Existence results of solitary wave solutions for a delayed Camassa-Holm-KP equation

This paper is concerned with the Camassa-Holm-KP equation, which is a model for shallow water waves. By using the analysis of the phase space, we obtain some qualitative properties of equilibrium points and existence results of solitary wave solutions for the Camassa-Holm-KP equation without delay. Furthermore we show the existence of solitary wave solutions for the equation with a special local delay convolution kernel by combining the geometric singular perturbation theory and invariant manifold theory. In addition, we discuss the existence of solitary wave solutions for the Camassa-Holm-KP equation with strength \begin{document}$ 1 $\end{document} of nonlinearity, and prove the monotonicity of the wave speed by analyzing the ratio of the Abelian integral.


1.
Introduction. The classical Korteweg-de Vries-type equations and Camassa-Holm-type equations are fundamental models for shallow water waves. Physical structures of such equations have been extensively studied by many authors [35,10,6,20,38,32,39,42,23,24]. In the past few decades remarkable progresses have been made in understanding the Korteweg-de Vries (KdV) equation, it can be considered as a paradigm in nonlinear science and has many applications in weakly nonlinear and weakly dispersive physical systems. The KdV equation u t + αuu x + u xxx = 0, which is first suggested by Korteweg and de Vries in 1895 [18] is a typical soliton equation where the balance between the nonlinear convection term uu x and the dispersion effect term u xxx gives rise to solitons, while dissipation effects are small enough to be neglected in the lowest order approximation. The KdV equation is a model that governs the one-dimensional propagation of small amplitude, weakly dispersive waves. The two best known models for the two-dimensional generalizations of the KdV equation are the Kadomtsev-Petviashivilli (KP) equation [17] (u t + auu x + u xxx ) x + u yy = 0, (1) and the Zakharov-Kuznetsov (ZK) equation [40] u t + auu x + (∇ 2 u) x = 0, where ∇ 2 = ∂ 2 x + ∂ 2 y + ∂ 2 z is the isotropic Laplacian. In mathematics and physics, the KP equation introduced by Kadomtsev and Petviashivilli is a partial differential equation to describe nonlinear wave motion. It has been studied in many papers, because it commonly appears in different physical applications, and it is integrable by means of the inverse scattering transform method. In addition to being used as a model for the evolution of surface waves [1], the KP equation has also been studied as a model for ion-acoustic wave propagation in isotropic media [31]. Saut and Tzvetkov [33] considered the Cauchy problems of KP equations in R d (d = 2, 3) with (u t + αu xxx + βu xxxx + uu x ) x + u yy = 0, u(0, x, y) = φ(x, y), in the two dimensional case, and (u t + αu xxx + βu xxxx + uu x ) x + u yy + u zz = 0, u(0, x, y, z) = φ(x, y, z), in the three dimensional case. The usual KP equations correspond to β = 0 and α = −1 (KP-I) or α = +1 (KP-II [15,14,28]). Molinet et al. [28] studied the initial value problem for the KP-II equation and they proved the global well-posedness of the Cauchy problem. Another shallow water wave equation different from the KP equation is the Camassa-Holm (CH) equation, which is an integrable, dimensionless and nonlinear partial differential equation where u(x, t) denotes the fluid velocity, or can also be interpreted as the height of the water's free surface above a flat bottom, and the constant k is related to the critical shallow water wave speed. In the special case that k is equal to zero, the CH equation has peakon solutions: solitons with a sharp peak, so with a discontinuity at the peak in the wave slope. This equation was first derived by Fuchssteiner and Fokas [10] as an abstract bi-Hamiltonian equation with infinitely many conservation laws, and later rederived by Camassa and Holm [6] from physical principles. Traveling wave solutions are basic patterns of different nonlinear differential equations. Lenells [20] provided a complete classification of all traveling wave solutions of the CH equation (4). Qu et al. [32] introduced a µ-version of the modified CH equation and investigated its integrability, well-posedness, wave breaking, and existence of peaked soliton and multi-peakon solutions. Novruzov and Hagverdiyev [29] considered Cauchy problem for the CH equation with dissipative term and compactly supported