Solitary wave solutions to the Isobe–Kakinuma model for water waves

arXiv:1910.06481v1 [math.AP] 15 Oct 2019

Solitary wave solutions to the Isobe–Kakinuma model for water waves

Mathieu Colin and Tatsuo Iguchi Abstract

We consider the Isobe–Kakinuma model for two-dimensional water waves in the case of the flat bottom. The Isobe–Kakinuma model is a system of Euler–Lagrange equations for a Lagrangian approximating Luke’s Lagrangian for water waves. We show theoretically the existence of a family of small amplitude solitary wave solutions to the Isobe–Kakinuma model in the long wave regime. Numerical analysis for large amplitude solitary wave solutions is also provided and suggests the existence of a solitary wave of extreme form with a sharp crest.

1 Introduction

In this paper we consider the motion of two-dimensional water waves in the case of the flat bottom. The water wave problem is mathematically formulated as a free boundary problem for an irrotational flow of an inviscid and incompressible fluid under the gravitational field. Lettbe the time and (x, z) the spatial coordinates. We assume that the water surface and the bottom are represented as z= η(x, t) and z=−h, respectively. As was shown by J. C. Luke [13], the water wave problem has a variational structure. His Lagrangian density is of the form

(1.1) L_Luke(Φ, η) =

Z η(x,t)

−h

∂_tΦ(x, z, t) +1

2 (∂_xΦ(x, z, t))²+ (∂_zΦ(x, z, t))² +gz

dz, where Φ = Φ(x, z, t) is the velocity potential and g is the gravitational constant. M. Isobe [7, 8] and T. Kakinuma [9, 10, 11] proposed a model for water waves as a system of Euler–

Lagrange equations for an approximate Lagrangian, which is derived from Luke’s Lagrangian by approximating the velocity potential Φ in the Lagrangian appropriately. In this paper, we adopt an approximation under the form

(1.2) Φ(x, z, t)≃

N

X

i=0

(z+h)^piφ_i(x, t),

where p₀, p₁, . . . , pN are nonnegative integers satisfying 0 = p₀ < p₁ < · · · < pN. Then, the corresponding Isobe–Kakinuma model in a nondimensional form is written as

(1.3)























H^pi∂tη+

N

X

j=0

∂x

1

pi+pj+ 1H^pi+pj+1∂xφj

−δ⁻² p_ip_j

pi+pj−1H^pi+pj−1φj

= 0 for i= 0,1, . . . , N,

N

X

j=0

H^pj∂_tφ_j +η+1 2











N

X

j=0

H^pj∂_xφ_j





2

+δ⁻²





N

X

j=0

p_jH^pj−1φ_j





2





= 0,

whereH(x, t) = 1 +η(x, t) is the normalized depth of the water andδa nondimensional param- eter defined by the ratio of the mean depth h to the typical wavelength λ. Here and in what

follows we use the notational convention ⁰₀ = 0. For the derivation and basic properties of this model, we refer to Y. Murakami and T. Iguchi [14] and R. Nemoto and T. Iguchi [15]. Moreover, it was shown by T. Iguchi [5, 6] that the Isobe–Kakinuma model (1.3) is a higher order shallow water approximation for the water wave problem in the strongly nonlinear regime. We note also that the Isobe–Kakinuma model (1.3) in the caseN = 0 is exactly the same as the shallow water equations. In the sequel, we always assumeN ≥1.

In this paper, we look for solitary wave solutions to this model under the form (1.4) η=η(x+ct), φj =φ_j(x+ct), j = 0,1, . . . , N,

wherec∈Ris an unknown constant phase speed. Plugging (1.4) into (1.3), we obtain a system of ordinary differential equations

(1.5)























cH^piη^′+

N

X

j=0

1

p_i+p_j + 1H^pi+pj+1φ^′_j ′

−δ⁻² pipj

p_i+p_j−1H^pi+pj−1φj

= 0 for i= 0,1, . . . , N, c

N

X

j=0

H^pjφ^′_j+η+1 2











N

X

j=0

H^pjφ^′_j





2

+δ⁻²





N

X

j=0

p_jH^pj−1φ_j





2





= 0.

As expected, this system has a variational structure, that is, the solution of this system is obtained as a critical point of the functional

L_IK(φ₀, . . . , φ_N, η) =cM_IK(φ₀, . . . , φ_N, η) +E_IK(φ₀, . . . , φ_N, η), where

M_IK(φ0, . . . , φ_N, η) = Z

R

η(x)∂x Φ^app(x, η(x)) dx, E_IK(φ₀, . . . , φN, η) = 1

2 Z

R

(Z _η(x)

−1

(∂xΦ^app(x, z))²+ (δ⁻¹∂zΦ^app(x, z))²

dz+η(x)² )

dx, and Φ^app is the approximate velocity potential defined by

Φ^app(x, z) =

N

X

i=0

(z+ 1)^piφi(x, t).

