Alicia Aloisi

Jean-Claude Berthet

Caroline Genre

Pierre Thuéry

Thibault Cantat

Complexes of the Tripodal Phosphine Ligands PhSi(XPPh2)3 (X = CH2, O): Synthesis, Structure and Catalytic Activity in the Hydroboration of CO2 †

Keywords: Tripodal ligands, Phosphorus ligands, Iron, Cobalt, Copper, Carbon dioxide reduction, Hydroboration

The coordination chemistry of the Fe 2+ , Co 2+ and Cu + ions was explored with the triphosphine and triphosphinite ligands PhSi{CH2PPh2}3 (1) and PhSi{OPPh2}3 (2), so as to evaluate the impact of the electronic properties of the tripodal phosphorus ligands on the structure and reactivity of the corresponding complexes. The synthesis and characterization of the complexes [Fe( 3 -PhSi{CH2PPh2}3)

5) and [Cu( 3 -PhSi{OPPh2}3)I] (5') were carried out. The crystal structures of 3, 3', 4, 4', and of the solvates 5•3THF and 5'•THF are reported. Complexes 3-5' were shown to promote the catalytic hydroboration of CO2 with (9-BBN)2 (9-BBN = 9-Borabicyclo[3.3.1]nonane). While the iron and cobalt complexes of the triphosphine 1 are more active than the analogous complexes with 2, the opposite trend is observed with the copper catalysts. Overall, the copper catalysts 5 and 5' are both more reactive and more selective than the Fe and Co catalysts, enabling the formation of the acetal H2C(OBBN)2 with a high molar ratio of H2C(OBBN)2:CH3OBBN up to 92:8. † CCDC reference numbers 1481874-1481881. For crystallographic data in CIF or other electronic format see DOI: 60 40 0:100 99 66 1.7

Reaction conditions : a) 0.18 mmol of B-H functionalities (0.09 mol of (9-BBN)2); b) % calculated relatively to the quantity of B-H functionalities; c) Yields and selectivity determined by { 1 H}NMR using mesitylene as internal standard; d)TON = [mmol(acetal) x 2 + mmol(methoxy) x 3]/mmol catalyst; e) TOF = TON/time(h).

Introduction

Tripodal ligands play a major role in transition metal chemistry, as they increase the stability of their complexes and enable a fine control of the reactivity at the metal center through their steric and electronic properties and their specific coordination geometry.

Within tripodal ligands, a few triphosphines have been utilized so far in coordination chemistry and catalysis and, among them, 1 triphos (MeC(CH2PPh2)3) has attracted considerable attention over the last years, being a valuable ligand for a number of late transition metals (with ca 700 crystal structures). The ability of triphos to stabilize low metal oxidation states and to adopt both  2 -and  3 -coordination modes has led to widespread and successful applications in catalysis, including CO2 hydrogenation, disproportionation of formic acid and reduction of amides. 2,3,4,5,6,7 The efficiency of catalytic reactions is generally strongly influenced by the nature of the ancillary ligand.

Because small changes within a ligand series, such as the pendant alkyl chains or the nature and/or the position of the linking groups, can induce different outcomes in a catalytic reaction, variations on the triphos backbone have been reported. Tripodal molecules such as PP3 (PP3 = P{CH2PR2}3), N{CH2PR2}3, R'Si{CH2PR2}3, etc. have been synthesized along with their coordination complexes. 8,9,10 In particular, R'Si(CH2PR2)3 ligands, featuring an organosilane linker, have been investigated and a number of their complexes exist with Ti, 11,12,13,14 V, 13,15,16,17, Cr, 18 Fe, [START_REF] Neumeyer | [END_REF]19,20,21,22,23 Cu, 24 Nb, 2,16,25 Mo, 26,27 Ru, [START_REF] Neumeyer | [END_REF]24,28,29,30,31 Rh, 24 Pd, 24 Ta, 2,16,32 W, 33 Os 1 and Pt. 24 Surprisingly, oxygen analogues of formula RSi(OPR'2)3 have remained elusive so far and, to our knowledge, only a single transition metal complex, eg [Mo(MeSi{OPOMe2}3)(CO)3], has been crystallographically characterized. 34 In this contribution we report the use of PhSi(CH2PPh2)3 (1) and its phosphinite derivative PhSi(OPPh2)3 (2) in the coordination chemistry of first-row transition metals, namely iron(II), cobalt(II) and copper(I) ions. The syntheses, NMR characterizations and crystal structures of pairs of complexes with the ligands 1 and 2 are detailed. To establish the influence of replacing the phosphines groups in 1 with phosphinites, the catalytic activity of the complexes was explored in the hydroboration of CO2 with (9-BBN)2. Because CO2 is a stable chemical waste, its selective multi-electron reduction under mild-conditions remains a challenge 35,[START_REF] Aresta | Carbon Dioxide as Chemical Feedstock[END_REF] and an ideal platform to establish structure/activity relationships for analogous catalysts.