initial data. They found a simple condition guaranteeing blow-up of the solution by using some properties of the solution generated by initial data. Wazwaz [36] considered the following CH-KP equations given by and where a > 0, k ∈ R and n is called the strength of the nonlinearity. The aforementioned variants (5) and (6) are developed similarly to the KP equation (1), derived from the modified CH equations Therefore, equations (5) and (6) are called CH-KP equations. By using the sinecosine method and the tanh technique, the solitons, compactons, solitary patterns and periodic solutions for equations (5) and (6) were obtained and expressed analytically. Lai et al. [19] studied the generalized forms of equations (5) and (6), which are written by and and derived families of exact traveling wave solutions of equations (5) and (6) by means of the mathematical method. Wei et al. [37] investigated the single peak solutions of the CH-KP equation for m = 2, n = 1, and k = 1, a = 1 under the boundary condition lim x,y→±∞ u(x, y) = A. Recently, Biswas [5] obtained an exact 1-solitons solution of the generalized CH-KP equation by the solitary wave ansatze. By using the bifurcation theory of planar dynamical systems to the generalized CH-KP equations, Zhang et al. [41] proved the existence of smooth and nonsmooth traveling wave solutions. Traveling wave solutions correspond to heteroclinic or homoclinic orbits of related ordinary differential equations. In general, such orbits can be found by means of geometric singular perturbation theory [9], which has been used by many researchers to obtain the existence of traveling waves for different nonlinear differential equations including generalized KdV equations [16], Fisher equations [2], perturbed BBM equation [7], FitzHugh-Nagumo equation [13,25], reaction-diffusion equations [30], tissue interaction model [3], predator-prey models and epidemiology [22,11], etc. The method has also received a great deal of interests in studying semilinear elliptic equations [4], slow-fast dynamic systems [26,27], Liénard equations [21], etc. Recently, Du et al. [8] discussed the existence of solitary wave solution for the delayed CH equation by using the method of dynamical system, especially the geometric singular perturbation theory and invariant manifold theory. Here f * u is the spatial-temporal convolution representing distributed delay and τ is a small constant.
In fact, geometric singular perturbation theory uses invariant manifolds in phase space to better understand the global structure of the phase space or to construct orbits with desired properties. The essential idea behind geometric singular perturbation theory is that this persistence can be established by showing that these singular structures correspond to transversal intersections of a pair of stable and unstable manifolds.
Motivated by above papers, this paper is to establish the existence of solitary wave solutions for the following CH-KP equation where the convolution f * u is defined by and the kernel f : [0, +∞) → [0, +∞) satisfies the normalization assumption f (t) ≥ 0 for all t ≥ 0, and ∞ 0 f (t)dt = 1. It is notable that the normalization assumption on f ensures that the uniform non-negative steady-state solutions are unaffected by the delay. The kernel are frequently used in the literature on delay differential equations. The first one is called the weak generic delay kernel and the second one is called the strong general delay kernel. In this paper, we discuss the strong general delay kernel, i.e., The remaining part of this paper is organized as follows. In Section 2, we present geometric singular perturbation theory which is important to obtain our main results. In Section 3, We prove the existence of solitary wave solutions for the the CH-KP equation (10) without perturbation and delay. In Section 4, we investigate the existence of solitary wave solutions for the CH-KP equation (10) with a special local delay convolution kernel by using the method of dynamical system, especially the geometric singular perturbation theory and invariant manifold theory. In Section 5, we discuss the existence of solitary wave solutions for equation (10) with n = 1, and prove the monotonicity of the wave speed by analyzing the ratio of the Abelian integral.

2.
Preliminaries. In this section, to present our results on the existence of traveling wave, we first introduce the geometric singular perturbation theory ( [9,16,12,34]).
