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

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.


INTRODUCTION
In this paper, we consider the motion of two-dimensional water waves in the case of a 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. Let be the time variable and ( , ) the spatial variables. We assume that the water surface and the bottom are represented as = ( , ) and = −ℎ, respectively. As was shown by Luke, 1 the water wave problem has a variational structure. His Lagrangian density is of the form where Φ = Φ( , , ) is the velocity potential and is the gravitational constant. Isobe 2,3 and Kakinuma 4-6 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 where 0 , 1 , … , are nonnegative integers satisfying 0 = 0 < 1 < ⋯ < . Then, the corresponding Isobe-Kakinuma model in a nondimensional form is written as where ( , ) = 1 + ( , ) is the normalized depth of the water and a nondimensional parameter defined by the ratio of the mean depth ℎ to the typical wavelength . Here and in what follows we use a notational convention 0 0 = 0. For the derivation and basic properties of this model, we refer to Murakami and Iguchi,7 and Nemoto and Iguchi. 8 It was also shown by Iguchi 9,10 that the Isobe-Kakinuma model (3) is a higher order shallow water approximation for the water wave problem in the strongly nonlinear regime. Moreover, Duchêne and Iguchi 11 found that the Isobe-Kakinuma model enjoys a Hamiltonian structure analogous to the one exhibited by Zakharov on the full water wave problem and that the Hamiltonian of the model is a higher order shallow water approximation to the one of the full water wave problem. We note also that the Isobe-Kakinuma model (3) in the case = 0 is exactly the same as the shallow water equations. In the sequel, we always assume ≥ 1. In this paper, we look for solitary wave solutions to this model under the form = ( + ), = ( + ), = 0, 1, … , , where ∈ is an unknown constant phase speed. Plugging (4) into (3), we obtain a system of ordinary differential equations As expected, this system has a variational structure, that is, the solution of this system is obtained as a critical point of the functional and Φ app is the approximate velocity potential defined by 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 (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 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 (5), we introduce norms ‖ ⋅ ‖ and ‖ ⋅ ‖ for = 0, 1, 2, …, by where ( ) is the th order derivative of . We also introduce function spaces and as closed subspaces of all even and odd functions ∈ ( ) satisfying ‖ ‖ < +∞, respectively, equipped with the norm ‖ ⋅ ‖ , and put ∞ = ∩ ∞ =0 for = , . The following theorem guarantees the existence of small amplitude solitary wave solutions to the Isobe-Kakinuma model in the long wave regime.
) , where is the gravitational constant and is 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 = 1 and 1 = 2, that is, (51). 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 Iguchi. 9,10 Numerical analysis in this paper suggests that there exists a critical value of given approximately by = 0.62633493 (8) such that for any ∈ (0, ), the Isobe-Kakinuma model (51) admits a smooth solitary wave solution, which decays exponentially at spatial infinity, and that this family of waves converges to a solitary wave of extreme form with a shape crest as ↑ . Moreover, the included angle of the crest in the physical space is given approximately by See Figure 1.
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 Friedrichs and Hyers. 12 Then, Amick and Toland 13 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' wave of extreme form as well as the sharp crest of the included angle 120 • was predicted by Stokes, 14 and then proved theoretically by Amick, Fraenkel, and Toland, 15 and Plotnikov. 16 An extension of these existence theories to the water waves with vorticity was first given by Groves and Wahlén, 17 and Hur 18 for small amplitude solitary waves. Then, Varvaruca 19 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 Korteweg-de Vries (KdV) 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 KdV equation, but again do not have any solitary wave of extreme form. Compared to these models, the Isobe-Kakinuma model even in the simplest case may catch the property on the existence of the solitary wave of extreme form, although the included angle of the crest is not 120 • and that the existence of the extreme wave is not yet proved theoretically. We also mention a result by Lannes and Marche, 20 where it was shown that extended Green-Naghdi equations for the water waves with a constant vorticity have a solitary wave solution of extreme form.
Among the long wave models for water waves that admit solitary waves of extreme form with a sharp crest, the Camassa-Holm equation is probably the most notable one. This equation was first derived by Fuchssteiner and Fokas 21 as an example of a system possessing a bi-Hamiltonian structure. Then, Camassa and Holm 22 derived this equation as a long wave model for water waves and showed that the equation in a special case has exact peaked solitons called nowadays peakons. Ehrnström and Wahlén 23 considered the Whitham equation and resolved Whitham's conjecture, that is, the existence of highest, cusped, and travelling wave solution of the equation. We also mention a result by Geyer and Pelinovsky, 24 where the uniqueness of the peaked periodic wave with a parabolic profile to the reduced Ostrovsky equation and its instability were demonstrated.
The remainder of this paper is organized 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 to 2 in the formal expansion (20) of the solitary wave solution to the Isobe-Kakinuma model (5). In Section 4, we reduce the problem by deriving equations for the remainder terms. More precisely, putting = 1 + 2 2 and defining the remainder terms ( , 0 , 1 , … , ) by where (0) ( ) = 4 sech 2 is a KdV soliton and − (0) ( ) = 4 tanh is an antiderivative of the KdV soliton, we derive equations for these remainder terms. In the following sections, we will solve the equations for ( , 0 , 1 , … , ) and show uniform boundedness of these remainder terms with respect to small . Here, for readers' convenience we sketch the proof. Linearized equations for the remainder terms ( , 0 , 1 , … , ) are given by where = ( 1 , … , ) T and ⊗ 0 = ( 0 ) T = ( 0 , … , 0 ) T is a tensor product. This is a system of second-order ordinary differential equations with variable coefficients, so that it is not so easy to construct directly Green's function of this system. Therefore, we will further reduce this linear system as follows. It follows from (10) that with a constant vector . As the last term in the right-hand side would be a harmless term for sufficiently small , we can regard this scalar equation as an equation for ′ 0 . This scalar equation was already analyzed by Friedrichs and Hyers 12 and, for readers' convenience, we reveiw a construction of Green's function in Section 5. It also follows from (10) that where the right-hand sides would be harmless terms again for sufficiently small . Once the right-hand sides are given, this system of equations determines not only but also ′′ 0 . Therefore, we have too many equations for 0 . To overcome this difficulty, we introduce a dummy variable = ′′ 0 and replace the above system with which is a system for and . As (12) has constant coefficients, it is straightforward to construct Green's function by Fourier transform. The construction and its estimates are also given in Section 5. Then, we obtain ( 0 , , ) a solution to (11)-(12) by using a method of successive approximation and it turns out that the solution satisfies the desired identity = ′′ 0 , see Lemma 1. In this way, we solve the system of linear equations (10) and this is carried out in Section 6. Then, we finish the proof of Theorem 1 by applying a fixed point procedure. 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.