Results and discussion

Synthesis of ligands 1 and 2

Scheme 1: Synthesis of the PhSi(XPPh2)3 ligands (X = O, CH2).

The syntheses of triphosphine PhSi(CH2PPh)3 (1) and its phosphinite analogue PhSi(OPPh)3 (2) are described in Scheme 1. Ligand 1 was obtained in a good 75 % yield, from a published protocol, [START_REF] Neumeyer | [END_REF][START_REF] Karsch | [END_REF]38 by reacting a toluene solution of Ph2PCH2Li(TMEDA) with PhSiCl3 in a 3:1 molar ratio. The new species 2 was prepared by analogy with the synthetic procedures reported for the phosphite esters RSi(OP{OR'})3 (R =Ph, Me; R' = OMe, OEt), by mixing PhSiCl3 with 3 equivalents of Ph2P(=O)H. Ph2P(=O)H is involved in an equilibrium with the phosphorus(III) derivatives Ph2P-OH and it readily adds to the chlorosilane, in the presence of NEt3 as a base, in toluene. 34,[39][40] After 5 h at room temperature, the solution was filtered to remove the HNEt3Cl deposit and 2 was obtained pure as a colorless oil, after vacuum drying (88 % yield). Compound 2 was characterized by its 1 H, 13 C and 31 P{ [START_REF] Neumeyer | [END_REF] H} NMR spectra in d8-THF. 2 exhibits a singlet at P100.93 for the P atoms which is shifted downfield in comparison to the chemical shift of -20.21 ppm in THF (-23.59 in benzene) for 1 (vs -27.3 ppm for triphos (toluene, -40°C)). [START_REF] Neumeyer | [END_REF]41 These P values for 1 and 2 fall in the range of classical phosphine and phosphinite groups, respectively. Importantly, the thermal stabilities of 1 and 2 strongly differ. Whereas a refluxing solution of 1 in benzene shows no sign of decomposition after 24 h, 2 rapidly degraded under the same conditions to a large number of unidentified phosphorus species.

Formation of iron complexes 3 and 3'

Treatment of FeCl2 with RSi(CH2PPh2)3 (R = Ph, i-Pr) [START_REF] Neumeyer | [END_REF] or the less sterically demanding ligands R'Si(CH2PMe2)3 (R' = tBu, Me) 20,22 has been previously reported, both in THF and CH2Cl2. In these weakly polar solvents, the colorless paramagnetic complexes [Fe( 2 -RSi{CH2PPh2}3)Cl2] (R = Ph, i-Pr) [START_REF] Neumeyer | [END_REF] or the purple, diamagnetic cationic [(Fe( 3 -RSi{CH2PMe2}3))2(-Cl)3][Cl] 20 complexes were formed. The pink iron(II) monocationic 22,23 were also obtained from the dinuclear species by addition of dmpe or dmpm (dmpe = dimethylphosphinoethane; dmpm = dimethylphosphinomethane). 22 These results provide a trend where the tridentate coordination mode of the RSi{CH2PR'2}3 ligands always affords diamagnetic and colored iron complexes, whereas colorless and paramagnetic derivatives are obtained with  2 -coordinated ligands.

derivatives [Fe( 3 -RSi{CH2PMe2}3)(-Cl)(Me2P(CH2)nPMe2)][Cl] (n= 1,2)
The polarity of the solvent and the nature of the X ligand in the FeX2 salt were found to influence the coordination mode (bidentate vs tridentate) of the RSi{CH2PR'2}3 triphosphine ligands. In addition, these parameters govern the crystallization of the complexes, which are obtained either in their neutral or cationic form. We thus considered the reaction of 1 and 2 with Fe(OTf)2 and acetonitrile (ε = 37.5 vs 7.5 for THF), aiming at preparing discrete cationic species with octahedral geometry. Reaction of 1 and 2 with Fe(OTf)2 in acetonitrile afforded orange (Y = O) and red (Y = CH2) solids, which crystallized readily in acetonitrile as 3 and 3', respectively, in moderate yields (55 and 52%, respectively). 3 and 3' underwent partial desolvation of the acetonitrile ligands during vacuum drying, and elemental analyses and 1 H NMR data collected for the resulting solids suggested a coordination of 2 to 3 acetonitrile molecules per iron ion. The iron complexes are soluble in THF yet insoluble in nonpolar solvents. 3 and 3' were characterized by 1 H, 13 C and 31 P{1H} NMR spectroscopy, from their acetonitrile solutions. Characteristic 31 P{ 1 H} resonance for 3 is a unique singlet at 30.15 ppm, shifted downfield with respect to the free ligand 1 (-20.21 ppm).