Consider the system where = d dt , u ∈ R k , and v ∈ R l with k, l ≥ 1. The parameter is a small parameter, which gives the system a singular character. The function f and g are assumed to be sufficiently smooth. Here 'sufficiently smooth' means at least C 1 in u, v and . In general, to obtain C r invariant manifolds, f and g should be C r+1 functions of u, v, , and the subset M 0 of {f (u, v, 0) = 0} we consider below should be a C r+1 submanifold of the phase space R k+l .
System (11) can be reformulated with a change of time-scale as where . = d dτ and τ = t. The time scale given by τ is said to be slow whereas that for t is fast. Thus we call (11) the fast system and (12) the slow system. As long as = 0 the two systems are equivalent. Each of the scalings is naturally associated with a limit as → 0. These limits are respectively given by and The former is called the layer problem and the latter is called the reduced system. The problem (13) and (14) are lower dimensional and can often be analysed in sufficient detail. By 'gluing' together fast and slow pieces of orbits, respectively obtained in the fast and slow limits, one can formally construct global singular structures, such as singular periodic orbits and singular homoclinic orbits. Moreover, the flow under (13) on the l-dimensional set f (u, v, 0) = 0 is trivial. On the other hand, (14) does prescribe a nontrivial flow on f (u, v, 0) = 0, but at the same time its validity is limited to only this set. The goal of geometric singular perturbation theory is to analyse the dynamics of system (11) with nonzero but small by suitably combining the dynamics of these two limits. If M 0 is an l-dimensional manifold contained in f (u, v, 0) = 0, and M 0 is normally hyperbolic, then Fenichel's first theorem is as follows.
Lemma 2.2 (Fenichel's first theorem [12]). Suppose M 0 ⊂ {f (u, v, 0) = 0} is compact, possibly with boundary, and normally hyperbolic, that is, the eigenvalues λ of the Jacobian ∂f ∂u (u, v, 0)| M0 all satisfy Re(λ) = 0. Suppose f and g are smooth. Then for > 0 and sufficiently small, there exists a manifold M , O( ) close and diffeomorphic to M 0 , that is locally invariant under the flow of the full problem (11).
Consider the equation (11). Suppose that, for = 0, the normally hyperbolic critical manifold M 0 ⊂ {f (u, v, 0) = 0} has an (l + m)-dimensional stable manifold W s (M 0 ) and an (l + n)-dimensional unstable manifold W u (M 0 ), with m + n = k. In other words, suppose that the Jacobian ∂f ∂u (u, v, 0)| M0 has m eigenvalues λ with Re(λ) < 0 and n eigenvalues λ with Re(λ) > 0. Then the following theorem holds.   i.e., Let where c = 0. Substituting (16) into (15), then we obtain the following solitary wave equation where = d dξ . It can be integrated twice to yield the equation which is equivalent to the following system of first-order equation It has first integral By (20) and the analysis of the phase space, we obtain the following two theorems when n is odd and even respectively. Theorem 3.1. When n is odd, then system (19) has two equilibrium points E 1 (0, 0) and E 2 (( (2k+1−c)(n+1) a ) 1 n , 0) in the (φ, ψ) phase plane and the following results hold. (i) If a > 0, c < 2k + 1, then E 1 is a center and E 2 is a saddle, and system (19) has a homoclinic orbit to the equilibrium point E 2 (Figure 1(a)).
(ii) If a < 0, c > 2k + 1, then E 1 is a saddle and E 2 is a center, and system (19) has a homoclinic orbit to the equilibrium point E 1 (Figure 1(b)).
(iii) If a > 0, c > 2k + 1, then E 1 is a saddle and E 2 is a center, and system (19) has a homoclinic orbit to the equilibrium point E 1 (Figure 1(c)).
(iv) If a < 0, c < 2k + 1, then E 1 is a center and E 2 is a saddle, and system (19) has a homoclinic orbit to the equilibrium point E 2 (Figure 1(d)).
