate to r. For any 0 ≤ δ < 1, we have that Choosing δ = r − 1 ∈ (0, 1) we find that δ 1+δ r ′ = r−1 r r ′ = 1 and the boundedness of r 2 φ 2 in L r ′ (R 2 ) follows from the L 2 -integrability of φ. Hence, statement (B) is shown, which concludes the proof of Theorem 1.2.
To visualize the decay rate of the solutions we consider the product of numerically generated lump solutions with 1/r 2 (x, y) = 1/(x 2 + y 2 ). Figure 7 shows x and y−cross sections of this product for α = 2, α = 1.7, and α = 1.35. As the decay rate is quadratic the result approaches to a constant value for increasing |x| and |y|, as expected. We observe that the behaviour is similar for all α values but the aforementioned constant becomes smaller for smaller values of α.   Due to the Coriolis parameter γ > 0, the symbol m is smooth at the origin, which allows lump solutions to decay exponentially at infinity (cf. [8,Theorem 1.6]). If the dispersive term βu xxx were replaced by the fractional term −β|D x | α u x , which would lead to an fractional rotation modified KP equation, we'd expect a decay of lump solutions, which depends on α (in a similar way as we see it for the fractional KdV equation [14,20]). .
Here β > 0 is the surface tension coefficient. Existence of lump solutions for full dispersion KP equation is shown in [11,12]. In the same way as for the fKP-I equation, the transverse direction induces a discontinuity at the origin of the symbol m(ξ 1 , ξ 2 ) = 1 c+l(ξ1,ξ2) . Therefore, the decay of lump solutions will also be at most quadratically.
Starting with T 1 we estimate .