We note that M_IK and E_IK represent the momentum in the horizontal direction and the total energy of the water, respectively. Both of them are conserved quantities for the Isobe–Kakinuma model (1.3). In this paper we do not use this variational structure to construct solitary wave solutions to the Isobe–Kakinuma model, whereas we use a perturbation method with respect to the small nondimensional parameter δ in the long wave regime.

In order to give one of our main results in this paper concerning the existence of a family of solutions to (1.5), we introduce normsk · k and k · k^k fork= 0,1,2, . . . ,by

(1.6) kuk= sup

x∈R

e^|x||u(x)|, kukk =

k

X

j=0

ku^(j)k,

whereu^(j)is thej-th order derivative ofu. We also introduce the function spacesB_e^k andB_o^k as closed subspaces of all even and odd functions u∈C^k(R) satisfying kukk <+∞, respectively, equipped with the norm k · k^k, and put B_α^∞ = ∩^∞k=0B_α^k for α = e, o. The following theorem guarantees the existence of small amplitude solitary wave solutions to the Isobe–Kakinuma model in the long wave regime.

Theorem 1.1. There exists a positive constant δ₀ such that for any δ ∈ (0, δ₀] the Isobe–

Kakinuma model (1.5) has a solution (c^δ, η^δ, φ^δ₀, . . . , φ^δ_N), which satisfies η^δ, φ^δ′₀ ∈ B_e^∞ and φ^δ₁, . . . , φ^δ_N ∈B_o^∞. Moreover, the solution satisfies c^δ= 1 + 2γδ² and

(kη^δ−4γδ²sech²xk^k+k(φ^δ₀+ 4γδ²tanhx)^′k^k ≤C_kδ⁴, kφ^δ_j −4γγjδ⁴sech²xtanhxkk≤C_kδ⁶ for j= 1, . . . , N

for any k ∈ N, where the constants γ, γ₁, . . . , γ_N are determined through p₁, . . . , p_N and the constant C_k does not depend onδ but on k.

Remark 1.2. The constantsγ andγ= (γ₁, . . . , γN)^Tin the statement of Theorem 1.1 are given by γ=A⁻¹₁ (1−a₀) and γ = (1−a₀)·A⁻¹₁ (1−a₀) with a N ×N matrix A₁ and a vector a₀ given by

(1.7) A₁ =

p_ip_j p_i+p_j−1

1≤i,j≤N

,

1 a^T₀ a₀ A₀

=

1 p_i+p_j+ 1

0≤i,j≤N

,

and1= (1, . . . ,1)^T. The matrices A₀ andA₁ are positive, so that the constantγ is also positive.

We will use these notations throughout this paper.

Remark 1.3. This theorem ensures the existence of a family of solitary wave solutions to the Isobe–Kakinuma model traveling to the left. We can also show a similar existence theorem to solitary wave solutions to the model traveling to the right.

Remark 1.4. By neglecting the term of order O(δ⁴), the solitary wave solutions in the dimen- sional form are given by















c≃ 1 + _2h^a√gh, η(x, t)≃asech²q _a

4γh³(x+ct) , φ₀(x, t)≃ −p

4γgah²tanhq _a

4γh³(x+ct) , φ_j(x, t)≃0 for j= 1, . . . , N,

where g is the gravitational constant and ais the amplitude of the wave.

In this paper we also analyze numerically the existence of large amplitude solitary wave solutions to the Isobe–Kakinuma model in a special case where we choose the parameters as N = 1 and p₁ = 2, that is, (7.1). We note that even in this simplest case the Isobe–Kakinuma model gives a better approximation than the well-known Green–Naghdi equations in the shallow water and strongly nonlinear regime. See T. Iguchi [5, 6]. Numerical analysis suggests that there exists a critical value ofδ given approximately by

(1.8) δ_c= 0.62633493

such that for any δ ∈ (0, δc), the Isobe–Kakinuma model (7.1) admits a smooth solitary wave solution and that this family of waves converges to a solitary one of extreme form with a shape

crest as δ ↑ δ_c. Moreover, the included angle θ of the crest in the physical space is given approximately by

(1.9) θ= 152.6^◦.

See Figure 1.1.

θ

Figure 1.1: Solitary wave of extreme form with the included angle θ= 152.6^◦.