Theorem 3.2. When n is even, the following results hold in the (φ, ψ) phase plane.
(ii) If a < 0, c > 2k + 1, then system (19) has three equilibrium points E 1 (0, 0), n , 0). E 1 is a saddle, E 2 and E 3 are center, and system (19) has periodic orbits to the equilibrium point E 1 which is connected by two homoclinic orbits (Figure 2(b)).  In this section, we establish the existence of solitary wave solutions for equation (10) with the condition a < 0, c > 2k + 1 and n is odd.
Consider the case of small positive averaging delay τ > 0. We search for solutions of (10) in the form u(x, y, t) = φ(ξ) with ξ = x + y − ct, then (10) is changed to where Then differentiating η(ξ) with respect to ξ, we obtain where Differentiating ζ with respect to ξ, we obtain Using the boundary condition at −∞, the equation (21) can be integrated once to yield the equation Therefore the solitary wave equation (24) is equivalent to the following system Note that when τ → 0, we get η → φ and arrive at the non-delay equation. We want to investigate the persistence of the homoclinic orbit for small τ > 0. By setting ξ = τ s, the system (25) becomes where˙= d ds . We refer to (25) as the slow system and (26) as the fast system. The above two systems are equivalent when τ > 0. If we set τ = 0 in (25), then the flow of system (25) is restricted to the set which is a three-dimensional manifold of equilibrium for (26) with τ = 0. Note that we have restricted the discussion in a neighborhood of the unperturbed homoclinic orbit. The linearization of (26) It can be easily seen that the matrix has five eigenvalues: 0, 0, 0, 1 c , 1 c , and thus we can conclude that M 0 is normally hyperbolic with two unstable normal directions. According to Lemma 2.2, there exists M τ for 0 < τ 1: where the functions g, h are smooth functions defined on a compact domain, and satisfy g(φ, ψ, ν, 0) = 0, h(φ, ψ, ν, 0) = 0.
Thus the function g, h can be expanded into the form of a Taylor series about τ Substituting η = φ + g(φ, ψ, ν, τ ), ζ = φ + h(φ, ψ, ν, τ ) into the slow system (25), we have An easy calculation shows the following asymptotic expansion: Therefore, restricted to M τ , (25) becomes the following system Because both normal directions are unstable, the persistent homoclinic orbit (if ever exists) must be contained totally in M τ . In addition, it is obvious that system (27) can be rewritten as the following form Now we consider the homoclinic orbit of system (28). n , ψ = 0}. We focus onM 0 , which is normally hyperbolic with one stable and one unstable normal direction. From Lemma 2.2, there existsM τ for 0 < τ 1, which is O(τ ) close toM 0 . Lemma 2.3 indicates the existence of two-dimensional stable and unstable manifold W s τ and W u τ ofM τ , being O(τ ) close to corresponding stable and unstable manifold W s 0 and W u 0 ofM 0 respectively.
Viewingν as parameter, (28) is a Hamiltonian system, with non-transversal intersection of W s 0 and W u 0 and a two-dimensional homoclinic manifold. We need to study the separation of W s τ and W u τ along the unperturbed homoclinic orbit Γ which satisfies Since the bounded parts of W s 0 and W u 0 coincide, the distance between the perturbed manifolds W s τ and W u τ is of order O(τ ) and may thus be computed via the adiabatic Melnikov integral where φ, ψ are evaluated along Γ, andν satisfies with φ, ψ evaluated along Γ. Obviously, solutions of the equation Q(c, τ ) = 0 correspond to homoclinic orbits of (28).
To compute M , we use integration by parts. By (29), one has i.e., From (29) and by using integration by parts again, we have We obtain Let Thus, we can solve the equation Q(c, τ ) = 0 locally for c = c 0 by the implicit function theorem. This indicates W s τ and W u τ intersect transversally and a homoclinic orbit to M τ persists. Remark 1. This paper only discusses the existence of solitary wave solution for the case that a < 0, c > 2k + 1, and n is odd. In fact, we could obtain the existence of solitary (traveling) wave solution for other cases by similar discussions, and we omit the details here.