A similar trend is noted for 3', which 31 P{1H} spectrum features a singlet at P 147.9, shifted +47 ppm downfield from the free ligand 2. The NMR data of 3 are similar to those observed for [Fe(PhCH2C{CH2PPh2}3)(NCMe)3][BF4]2 (P 33.2) and [Fe(triphos)(NCMe)3][BF4]2 (P 32.3). 42,43 Large red and orange crystalline platelets of 3 and 3' were obtained by layering diethyl ether on an acetonitrile solution of the compounds. Both complexes crystallized as discrete cation-anion pairs and views of the cations of 3 and 3' are shown in Figure 1, respectively, while selected bond lengths and angles are gathered in Table 1. The 31 P{ 1 H} NMR spectra of 5 and 5' in dichloromethane at 21°C (Figure 3) exhibit a quartet centered at -33 and + 80 ppm, respectively. A JP-Cu coupling constant of 860 Hz is measured for 5' while the 31 P{ 1 H} NMR spectrum of 5 exhibits two nearly superimposed quartets related to the phosphorus coupling with each copper isotope (JP-63Cu ~ 860 Hz, JP-65Cu ~ 900 Hz). Classically, the 31 P{ 1 H} NMR spectra of copper-phosphine complexes present broad signals and the coupling between the 31 P and the two 63 Cu and 65 Cu isotopes is rarely observed in solution because the isotopes of copper are both quadrupolar nuclei. 46 In fact, the relaxation rate of quadrupolar nuclei is higher in solution than in the solid state and fluxional behaviors such as ligands exchanges might occur in solution, thereby precluding the observation of fine NMR structures. The observation of P-Cu coupling for 5 and 5' might point to a strong coordination of the tripodal ligands to the Cu + ion and a highly symmetric structure that decreases the electric field gradient at the tetrahedral center, thus reducing the quadrupolar relaxation rate 47 . On the contrary, the 31 P{ 1 H} NMR spectrum of 5'' in THF at 21°C shows a very broad signal at +81 ppm (Fig. S13), indicative of a stronger electric field gradient induced by the combination of a bromide and phosphinite ligand.

M(CO)4(PPh2POPPh2) (M = Mo, Cr) 48,49 [RuCl2(Ph2PO2PPh2)(PPh3)({Ph2PO2}2C2H4)], 50 or [Pt3(PPh2POPPh2)(PPh2)(C6F5)2]
. 51 These examples were often obtained inadvertently, with the P-O-P fragments resulting from the evolution of R2POH and/or R2(O=)P-PR2 molecules due to isomerisation of P=O bonding in the presence of M 2+ ions (Cu, Co…). [52][53][54] Recently Cu(I) and Cu(II) salts were also found to selectively catalyse the dehydrogenative coupling of R2P(=O)H molecules to R2(O=)P-O-R2P(=O) derivatives, under aerobic conditions. 55 Here, the Ph2POPPh2 ligand in 6 might derive from degradation of 2 due to reaction with adventitious traces of water, after prolonged heating. They would regenerate the phosphinous acid Ph2POH and its evolution toward Ph2POPPh2. 55,49,56 Complex 7 presents the coordination of a novel type of triphosphinite ligand in which the PPh2 moieties are linked through a [PhSiO]3 unit, similar to a silsesquioxane arrangement (Scheme 3). To our knowledge, such an arrangement is unknown and reminiscent of mesitylene based tripodal phosphorus ligands used to form mono and oligomeric assemblages with a variety of d and f metal ions. [57][58][59][60][61][62][63][64] Although ligand 2 is sensitive to temperature, it is notable that its complexes (e.g. 3', 4', 5' and 5") are more robust and solutions of the complexes in THF show no sign of decomposition after 24 h at 60°C. 67 The averages for the P-Cu-P and P-Cu-X angles, respectively 100.6 (12) and 117(2)° in 5 and 96.55 (9) and 120.5(18)° in 5', are quite distinct and reflect noticeable deviation from an ideal tetrahedral geometry. These angles are similar to those found in Cu( 2 -triphos)Br. 18 In all the complexes 3, 3', 4, 4', . 69 The Cu---Cu distance of 2.8024(5) Å is classical and equals the sum of the Van der Waals radii (2.80 Å). 70,71 It [START_REF] James | Private Communication[END_REF] The corresponding distance in the twofold bridged dinuclear complex [START_REF] Naik | [END_REF] 76 the Cu-P bond length is also similar to those measured in 5 and 5'.