CONSERVATION LAWS
As was explained in the previous section, for the Isobe-Kakinuma model (3), the mass, the momentum in the horizontal direction, and the total energy are conserved. In the numerical analysis that will be carried out in Section 7, we also need to know corresponding flux functions. The following proposition gives such flux functions.
Proof. Conservation law of mass (13) is nothing but the first equation in (3) with = 0. It follows from the equations in (3) that which gives conservation law of momentum (14). We see also that which gives conservation law of energy (15). ■ These conservation laws provide directly the following proposition.
) of class 2 and satisfying the condition at spatial infinity we have and

APPROXIMATE SOLUTIONS
We first transform the Isobe-Kakinuma model (5) into an equivalent system. Throughout this and the following sections, we always consider classical solutions to the Isobe-Kakinuma model (5), except in the argument of extreme waves in the last section. The first equation in (5) with = 0 can be integrated under the condition at spatial infinity (16) so that we have (17), particularly, Plugging this into the first equation in (5) with = 1, … , to eliminate , we have which are equivalent to where = 1 + . To simplify the notation, we put = ( 1 , … , ) T . Suppose that ( , , 0 , ) is a solution to (19) and can be expanded with respect to 2 as for ∈ .
Remark 4. In this expansion, we assumed a priori that , 0 , and are of order ( 2 ). This is essentially the assumption that the solution is in the long wave regime.
Then, plugging (20) into (19) and equating the coefficients of the lowest power of 2 , we have 1 (0) = so that (0) = , because 1 is nonsingular. Equating the coefficients of 2 , we obtain It follows from the first and the third equations of the above system that 2 (0) = 1 in order to obtain a nontrivial solution. In the following, we will consider the case (0) = 1. Then, we have ′ 0(0) = − (0) , (1) where is the constant vector defined in Remark 1. Next, equating the coefficients of 4 in the first and the third equations in (19), we have which together with (21) yield where is the positive constant defined in Remark 1.
As is well known, (22) is nothing but the KdV equation for traveling waves, and it has a family of solutions of the form with a parameter > 0. Then, it follows from (21) that Therefore, it is natural to expect that (5) has a family of solutions of the form with parameters > 0 and 0 < ≪ 1.
In view of (31), we proceed to consider the following system of linear ordinary differential equations for unknowns ( , 0 , ). We can rewrite the third equation in (32) as , where 2 = −1 1 ( 0 − ⊗ 0 ). Plugging this and (33) into the first equation in (32), we obtain where = T 2 ( − 0 ). This is an equation for 0 . In order to take into account the fact that the first two equations in (32) are equations for ( 0 , ) and that we have already an equation for 0 , we introduce a new variable = ′′ 0 and solve This is a system of equations for ( , ). Of course, is expected to be ′′ 0 , but we do not know it a priori. However, we have the following lemma. Plugging this into the first equation in (35), we have Comparing this with (34), we obtain ( − ′′ 0 ) ′ = 0, which gives the desired results. ■