Here, we mention the related results on the existence of the solitary wave solutions to the water wave problem. The existence of small amplitude solitary waves for the water wave problem was first given by K. O. Friedrichs and D. H. Hyers [3]. Then, C. J. Amick and J. F. Toland [2]

proved the existence of solitary wave solutions of all amplitudes from zero up to and including that of the solitary wave of greatest height. On the other hand, a periodic wave of permanent form is called Stokes’ wave. The existence of Stokes’s wave of extreme form as well as the sharp crest of the included angle 120^◦ was predicted by G. G. Stokes [4], and then proved theoretically by C. J. Amick, L. E. Fraenkel, and J. F. Toland [1]. An extension of these existence theories to the water waves with vorticity was given by E. Varvaruca [16]. The author has proved the existence of the solitary wave as well as Stokes’ wave of greatest height with a shape crest of the same included angle 120^◦as in the irrotational case. As for the model equations for water waves, it is well-known that the Korteweg–de Vries equation has solitary wave solutions of arbitrarily large amplitude and does not have any wave of extreme form. The Green–Naghdi equations are known as higher order shallow water approximate equations for water waves in the strongly nonlinear regime and have the same solitary wave solutions as those of the Korteweg–de Vries equation, but again does not have any solitary wave of extreme form. Compared to these models, the Isobe–Kakinuma model even in the simplest case catches the property on the existence of the solitary wave of extreme form, although the included angle of the crest is not 120^◦. We also mention a result by D. Lannes and F. Marche [12], where it was shown that an extended Green–

Naghdi equations for the water waves with a constant vorticity has a solitary wave solution of extreme form.

The contents of this paper are as follows. In Section 2, we give conservation laws for the Isobe–Kakinuma model, which will be used in the numerical analysis in Section 7. In Section 3, by using formal asymptotic analysis we calculate the first order approximate term with respect toδ² in the expansion (3.2) of the solitary wave solution to the Isobe–Kakinuma model (1.5). In Section 4, we reduce the problem by deriving equations for the remainder terms. In Section 5, we construct Green’s functions for the linearized equations together with their estimates. In Section 6, we first prove an existence theorem for the system of linear equations derived in Section 4 and then prove the existence of small amplitude solitary wave solutions to the Isobe–Kakinuma model. Finally, in Section 7, we analyze numerically large amplitude solitary wave solutions and calculate the solitary wave of extreme form together with the included angle.

Acknowledgement

T. I. was partially supported by JSPS KAKENHI Grant Number JP17K18742 and JP17H02856.

2 Conservation laws

As was explained in the previous section, for the Isobe–Kakinuma model (1.3), the mass, the momentum in the horizontal direction, and the total energy are conserved. In the numerical analysis which will be carried out in Section 7, we also need to know corresponding flux functions.

The following proposition gives such flux functions.

Proposition 2.1. Any regular solution (η, φ₀, . . . , φN)to the Isobe–Kakinuma model (1.3)sat- isfies the conservation laws

(2.1) ∂tη+∂x





N

X

j=0

1

p_j + 1H^pj+1∂xφj



= 0,

∂t

( η∂x

N

X

i=0

H^piφi

!) +∂x

(

−η∂t N

X

i=0

H^piφi

!

−1 2η² (2.2)

+1 2

N

X

i,j=0

1

pi+pj + 1H^pi+pj+1(∂xφ_i)(∂xφ_j)−δ⁻² p_ip_j

pi+pj−1H^pi+pj−1φ_iφ_j







= 0, and

∂t





 1 2η²+1

2

N

X

i,j=0

1

pi+pj+ 1H^pi+pj+1(∂xφi)(∂xφj) +δ⁻² p_ip_j

pi+pj−1H^pi+pj−1φiφj





 (2.3)

−∂_x







N

X

i,j

1

pi+pj+ 1H^pi+pj+1(∂xφ_i)(∂tφ_j)







= 0.

Proof. Conservation law of mass (2.1) is nothing but the first equation in (1.3) with i= 0. It

follows from the equations in (1.3) that

∂_t (

η∂_x

N

X

i=0

H^piφ_i

!)

−∂_x (

η∂_t

N

X

i=0

H^piφ_i

!)

= (∂tη)

N

X

i=0

H^pi∂_xφ_i−(∂xη)

N

X

i=0

H^pi∂_tφ_i

=

N

X

i=0



−

N

X

j=0

∂_x

1

p_i+p_j+ 1H^pi+pj+1∂_xφ_j

−δ⁻² p_ip_j

p_i+p_j −1H^pi+pj−1φ_j



∂_xφ_i

+ (∂_xη)



η+1 2











N

X

j=0

H^pj∂_xφ_j





2

+δ⁻²





N

X

j=0

p_jH^pj−1φ_j





2









=−

N

X

i,j=0

1

p_i+p_j + 1H^pi+pj+1(∂xφi)(∂_x²φj)−δ⁻² p_ip_j

p_i+p_j−1H^pi+pj−1φj∂xφi

+ (∂xη)