5
is similar to that found in the cationic dimers [Cu( 2 -dcpm)(NCMe)]2 [BF4]2 [2.809 Å] (dcpm = bis(dicyclohexylphosphino)methaneb ) 72 or [Cu( 2 -Ph2P(CH2)2PPh2)(NCMe)2]2 [PF6]2 [2.873 Å].
[Cu( 2 -X)(NCMe)]2( 2 -(R2P)2(N-C6H4-N)2(R2P)2)[Cu( 2 -X)(NCMe)]2
Catalytic hydroboration of CO2 with 3, 3', 4, 4', 5 and 5'

Having in hand a series of first row transition metal complexes, namely 3, 3', 4, 4', 5

and 5', their catalytic properties were explored in the field of CO2 transformation. CO2 conversion is currently dominated by non-redox processes that enable the functionalization of CO2 to carbonates, carbamates and urethanes. 35,36, 77-7835, 79,80 Recent progresses have enabled the coupling of CO2 and epoxides to either cyclic or polymeric carbonates, with a high regio-and stereoselectivity and using earth abundant metal catalysts. [81][82][83][84][85][86] These methods are progressively turning into efficient technologies and some of them are currently under pilot development. 80,87 In contrast to functionalization strategies, the reduction of CO2 offers an access to more energetic C1 products such as formic acid, carbon monoxide or methanol, with carbon atoms at the +II and -II oxidation states. 36, 77-78,64a,68,88 So far, the electro-reduction of CO2 is mostly limited to two electrons processes with the formation of HCOOH and CO, whereas the production of methanol by electrolysis still suffers from low Faraday yields. [89][90][91][92][93] Hydrogenation methodologies nicely supplement electrolysis and, for example, CO2 hydrogenation to methanol can be promoted by solid copper-based catalysts, on the industrial scale. 80,[87][88] In solution, several molecular catalysts have been proposed for the hydrogenation of CO2 to formic acid or methanol. 6, 94-97 So far, only a single molecular catalyst has been reported for the reduction of CO2 to methanol with H2, based on ruthenium(II) complex supported by the triphos ligand. 96 Replacing H2 with hydrosilanes (R3SiH) or hydroboranes (R2BH) enables the use of extremely mild reaction conditions for the reduction of CO2, because the Si-H and B-H functionalities are polarized and more reactive. [98][99][100][101][102][103][104][105][106][107][108][109][110] While hydrogenation of CO2 imposes the use of metal catalysts working at high temperature (>120 °C) and high pressure (>80 bars), hydrosilylation and hydroboration reactions can be performed under ambient conditions (25 °C, <1 bar CO2) with first row transition metal or metal free catalysts. These advantages have been exemplified with the recent development of novel diagonal reactions 79 leading to the reductive functionalization of CO2 to formamides, formamidines, methylamines, etc. 111 Although hydrosilanes and hydroboranes will not be suitable for large scale applications in the short runbecause of cost and atom efficiencythey offer an increased chemoselectivity in the reduction of CO2. In particular, C 0 derivatives such as acetals and aminals have been successfully obtained from CO2 and hydrosilanes or hydroboranes, while the corresponding formaldehyde state has been elusive so far when H2 is utilized in the reduction of CO2. 112,113 Ligands 1 and 2 are synthetic derivatives of the well-known tripodal triphos and PP3

(tris[2-(diphenylphosphino)ethyl]phosphine) ligands, which d transition metal (Ru, Fe, Cu, etc.) complexes have been successfully utilized in hydrogen transfer catalysis, 6, 114 specifically for the reduction of carbonyl compounds and CO2 or the generation of H2 from formic acid. 115,116,117 The catalytic activity of 3, 3', 4, 4', 5 and 5' was thus evaluated in the hydroboration of CO2 so as to determine the influence of the metal ion and the electronic nature of the ligand in this model reaction.

Exposing an acetonitrile solution of 9-BBN and 1.5 mol% 5 to an excess CO2 (1 bar) resulted in the formation of the boryl acetal H2C(OBBN)2 and methoxyborane CH3OBBN products in 32 and 13 % yield, after 24 h at 25 °C (Entry 3 in Table 4). This result corresponds to a conversion of 45 % of the B-H functionalities and to an H2C(OBBN)2/CH3OBBN molar ratio of 71:29. Interestingly, the triphosphinite complex 5' exhibits a higher catalytic activity and a conversion of 69 % is reached after only 4 h at 25 °C, while full completion is noted after 17 h (Entries 4 and 5 in Table 4). 5' is also Importantly, whereas (NHC)CuX (X= OCHO, Ot-Bu) catalysts selectively yield formate derivatives, 5 and 5' converge directly to the acetal product showing that the hydroboration of the latter to the methoxyborane is kinetically limiting. This selectivity is rather unusual and has been described, so far, only with (dmpe)2FeH2, as reported recently by the group of Sabo-Etienne and Bontemps. 113 In contrast, the iron and cobalt complexes 3, 3', 4 and 4' show no catalytic activity in the hydroboration of CO2 with 9-BBN at room temperature and raising the reaction temperature to 60 °C is needed to observe the reduction of CO2. Using the triphosphine iron complex 3, 45 % of the hydroborane is converted after 24 h at 60 °C to a 19:81 mixture of H2C(OBBN)2 and CH3OBBN (Entry 10 in Table 4). Interestingly, the triphosphinite analogue 3' displays a lower catalytic activity and a conversion of only 4).