GREEN'S FUNCTIONS
In view of (34), we first consider the solvability of the equation for an unknown − ′′ + (4 − 3 (0) ) = under the condition ( ) → 0 as → ±∞. We remind that (0) ( ) = 4 sech 2 , so that this equation does not depend essentially on the positive constant . This equation has already been analyzed by Friedrichs and Hyers. 12 Let us recall briefly their result. In order to construct Green's function, we consider the initial value problems with homogeneous equation: The solutions of these initial value problems have the form We note that these fundamental solutions satisfy for = 0, 1, 2, …. As the Wronskian is ′ 1 ( ) 2 ( ) − 1 ( ) ′ 2 ( ) ≡ 1, the solution of (36) can be written as where 1 and 2 are arbitrary constants. In order that this solution satisfies the condition ( ) → 0 as → ±∞, as necessary conditions, we obtain so that we have a necessary condition for the existence of the solution, and that the solution has the form Nonuniqueness of the solution comes from the translation invariance of the original equations. When is an even function, the necessary condition (38) is automatically satisfied and the solution that is an even function is uniquely determined by the above formula with 1 = 0. As a result, we can show easily the following proposition.
where is a positive constant depending on while 0 does not.
In view of (35), we then consider the solvability of the system of ordinary differential equations with constant coefficients { ( − 0 ) ⋅ = 0 , for unknowns 0 and = ( 1 , … , ) T while 0 and = ( 1 , … , ) T are given functions. We proceed to construct Green's function to this system. By taking the Fourier transform of (39), we obtain Now, we need to show that the coefficient matrix is invertible. Put Then, we have the following lemma.
Proof. It is easy to see that ) .
Here, we will show that the symmetric matrix 0 − 0 ⊗ 0 is positive. In fact, for any = ( 1 , … , ) T ∈ , we see that by the Cauchy-Schwarz inequality. Moreover, the equality holds if and only if ∑ =1 is constant for ∈ [0, 1], that is, = . This shows that 0 − 0 ⊗ 0 is positive.
In view of (42), we consider a function ( ) defined by where ( 2 ) and ( 2 ) are polynomials in 2 of degree and of degree less than − 1, respectively, and ( 2 ) ≥ 0 > 0. It is easy to see that ( ) is a real valued, even, and continuous function on . As ( 2 ) is a polynomials in 2 of degree − 1 with real coefficients and positive definite, roots of ( 2 ) = 0 have the form Therefore, by the residue theorem, we have an expression with some complex constants ± 1 , … , ± −1 . By taking into account the continuity at = 0, we have also We note that the second derivative ′′ ( ) contains in general the Dirac delta function. Thanks to these expressions, we have the following lemma.

Lemma 4. Let ( ) be defined by
Proof. It follows from Lemma 3 that In view of ( * ) ′ = ′ * , a similar estimate holds for ( * ) ′ . Moreover, it is easy to see that if is even or odd, then so is * , respectively. Therefore, we obtain the desired result. ■ In order to give an estimate for the solution ( 0 , ) to (39), it is convenient to introduce the following weighted norm for = 0, 1, 2, …. By the above arguments, we obtain the following proposition.

EXISTENCE OF SMALL AMPLITUDE SOLITARY WAVES
We begin with an existence theorem of the solution ( , 0 , ) to the system of linear ordinary differential equations (32).