 η−1

2

N

X

i,j=0

H^pi+pj(∂xφi)(∂xφj)−δ⁻²pipjH^pi+pj−2φiφj







=∂_x





 1 2η²−1

2

N

X

i,j=0

1

p_i+p_j+ 1H^pi+pj+1(∂xφ_i)(∂xφ_j)−δ⁻² p_ip_j

p_i+p_j −1H^pi+pj−1φ_iφ_j





 , which gives conservation law of momentum (2.2). We see also that

∂t





 1 2η²+ 1

2

N

X

i,j=0

1

p_i+p_j+ 1H^pi+pj+1(∂xφi)(∂xφj) +δ⁻² pipj

p_i+p_j−1H^pi+pj−1φiφj







= (∂tη)



η+1 2











N

X

j=0

H^pj∂xφj





2

+δ⁻²





N

X

j=0

pjH^pj−1φj





2









+

N

X

i,j=0

1

p_i+p_j+ 1H^pi+pj+1(∂xφ_i)(∂t∂_xφ_j) +δ⁻² p_ip_j

p_i+p_j−1H^pi+pj−1φ_i∂_tφ_j

=−(∂tη)

N

X

j=0

H^pj∂_tφ_j+∂_x







N

X

i,j=0

1

p_i+p_j+ 1H^pi+pj+1(∂xφ_i)(∂tφ_j)







+

N

X

i,j=0

∂_x

1

p_i+p_j+ 1H^pi+pj+1∂_xφ_i

−δ⁻² pipj

p_i+p_j−1H^pi+pj−1φ_i

∂_tφ_j

=∂_x







N

X

i,j=0

1

p_i+p_j+ 1H^pi+pj+1(∂_xφ_i)(∂_tφ_j)





 , which gives conservation law of energy (2.3).

These conservation laws provide directly the following proposition.

Proposition 2.2. For any regular solution (η, φ₀, . . . , φ_N) to the Isobe–Kakinuma model (1.5) satisfying the condition at spatial infinity

(2.4) η(x), φ^′₀(x), . . . , φ^′_N(x), φ1(x), . . . , φN(x)→0 as x→ ±∞, we have

(2.5) cη+

N

X

j=0

1

p_j+ 1H^pj+1φ^′_j = 0 and

(2.6) 1

2η²−1 2

N

X

i,j=0

1

p_i+p_j+ 1H^pi+pj+1φ^′_iφ^′_j−δ⁻² p_ip_j

p_i+p_j−1H^pi+pj−1φ_iφ_j

= 0.

3 Approximate solutions

We first transform the Isobe–Kakinuma model (1.5) into an equivalent system. The first equation in (1.5) withi= 0 can be integrated under the condition at spatial infinity (2.4) so that we have (2.5), particularly,

cη=−

N

X

j=0

1

pj+ 1H^pj+1φ^′_j.

Plugging this into the first equation in (1.5) with i= 1, . . . , N to eliminate cη, we have H^pi

N

X

j=0

1

pj+ 1H^pj+1φ^′_j ′

=

N

X

j=0

1

pi+pj+ 1H^pi+pj+1φ^′_j ′

−δ⁻² p_ip_j

pi+pj−1H^pi+pj−1φ_j

, which are equivalent to

N

X

j=0

1

p_j+ 1− 1 p_i+p_j+ 1

H^pj+2φ^′′_j +δ⁻²

N

X

j=0

p_ip_j

p_i+p_j −1H^pjφ_j = 0

fori= 1, . . . , N. Therefore, (1.5) under the condition (2.4) is transformed equivalently into

(3.1)





























 cη+

N

X

j=0

1

pj+ 1H^pj+1φ^′_j = 0,

N

X

j=0

1

pj+ 1 − 1 pi+pj+ 1

H^pj+2φ^′′_j +δ⁻²

N

X

j=0

p_ip_j

pi+pj−1H^pjφ_j = 0 for i= 1, . . . , N, c

N

X

j=0

H^pjφ^′_j +η+1 2











N

X

j=0

H^pjφ^′_j





2

+δ⁻²





N

X

j=0

p_jH^pj−1φ_j





2





= 0, whereH = 1 +η.

To simplify the notation, we put φ= (φ₁, . . . , φ_N)^T. Suppose that (c, η, φ₀,φ) is a solution to (3.1) and can be expanded with respect to δ² as

(3.2)















c=c₍₀₎+δ²c₍₁₎+δ⁴c₍₂₎+· · ·, η=δ²(η₍₀₎+δ²η₍₁₎+δ⁴η₍₂₎+· · ·), φ₀ =δ²(φ₀₍₀₎+δ²φ₀₍₁₎+δ⁴φ₀₍₂₎+· · ·), φ=δ²(φ₍₀₎+δ²φ₍₁₎+δ⁴φ₍₂₎+· · ·).