To the best of our knowledge, complexes 4 and 4' are the first examples of cobalt catalysts active in the hydroboration. Overall, complexes 3, 3', 4, 4', 5 and 5' complete the small series of catalysts reported for this transformation, based on nickel, ruthenium, iron and copper complexes. 107,108,113,118,119 Although the most reactive catalysts in the hydroboration of CO2 remain based on nickel pincer structures and organic phosphorus bases, the present copper system has a catalytic activity and a selectivity close to (dmpe)2FeH2. 106, 109, 113

Conclusion

In conclusion, the synthesis and characterization (crystal structure, NMR spectroscopy) of metal complexes of the first row Fe 2+ , Co 2+ , and Cu + ions have been [Co(PhSi{CH2PPh2}3)Cl2] (4) : A 25 mL round bottom flask was charged with CoCl2 (15 mg, 0.12 mmol) and a solution of 1 (100 mg, 0.14 mmol) in 10 mL THF. The resulting blue solution was stirred for 24 h at room temperature. The solvent was evaporated off affording a blue powder which was washed with pentane (3 x 5mL) and then dried under vacuum to yield pure 4 as a blue solid (56 mg, 49%).

Blue crystals of 4 were obtained by slow diffusion of pentane into a THF solution of the isolated product. [Cu(PhSi{CH2PPh2}3)Br] (5) : A 25 mL round bottom flask was charged with CuBr (17 mg, 0.12 mmol) and a solution of 1 (100 mg, 0.14 mmol) in 10 mL of THF. The resulting colorless solution was stirred for 24 h at room temperature without change of color. The solution was concentrated to 3 mL and the addition of 6 mL pentane induced precipitation of a white solid of 5 which was isolated pure after filtration and drying under vacuum (77 mg, 76 %). Colorless crystals of the solvate 5•(THF)3 were obtained by diffusion of pentane into a THF solution of the product. [Cu(PhSi{OPPh2}3)Br] (5'') : A 25 mL round bottom flask was charged with CuBr (20 mg, 0.14 mmol) and a solution of 2 (110 mg, 0.15 mmol) in 10 mL THF. The initially colorless solution was stirred for 24 h at room temperature without change of color.

The solution was concentrated to 3 mL and the addition of 6 mL pentane induced precipitation of a white solid of 5'' which was isolated pure after filtration and drying under vacuum (98 mg, 82 %). Colorless crystals of the solvate 5'•(THF) were obtained by diffusion of pentane into a THF solution of the product. [Cu(PhSi{OPPh2}3)I] (5') : A 25 mL round bottom flask was charged with CuI (30 mg, 0.16 mmol) and a solution of 2 (150 mg, 0.21 mmol) in 10 mL THF. The initially colorless solution was stirred for 24 h at room temperature without change of color.

The solution was concentrated to 3 mL and the addition of 6 mL pentane induced precipitation of a white solid of 5' which was isolated pure after filtration and drying under vacuum (125 mg, 88 %). Colorless crystals of the solvate 5'•(THF) were obtained by diffusion of pentane into a THF solution of the product.

Catalytic experiments

Catalytic reactions have been performed in a J-Young NMR tube equipped with valve.

In an NMR tube, the borane (9-BBN)2 (ca 0.09 or 0.20mmole) and the catalyst (1.5 mol%) were weighted and dissolved in 0.4 mL of d3-acetonitrile, then 7 µL (0.05 mmol)

of mesitylene were added. The tube was then fitted onto a vacuum line, degassed under vacuum after having freeze the solution and filled with 1 atmosphere of pure CO2 (quality 4.5, ca 0.08 mmole).

Crystallography

The data were collected at 150(2) K on a Nonius Kappa-CCD area detector diffractometer 120 using graphite-monochromated Mo-K radiation ( = 0.71073 Å). The crystals were introduced into glass capillaries with a protective coating of Paratone-N oil (Hampton Research). The unit cell parameters were determined from ten frames, then refined on all data. The data (combinations of -and -scans with a minimum redundancy of at least 4 for 90% of the reflections) were processed with HKL2000. 121 Absorption effects were corrected empirically with the program SCALEPACK. 121 The structures were solved by intrinsic phasing with SHELXT 122 expanded by subsequent difference Fourier synthesis and refined by full-matrix least-squares on F 2 with SHELXL-2014. [123][124] All non-hydrogen atoms were refined with anisotropic displacement parameters.