NUMERICAL ANALYSIS FOR LARGE AMPLITUDE SOLITARY WAVES
In the previous section, we proved the existence of small amplitude solitary wave solutions to the Isobe-Kakinuma model (3). In the present section, we will analyze numerically large amplitude solitary wave solutions to the model in the special case where the parameters are chosen as = 1 and 1 = 2. Even in this special case, the Isobe-Kakinuma model gives a better approximation than the Green-Naghdi equations in the shallow water and strongly nonlinear regime. Therefore, we will consider the equations where = 1 + . In this case, the constant in Theorem 1 is given by = 1 3 so that we can put For numerical analysis, it is convenient to rewrite the equations in (51) as a system of ordinary differential equations of order 1. To this end, we suppose that ( , 0 , 1 ) is a classical solution to (51)-(53) and introduce a new unknown function which is the horizontal component of the velocity on the water surface. Observe that the first equation in (51) and (56) can be rewritten into a system ( or equivalently . (57) Differentiating the first equation in (51) and (56), we obtain ( which appears in the denominator of the right-hand sides in (61).
This is a quartic equation in (0) so that for each given we have four roots. Generally, two of them are complex numbers and one real root does not give the correct initial data for the solitary wave solution, so that we can determine the initial data ( (0), (0), 1 (0)) for appropriately chosen . We proceed to compare solitary wave solutions to the Isobe-Kakinuma model (51)-(54) calculated numerically as above with the classical solitons of the KdV equation which is the first approximation for small amplitude solitary wave solutions to the Isobe-Kakinuma model as was guaranteed by Theorem 1. In Figure 2, we plot the surface profile ( ) of the solitary In Figures 3 and 4, we plot also the surface profiles of the solitary wave solutions to the Isobe-Kakinuma model for several values of in the same figures. We observe that the wave height is monotonically increasing as increases and that a sharp crest is formed as approaches the critical  value . We will analyze more precisely this formation of a sharp crest. To this end, we calculate numerically the curvature at the crest of the surface profile, which is defined by In Table 1, we list the wave height, the curvature at the crest, and the value of the denominator defined by (64) at the crest of the solitary wave solutions to the Isobe-Kakinuma model for several values of . We observe that as approaches some critical value , the wave height converges toward a maximum wave height (0), the curvature at the crest is blowing up, and the denominator at the crest is going to vanish. This consideration suggests strongly the existence of solitary wave of extreme form as well as a sharp crest to the Isobe-Kakinuma model and that the critical value would be obtained by the equation (0) = 0, that is, 6 (0)( + (0)) 2 − 3( + (0))( (0) + (0) (0)) − (0) 2 = 0, where (0) = 1 + (0) and is given by (54). In fact, we can calculate this critical value , the maximum wave height (0), the horizontal velocity (0) of the water at the crest, and the critical In Figures 5 and 6, we plot the surface and horizontal velocity profiles of the solitary wave of extreme form to the Isobe-Kakinuma model. It follows from (70) that + (0) > 0, which means that the crest of the solitary wave of extreme form is not the stagnation point unlike the full water wave problem. We also note that for > the quartic equation (67) for (0) does not have any real root, which suggests nonexistence of the solitary wave higher than the extreme wave. We proceed to calculate the angle of the crest of the solitary wave of extreme form. To this end, it is sufficient to evaluate ′ (±0), where ( , , 1 ) is the solution to the Isobe-Kakinuma model (61) in the critical case = . We denote by the corresponding denominator defined by (64) so that by the first equation in (61), we have Differentiating (64) and using 1 (0) = 0, we have ′ (±0) = ( 3 (0) 2 − 2 (0) ) ′ (±0) + 3 ( 2 (0) (0) + ) ′ (±0).
It follows from (60) in the critical case = that which gives the angle of the crest.
Here, note that we have rewritten all physical quantities in a nondimensional form. In order to calculate the angle of the crest in the physical space, we have to work with dimensional variables. Let * and * be the horizontal spatial coordinate and the surface elevation in the physical space, respectively, so that we have * = and * = ℎ , where ℎ is the mean depth of the water and the typical wavelength. The angle of the crest in the physical space should be calculated from * ′ (±0). In view of the relation * ( * ) = ℎ ( −1 * ), we have * ′ ( * ) = ′ ( ) so that * ′ (±0) = ′ (±0), that is, * ′ (±0) = ∓ We are now able to give a numerical approximation of the included angle of the shape crest in the physical space, which is approximately equal to = 152.6 • . Let us note that it is larger than the included angle 120 • of the sharp crest of the solitary wave solution of extreme form to the full water wave problem. ACKNOWLEDGMENT T. I. was partially supported by JSPS KAKENHI Grant Number JP17K18742 and JP17H02856.