Remark 3.1. In this expansion, we assumed a priori that η, φ₀, and φ are of order O(δ²).

This is essentially the assumption that the solution is in the long wave regime.

Then, equating the coefficients of the lowest power of δ² we have A₁φ₍₀₎ =0 so that φ₍₀₎ =0, becauseA₁ is nonsingular. Equating the coefficients of δ² we have











c₍₀₎η₍₀₎+φ^′₀₍₀₎ = 0,

(1−a₀)φ^′′₀₍₀₎+A₁φ₍₁₎=0, c₍₀₎φ^′₀₍₀₎+η₍₀₎ = 0.

It follows from the first and the third equations of the above system that c²₍₀₎ = 1 in order to obtain a nontrivial solution. In the following we will consider the case c₍₀₎= 1. Then, we have (3.3) φ^′₀₍₀₎ =−η₍₀₎, φ₍₁₎ =−γφ^′′₀₍₀₎ =γη₍₀₎^′ ,

where γ is the constant vector defined in Remark 1.2. Next, equating the coefficients ofδ⁴ in the first and the third equations in (3.1) we have

(η₍₁₎+φ^′₀₍₁₎+c₍₁₎η₍₀₎+η₍₀₎φ^′₀₍₀₎+a₀·φ^′₍₁₎= 0, φ^′₀₍₁₎+η₍₁₎+c₍₁₎φ^′₀₍₀₎+1·φ^′₍₁₎+¹₂(φ^′₀₍₀₎)² = 0, which together with (3.3) yield

(3.4) 2c₍₁₎η₍₀₎−3

2η²₍₀₎−γη₍₀₎^′′ = 0, whereγ is the positive constant defined in Remark 1.2.

As is well known, (3.4) has a family of solutions of the form c₍₁₎ =α, η₍₀₎(x) = 2αsech² q_α

2γx with a parameterα >0. Then, it follows from (3.3) that







φ₀₍₀₎(x) =−2√

2αγtanh q_α

2γx

+ const., φ₍₁₎(x) =−4αγq_α

2γtanh q_α

2γx

sech² q_α

2γx .

Therefore, it is natural to expect that (1.5) has a family of solutions of the form

(3.5)















c= 1 +αδ²+O(δ⁴), η(x) = 2αδ²sech² q_α

2γx

+O(δ⁴), φ₀(x) =−2√

2αγδ²tanh q

α 2γx

+O(δ⁴), φ(x) =−4αγq

α

2γδ⁴tanh q

α 2γx

sech² q

α 2γx

+O(δ⁶) with parameters α >0 and 0< δ≪1.

4 Reduction of the problem

Without loss of generality, we can assume that α = 2γ. We denote the approximate solitary wave solution obtained in the previous section byη₍₀₎ andφ₍₀₎, that is,

(4.1) η₍₀₎(x) = 4γsech²x, φ₍₀₎(x) =−4γtanhx.

Then, we will look for solutions to the Isobe–Kakinuma model (1.5) in the form (4.2) c= 1 + 2γδ², η=δ²η₍₀₎+δ⁴ζ, φ₀ =δ²φ₍₀₎+δ⁴ψ₀, φ=δ⁴γη^′₍₀₎+δ⁶ψ.

Plugging these into the first and the third equations in (3.1) we obtain (4.3)

(ζ+ψ^′₀+G₁+δ² (2γ−η₍₀₎)ζ+η₍₀₎ψ₀^′ +a₀·ψ^′+G₂

+δ⁴F₁ = 0, ζ+ψ^′₀+G₃+δ² (2γ−η₍₀₎)ψ₀^′ +1·ψ^′+G₄

+δ⁴F₂ = 0, where















G₁ = 2γη₍₀₎−η₍₀₎² +κ₁η^′′₍₀₎, G₂ =κ₂η₍₀₎η₍₀₎^′′ ,

G₃ =−2γη₍₀₎+ ¹₂η²₍₀₎+κ₂η^′′₍₀₎,

G₄ =κ₂(2γ−η₍₀₎)η₍₀₎^′′ +κ₃η₍₀₎η₍₀₎^′′ +¹₂κ²₃(η^′₍₀₎)², withκ₁=a₀·γ,κ₂ =1·γ, and κ₃ = (p₁, . . . , pN)^T·γ and