In compound 3, one of the trifluoromethanesulfonate counter-ions is disordered over two positions sharing two oxygen and two fluorine atoms, which were refined with occupancy parameters constrained to sum to unity and restraints on bond lengths and displacement parameters.

In compound 4, the displacement parameters of several carbon atoms in the phenyl rings are highly anisotropic, which is probably indicative of unresolved disorder.

In compound 4', the phenyl ring bound to Si1 is rotationally disordered over two positions sharing two carbon atoms, which were refined with occupancy parameters constrained to sum to unity; one of these positions (A) was refined as an idealized hexagon and the other (B) with restraints on bond lengths and angles.

Complex 5•3THF was refined as a two-component inversion twin, with a Flack parameter of 0.479 (13).

In compound 7, some voids in the lattice indicate the presence of unresolved solvent molecules, but, the residual electronic density in these voids being quite low, the SQUEEZE software (PLATON) 125 did not improve the data significantly. These solvent molecules may have at least partially left the lattice, as suggested by the partly opaque aspect of all crystals.

In all compounds, the hydrogen atoms were introduced at calculated positions and they were treated as riding atoms with an isotropic displacement parameter equal to 1.2 times that of the parent atom (1.5 for CH3, with optimized geometry). Crystal data and structure refinement parameters are given in Table 3. The molecular plots were drawn with ORTEP-3. 126

Captions to Figures

Scheme 1:

Scheme 2 :

 2 Scheme 2: Synthesis of the Fe 2+ , Co 2+ and Cu + complexes with the Ph-Si(YPPh2)3 ligands (Y = O, CH2).

3 and 3 '

 3 are isostructural with the iron ion lying in a distorted octahedral environment, formed by three acetonitrile molecules and the  3 -coordinated tripodal ligand (1 or 2). NCMe)3][BF4]2 complex, reported during the course of the present study,21 and to that of [Fe( 3 -RC{CH2PPh2}3)(NCMe)3][BF4]2 (R = Me, PhCH2

Formation of the copper complexes 5, 5' and 5 "

 5 Addition of CuBr to a THF solution of 1 afforded [Cu( 3 -PhSi{CH2PPh2}3)Br] (5), after 24 h reaction at room temperature and washings with pentane. Repeating the same procedure with 2 and CuX (X= I, Br), [Cu( 3 -PhSi{OPPh2}3)I] (5') and [Cu( 3 -PhSi{OPPh2}3)Br] (5") could be isolated in good 88 and 82 % yields, respectively. The 1 H,13 C and 31 P{ 1 H} NMR spectra recorded at 21°C are consistent with complexes having a C3 symmetry for the three complexes, which was confirmed by X-ray diffraction analyses. Elemental analyses were carried out on the bromide complexes and, because [Cu( 3 -PhSi{OPPh2}3)Br] could not be crystallized, its iodide counterpart 5' was further considered in this work. Crystals of the adducts 5•3THF and 5'•THF were grown by diffusion of pentane into THF solutions of 5 and 5'.

Figure 3 :

 3 Figure 3: 31 P{ 1 H} NMR spectra of 5 (left) and 5' (right) in d2-CH2Cl2 at 21°C.

  and 5', the Si-O and Si-CH2 distances are insensitive to the nature of the metal ions and the coordination mode of the ligand. Complex 6, [Cu 2 -I)2(NCMe)]2( 2 -(Ph2POPPh2)), crystallizes as a solvate with a single acetonitrile molecule in the cell and it presents a two-fold axis of symmetry, defined by the O1 atom and the center of the Cu---Cu fragment. The Cu2I2 core adopts a butterfly shape with the I -atoms at the wingtips. The two iodide ions and the Ph2POPPh2 ligands are bridging and one acetonitrile molecule completes the coordination sphere of the metal ion which is in a pseudo-tetrahedral environment with a mean ligand-Cu-ligand angle of 109(7)°, close to the ideal value. The average Cu-I bond length of 2.701(7) Å is close to mean values in the 4-coordinate phosphite and phosphine derivatives [Cu( 2 -I)2(P{OPh}3)(NCMe)]2 [2.65(2) Å] 68 and [Cu( 2 -I)(NCMe)]( 2 -(C20H12O2SP}2NPh) [2.68(4) Å]

more selective than 5

 5 and provides a selectivity of H2C(OBBN)2/CH3OBBN= 80:20 at full conversion (92:8 after 4 h). Complexes 5 and 5' are only the second examples of copper complexes able to catalyze the hydroboration of CO2. In 2013, Shintani and Nozaki indeed reported the use of copper(I) N-heterocyclic carbene (NHC) complexes able to reduce CO2 to formoxyboranes HCOOBR2 at low temperature (35 °C). 118