5.
Monotonicity of wave speed for CH-KP equation with n = 1. In this section, we will study the monotonicity of the wave speed for CH-KP equation with n = 1. To do this, we will use another convenient way to obtain the existence of solitary wave solutions for (10) with n = 1. In fact, this verifies again the result for n = 1 in previous section. Then, by analyzing the ratio of Abelian integrals, we obtain the monotonicity of the wave speed and give the upper and lower bounds of the limit wave speed. When n = 1, equation (10) becomes First, we use the Hamiltonian function to discuss the equation (32) without delay, i.e., Substituting u(x, y, t) = φ(ξ), ξ = x + y − ct into (33), then we obtain where = d dξ . It can be integrated twice to yield the equation Taking the transformations φ 1 = φ c−2k−1 and z = c−2k−1 c ξ, we obtain which is equivalent to the following system of first-order equation where˙= d dz . It has first integral From (38) and easy analysis of the phase space, we obtain the following theorem.
Consider the case of small positive averaging delay τ > 0. We search for solutions of (32) in the form u(x, y, t) = φ(ξ), ξ = x + y − ct, then (32) is changed to where Using the boundary condition at −∞, equation (39) can be integrated once to yield the equation Taking the transformations φ 1 = φ c−2k−1 and z = c−2k−1 c ξ, we obtaiṅ where Therefore the solitary wave equation (41) is equivalent to the following system where By setting z = τ s, the system (42) becomes If τ is set to zero in (42), then the flow of that system is restricted to the set which is a three-dimensional manifold of equilibrium for (43) with τ = 0. The linearization of (43) with τ = 0 is It can be easily seen that the matrix has five eigenvalues: 0, 0, 0, , and thus we can conclude that M 0 is normally hyperbolic with two unstable normal directions. According to Lemma 2.2, there exists M τ for 0 < τ 1: where the functions g, h are smooth functions defined on a compact domain, and satisfy g(φ 1 , ψ 1 , ν 1 , 0) = 0, h(φ 1 , ψ 1 , ν 1 , 0) = 0. Easy calculation shows the following asymptotic expansion: Therefore restricted to M τ , (42) is the following system Because both normal directions are unstable, the persistent homoclinic orbit (if ever exists) must be contained totally in M τ . In addition, it is obvious that system (44) can be rewritten as the following form Now we consider the homoclinic orbit of system (45).
Thus, we can solve the Melnikov distance Q(c, τ ) = 0 locally for c = c 0 by the implicit function theorem. This indicates W s τ and W u τ intersect transversally and a homoclinic orbit to M τ persists.
In the following, we discuss the monotonicity of the wave speed c and the upper and lower bounds of the wave speed by analyzing the ratio of the Abelian integral. First of all, let Q and R be  The orbit (φ 1 (z), ψ 1 (z)) is on the level curve H = M 2 , therefore, by (37) we have where E(φ 1 ) = a 3 φ 3 1 + φ 2 1 − M . Now Q and R are the functions of M only. The purpose of this section is to prove the following proposition which will assert the monotonicity of the speed on M . In order to prove the proposition, it is convenient to represent Q and R by the following integrals J n (M ) = β α φ n 1 E(φ 1 )dφ 1 , n = 0, 1, 2, · · · .
Therefore, Q and R are represent as follows: and R = J 0 . First, let us study the basic properties of J 0 and J 1 by the following four lemmas. Proof. From the relation E 2 = a 3 φ 3 1 + φ 2 1 − M, we have 2E dE du = 2φ 1 + aφ 2 1 . J 0 can be calculated as Thus J 0 = 6 5 M J 0 + 4 5a J 1 .
On the other hand, J 1 is also calculated as