F₁ =ζψ^′₀+η₍₀₎

N

X

j=1

ψ^′_j+

N

X

j=1

1

p_j+ 1δ⁻⁴ H^pj+1−1−δ²(p_j+ 1)η₍₀₎

(γ_jη^′′₍₀₎+δ²ψ^′_j), F₂ =F₃+ 2γF₄+1

2(ψ₀^′ +κ₂η₍₀₎^′′ )² +1

2δ⁻⁴

φ^′₍₀₎+δ²(ψ^′₀+κ₂η₍₀₎^′′ ) +δ⁴F₄2

− φ^′₍₀₎+δ²(ψ₀^′ +κ₂η^′′₍₀₎)2

+1 2δ⁻²











N

X

j=1

pjH^pj−1(γjη₍₀₎^′ +δ²ψj)





2

−





N

X

j=1

pjγjη^′₍₀₎





2



 , withH = 1 +δ²η₍₀₎+δ⁴ζ and

F₃ =η₍₀₎

N

X

j=1

pjψ_j^′ +

N

X

j=1

δ⁻⁴(H^pj−1−δ²pjη₍₀₎)(γjη₍₀₎^′′ +δ²ψ^′_j),

F₄ =

N

X

j=1

ψ^′_j+

N

X

j=1

δ⁻²(H^pj−1)(γjη^′′₍₀₎+δ²ψ_j^′).

In the above calculations, we have used the identities

N

X

j=0

H^pjφ^′_j =δ²φ^′₍₀₎+δ⁴(ψ^′₀+κ₂η^′′₍₀₎) +δ⁶



κ₃η₍₀₎η^′′₍₀₎+

N

X

j=1

ψ^′_j



+δ⁸F₃

=δ²φ^′₍₀₎+δ⁴(ψ^′₀+κ₂η^′′₍₀₎) +δ⁶F₄.

Since η₍₀₎ satisfies (3.4) with c₍₁₎= 2γ, we have G₁ =G₃, so that (4.3) is equivalent to (4.4)

((η₍₀₎−2γ)ζ+ 2(γ−η₍₀₎)ψ₀^′ + (1−a₀)·ψ^′ =g₀+δ²f₀, ζ+ψ^′₀=gN+1+δ²f_N+1,

with g₀ =G₂−G₄, g_N+1 =G₁,f₀ =F₁−F₂, and f_N+1 = (η₍₀₎−2γ)ψ₀^′ −1·ψ^′−G₄−δ²F₂. On the other hand, plugging (4.2) into the second equation in (3.1), we obtain

(4.5) (1−a₀)ψ^′′₀−δ²(A₀−1⊗a₀)ψ^′′+A₁ψ=g+δ²f, where

g= 2(1−a₀)−A₁diag(p₁, . . . , p_N)γ

η₍₀₎η^′₍₀₎+ (A₀−1⊗a₀)γη₍₀₎^′′′ , (4.6)

f = (1−a₀) δ⁻⁴(H²−1−2δ²η₍₀₎)η₍₀₎^′ +δ⁻²(H²−1)ψ^′′₀ (4.7)

−(A₀−1⊗a₀)δ⁻² diag(H^p1+2−1, . . . , H^pN+2−1)

(γη₍₀₎^′′′ +δ²ψ^′′)

−A₁

δ⁻⁴ diag(H^p1−1−δ²p₁η₍₀₎, . . . , H^pN −1−δ²p_Nη₍₀₎) γη₍₀₎^′ +δ⁻² diag(H^p1−1, . . . , H^pN−1)

ψ . To summarize, the Isobe–Kakinuma model (3.1) is reduced to

(4.8)







(η₍₀₎−2γ)ζ+ 2(γ −η₍₀₎)ψ₀^′ + (1−a₀)·ψ^′=g₀+δ²f₀, (1−a₀)ψ₀^′′−δ²(A0−1⊗a₀)ψ^′′+A₁ψ=g+δ²f, ζ+ψ₀^′ =g_N+1+δ²f_N+1

for the remainder terms (ζ, ψ₀,ψ). We note thatg₀ andg_N+1are polynomials in (η₍₀₎, η^′₍₀₎, η₍₀₎^′′ ), that g is a polynomial in (η₍₀₎, η₍₀₎^′ , η^′′′₍₀₎), and that g₀, g_N+1 ∈ B_e^∞ and g ∈ B_o^∞. More- over, f₀ and fN+1 are polynomials in (η₍₀₎, η₍₀₎^′ , η^′′₍₀₎, ζ, ψ₀^′,ψ,ψ^′), and f is a polynomial in (η₍₀₎, η^′₍₀₎, η₍₀₎^′′′ , ζ, ψ^′′₀,ψ, δ²ψ^′′), whose coefficients are polynomials in δ.