3 %

 3 under identical conditions (Entry 12). The cobalt complexes 4 and 4' exhibit a slightly enhanced reactivity, with a conversion of 63 and 6 % with 4 and 4', respectively, after 24 h at 60 °C. The iron and cobalt complexes thus differ from copper catalysts 5 and 5' as they show a lower activity in hydroboration for the phosphinite derivatives (3' and 4') than the phosphine complexes (3 and 4). In addition, different selectivities are observed between the two groups of catalysts. Whereas iron and cobalt complexes 3, 3', 4 and 4' produce the methoxyborane as a major product (with a H2C(OBBN)2/CH3OBBN molar ratio of ca. 20:80), even at low conversions. The selectivity of 3 and 4 toward the methoxyborane is expectedly more pronounced with an excess of (9-BBN)2 (Entries 20-23), because CH3OBBN is the thermodynamic product. The copper complexes 5 and 5' favor the formation of the C 0 product (H2C(OBBN)2/CH3OBBN ratio of ca. 80:20), which is only slowly converted to the methoxide level. These trends reveal that different mechanisms are at play in the metal-catalyzed hydroboration of CO2. Presumably, the copper complexes act as Lewis acids in this transformation and their catalytic activity is enhanced when the Cu + ion is coordinated to the less electron donating ligand 2. These different mechanisms also translate into different reactivities of the catalysts in the presence of a high concentration of hydroborane. Indeed, while catalysts 3 and 4 are more active with an excess hydroborane, doubling the initial quantity of 9-BBN poisons 5' which Turn Over Frequency (TOF) drops from 11.5 to 1.6 h -1 (Entries 19─23 in Table

2 ( 3

 23 described with the tripodal ligands PhSi(YPPh)3 (Y = CH2 (1), O (2)). Replacing the CH2 group with an O atom affects the coordination properties of the molecule favoring   -PhSi(OPPh)3 versus   -PhSi(CH2PPh)3 coordination onto CoCl2. All the complexes are active in the catalytic hydroboration of CO2 with (9-BBN)2. While the PhSi(CH2PPh)3 (1) ligand favors the catalytic performances of Fe 2+ and Co 2+ ions over its oxygen analogue PhSi(OPPh)3 (2), the reverse trend is observed with the Cu + complexes 5 and 5' which proved the most active catalysts. The activity of the triphosphinite complex 5' is higher than that of 5. Both copper catalysts favor the reduction of CO2 to the C 0 acetal product, and the selective and quantitative conversion of (9-BBN)2 into a 80:20 mixture of H2C(OBBN)2 and CH3OBBN is obtained at room temperature. The iron and cobalt catalysts are more sluggish and require heating at 60 °C to perform the hydroboration of CO2. They also exhibit a selectivity different from the copper complexes and facilitate the reduction of CO2 to methoxyboranes, preferentially. The different mechanisms at play in these catalysts classes are currently under scrutiny in our laboratory, to highlight the different active forms of the metal catalysts.ExperimentalThe complexes described herein are moisture sensitive. Syntheses and manipulations of the compounds were conducted under ultra-high purity argon atmosphere with rigorous exclusion of air and water (< 5 ppm oxygen or water), using Schlenk-vessel and vacuum-line techniques or in recirculating mBraun LabMaster DP gloveboxes.Glassware was dried overnight at 70 ºC before use. All NMR spectra were obtained using a Bruker DPX 200 MHz spectrometer. Chemical shifts for 1 H and 13 C{1H} NMR spectra were referenced to solvent impurities. The solvents toluene, pentane, diethyl ether and tetrahydrofuran were dried over a mixture of sodium-benzophenone, and pyridine over potassium and they were distilled immediately before use. Deuterated solvents (Eurisotop), tetrahydrofuran (d8-THF), benzene and toluene were dried over a sodium/benzophenone mixture and acetonitrile (d3-MeCN) over KH; they are distilled and kept under argon in the gloves box. The commercially available compounds trichlorophenylsilane, diphenylphosphine oxide were obtained from Aldrich, used as received and stored in the gloves box. The amine NEt3 (Aldrich) was dried over molecular sieve before use, then distilled under vacuum and kept on 4 Å molecular sieve under inert atmosphere. The metal salts CoCl2, CuX (X = I, Br) and Fe(OTf)OTf = O3SCF3) were purchased from Strem or Aldrich, degassed and dried under vaccum for 15h and stored under argon. Synthesis of PhSi(CH2PPh)3 (1) (sticky beige solid) has already been reported[START_REF] Neumeyer | [END_REF]25 and the new ligand PhSi(OPPh)3 was prepared from a derived procedure. (2) : A 10 mL round-bottom flask was charged with diphenylphosphine oxide (230 mg, 1.1 mmol), NEt3 (175 µL, 1.2 mmol), PhSiCl3 (60 µL, 0.37 mmol) and freshly distilled toluene (5 mL) was introduced. After 5 h at room temperature the [HNEt3]Cl salt which deposited from the colorless solution was filtered off on a frit in the gloves box and further washed with 5 mL diethyl ether. The solvents of the combined solutions were evaporated under vacuum affording pure