In view of (4.8) we proceed to consider the following system of linear ordinary differential equations

(4.9)







(η₍₀₎−2γ)ζ+ 2(γ−η₍₀₎)ψ^′₀+ (1−a₀)·ψ^′ =F₀, (1−a₀)ψ₀^′′−δ²(A0−1⊗a₀)ψ^′′+A₁ψ =F, ζ+ψ^′₀ =F_N+1

for unknowns (ζ, ψ₀,ψ). We can rewrite the third equation in (4.9) as

(4.10) ζ =−ψ^′₀+F_N+1,

which enable us to remove the unknown variable ζ from the equations. It follows from the second equations in (4.9) that

ψ =−γψ₀^′′+δ²A₂ψ^′′+A⁻¹₁ F,

where A₂ = A⁻¹₁ (A₀−1⊗a₀). Plugging this and (4.10) into the first equation in (4.9), we obtain

(4.11) −γψ₀^′′′+ (4γ−3η₍₀₎)ψ₀^′ =F₀+ (2γ−η₍₀₎)F_N+1−γ·F^′−δ²β·ψ^′′′,

whereβ=A^T₂(1−a₀). This is the equation for ψ₀. In order to derive equations to solveψ, we putχ=ψ^′′₀. Then, it follows from (4.9) that

(4.12)

((1−a₀)·ψ^′ =F₀+ (2γ −η₍₀₎)F_N+1+ (3η₍₀₎−4γ)ψ₀^′, (1−a₀)χ−δ²(A₀−1⊗a₀)ψ^′′+A₁ψ=F,

This is the equations for (χ,ψ). Of course, χ is expected to be ψ^′′₀, but we do not know it a priori. However, we have the following lemma.

Lemma 4.1. If ζ, ψ₀,ψ, and χ solve (4.10)–(4.12) and satisfies χ(0) = ψ₀^′′(0), then we have χ=ψ^′′₀. Particularly, ζ, ψ₀, and ψ solve (4.9).

Proof. It follows from the second equation in (4.12) that ψ=−γχ+δ²A₂ψ^′′+A⁻¹₁ F. Plugging this into the first equation in (4.12) we have

−γχ^′+ (4γ−3η₍₀₎)ψ₀^′ =F₀+ (2γ −η₍₀₎)FN+1−γ·F^′−δ²β·ψ^′′′. Comparing this with (4.11) we obtain (χ−ψ₀^′′)^′ = 0, which gives the desired results.

5 Green’s functions

In view of (4.11), we first consider the solvability of the equation for an unknownu

(5.1) −γu^′′+ (4γ −3η₍₀₎)u=f

under the condition u(x) → 0 as x → ±∞. We remind that η₍₀₎(x) = 4γsech²x, so that this equation does not depend essentially on the positive constantγ. This equation has already been analyzed by K. O. Friedrichs and D. H. Hyers [3]. Let us recall briefly their result. In order to construct Green’s function, we consider the initial value problems with homogeneous equation:

(−γu^′′₁+ (4γ−3η₍₀₎)u₁= 0, u₁(0) = 0, u^′₁(0) = 1,

(−γu^′′₂ + (4γ−3η₍₀₎)u₂ = 0, u₂(0) = 1, u^′₂(0) = 0.

The solutions of these initial value problems have the form (u₁(x) = sech²xtanhx,

u₂(x) = ¹₈(−6−cosh(2x) + 15sech²x−15xsech²xtanhx).

We note that these fundamental solutions satisfy

(5.2) |u^(k)₁ (x)| ≤C_ke^−2|x|, |u^(k)₂ (x)| ≤C_ke^2|x|

fork= 0,1,2, . . .. Since the Wronskian isu^′₁(x)u₂(x)−u₁(x)u^′₂(x) ≡1, the solution u of (5.1) can be written as

u(x) =γ⁻¹

C₁− Z x

0

u₂(y)f(y)dy

u₁(x) +γ⁻¹

C₂+ Z x

0

u₁(y)f(y)dy

u₂(x),

where C₁ and C₂ are arbitrary constants. In order that this solution satisfies the condition u(x)→0 as x→ ±∞, as necessary conditions, we obtain

C₂+ Z _∞

0

u₁(y)f(y)dy=C₂− Z ₀

−∞

u₁(y)f(y)dy = 0, so that we have a necessary condition

(5.3)

Z ∞

−∞

u₁(y)f(y)dy= 0

for the existence of the solution, and that the solution has the form u(x) =γ⁻¹u₁(x)

C₁−

Z x 0

u₂(y)f(y)dy

−γ⁻¹u₂(x) Z ∞

x

u₁(y)f(y)dy.

Nonuniqueness of the solution comes from the translation invariance of the original equations.

Proposition 5.1. Let γ >0. For any f ∈B_e^∞, there exists a unique solution u∈B_e^∞ to (5.1).

Moreover, for anyk= 0,1,2, . . ., the solution satisfies

kukk+2 ≤C₀kfkk+C_kkfk0, where C_k is a positive constant depending on k while C₀ does not.

Proof. Since f is even and u₁ is odd, the necessary condition (5.3) for the existence of the solution is automatically satisfied. Moreover, we restricted ourselves to a class of solutions which are even so that the solution is unique and given by

u(x) =−γ⁻¹u₁(x) Z x

0

u₂(y)f(y)dy−γ⁻¹u₂(x) Z _∞

x

u₁(y)f(y)dy.

Using this solution formula, we easily obtain kukk+2 ≤C_kkfk^k.

We proceed to improve this estimate. Differentiating (5.1)j-times, we have

−γu^(j)′′+ (4γ−3η₍₀₎)u^(j)=f^(j)+ 3[(_dx^d)^j, η₍₀₎]u

Applying the previous estimate withk= 0 tou^(j)and adding them forj= 1,2, . . . , k, we obtain kukk+2 ≤ Ckfkk+C_kkukW^k,∞, which together with the interpolation inequality kukW^k,∞ ≤ ǫkukW^k+2,∞ +C_ǫkuk^L∞ ≤ǫkukk+2+C_ǫkuk2 for any ǫ >0 yields the refined estimate.

In view of (4.12), we then consider the solvability of the system of ordinary differential equations with constant coefficients

(5.4)

((1−a₀)·ϕ=f₀,

(1−a₀)ϕ₀−δ²(A₀−1⊗a₀)ϕ^′′+A₁ϕ=f,

for unknowns ϕ₀ and ϕ = (ϕ₁, . . . , ϕN)^T while f₀ and f = (f₁, . . . , fN)^T are given functions.

We procced to construct Green’s function to this system. By taking the Fourier transform of (5.4), we obtain

0 (1−a₀)^T

1−a₀ (δξ)²(A₀−1⊗a₀) +A₁ ˆ ϕ₀

ˆ ϕ

= fˆ₀

fˆ

. Now, we need to show that the coefficient matrix is invertible. Put

(5.5) q(ξ²) =−det

0 (1−a₀)^T 1−a₀ ξ²(A₀−1⊗a₀) +A₁

. Then, we have the following lemma.

Lemma 5.2. q(ξ²) is a polynomial in ξ² of degree N −1. Moreover, there exists a positive constant c₀ such that for any ξ ∈R we have q(ξ²)≥c₀>0.

Proof. It is easy to see that

q(ξ²) =−det

0 (1−a₀)^T 1−a₀ ξ²(A₀−a₀⊗a₀) +A₁

.

Here, we will show that the symmetric matrix A₀−a₀⊗a₀ is positive. In fact, for any φ = (φ₁, . . . , φN)^T∈R^N, we see that

φ·(A₀−a₀⊗a₀)φ=

N

X

i,j=1

1

pi+pj+ 1φ_iφ_j− N

X

j=1

1 pj+ 1φ_j

2

= Z 1

0

N

X

j=1

φ_jz^pj 2

dz− Z 1

0 N

X

j=1

φ_jz^pjdz 2

≥0

by the Cauchy–Schwarz inequality. Moreover, the equality holds if and only if PN

j=1φ_jz^pj is constant forz∈[0,1], that is, φ=0. This shows that A₀−a₀⊗a₀ is positive.

Next, we note that for any invertible matrix A and any vector xwe have det

0 x^T x A

=−(detA)x·A⁻¹x.

This equality comes easily from the identity 1 −x^TA⁻¹

0 Id

0 x^T x A

=

−x·A⁻¹x 0^T

x A

.

Since A₁ is also positive, for each ξ ∈ R the matrix ξ²(A₀ −a₀ ⊗a₀) +A₁ is positive too.

Therefore, we have q(ξ²)>0, which yields the second assertion of the lemma.

It is easy to see that q(ξ²) is a polynomial in ξ² of degree less than or equal to N −1 and that the coefficient of ξ^2(N−1) is

−

0 (1−a₀)^T 1−a₀ A₀−a₀⊗a₀

,

which is positive due to the positivity A₀−a₀⊗a₀. Therefore, we obtain the first assertion of the lemma.

We denote the inverse matrix by 0 (1−a₀)^T

1−a₀ (δξ)²(A0−1⊗a₀) +A₁ −1

=

q((δξ)²) q((δξ)²)^T q((δξ)²) Q((δξ)²)

,

whereq(ξ²) = (q₁(ξ²), . . . , qN(ξ²))^T andQ(ξ²) = (qij(ξ²))_1≤i,j≤N. Then, the solution (ϕ₀,ϕ) to (5.4) can be expressed as

(5.6)

(ϕˆ₀=q((δξ)²) ˆf₀+q((δξ)²)·fˆ, ˆ

ϕ=q((δξ)²) ˆf₀+Q((δξ)²)ˆf.