[

  31 P{1H} NMR ( d8-THF): 100.93. Fe(PhSi{CH2PPh2}(MeCN)3][OTf]2 (3) : A 25 mL round bottom flask was charged with Fe(OTf) (65 mg, 0.18 mmol) and a solution of 1 (140 mg, 0.20 mmol) in acetonitrile (10 mL). The red solution was stirred for 24 h at room temperature and the solvent evaporated off. The solid residue was washed with pentane (3 x 5 mL) and dried under vacuum to give [Fe(PhSi{CH2PPh2}3)(MeCN)x][OTf]2 (x = 2-3) as a red solid (117 mg ,55 %). Dark red crystalline needles of [Fe(PhSi{CH2PPh2}3)(MeCN)3][OTf]2 (3) were obtained by slow diffusion of diethyl ether into an acetonitrile solution of the isolated product.

  ]2 (3') : A 25 mL round bottom flask was charged with Fe(OTf)2 (100 mg, 0.28 mmol) and a solution of 2 (215 mg, 0.30 mmol) in acetonitrile (10 mL). The orange solution was stirred for 24 h at room temperature and the solvent evaporated off. The orange solid residue was successively washed with diethyl ether (3 x 5 mL) and then pentane (3 x 5 mL) and dried under vacuum to yield [Fe(PhSi{OPPh2}3)(MeCN)x][OTf]2 (x = 2-3) as an orange solid (172 mg, 52 %). Orange crystals of [Fe(PhSi{OPPh2}3)(MeCN)3][OTf]2 (3') were obtained by slow diffusion of diethyl ether into an acetonitrile solution of the isolated product.

Scheme 2 :Scheme 3 : 2 Figure 1 .

 2321 Scheme 1: Synthesis of the PhSi(XPPh2)3 ligands (X = O, CH2).

Figure 2 .For 4 'Figure 3 .Figure 4 .

 2434 Figure 2. Views of complexes 4 (left) and 4' (right). Displacement ellipsoids are drawn respectively at the 30% and 40% probability level. The hydrogen atoms are omitted. For 4' only one position of the disordered aromatic ring is represented

Figure 5 .

 5 Figure 5. View of complex 6. Displacement ellipsoids are drawn at the 50% probability level. The solvent molecule and hydrogen atoms are omitted. Symmetry code: i = 1x, y, 3/2z.

Figure 6 .

 6 Figure 6. View of complex 7. Displacement ellipsoids are drawn at the 30% probability level. The hydrogen atoms are omitted. Symmetry codes: i = z, x, y; j = y, z, x.

Figure 1 .

 1 Figure 1.

Figure 2 .

 2 Figure 2.

Figure 3 .

 3 Figure 3.

Figure 4 . 32 Figure 5 .

 4325 Figure 4.

Figure 6 .

 6 Figure 6.

Formation of the cobalt complexes 4 and 4'

	the more electron rich triphosphine 1 is favored, resulting in the formation of the four-
	coordinate species 4.
	Cobalt(II) chloride reacted readily with 1 in THF to afford a blue solution from which Within each complex, the two Co-Cl bond lengths differ by less than 0.03 Å, thus complex 4 was isolated as a solid (49 % yield), after several washings with pentane. confirming the neutral nature of these species. The Co-Cl distances are shorter in 4 Treating a THF solution of CoCl2 with 2 yielded a red solution from which 4' was than in 4', which reflects the lower coordination number at the Co 2+ ion in 4. In addition, isolated as a red solid (36 % yield). Paramagnetism of the 3d 7 Co 2+ ion impeded any Co-P distances in 4 are ca 0.12 Å longer than in 4', with average values of 2.3483(6) NMR characterization and the compounds were characterized by X-ray diffraction and and 2.22(4) Å in 4 and 4', respectively. These structural features are consistent with a elemental analyses. Blue crystals of 4 and dark red crystals of 4' were obtained by chelating effect resulting in a stronger metal-ligand interaction in 4' and the increased slow diffusion of pentane into a THF and a dichloromethane solution of the complexes, 3s character of the P lone pairs in 2. The Cl-Co-Cl angle is strongly related to the respectively. These distinct colors suggest different coordination geometries around steric environment of the metal ion. A decreasing coordination number allows the Cl-the Co 2+ ion in 4 and 4'. It is well established that tetrahedral Co 2+ complexes are generally blue, while they are pink-red when the metal ion adopts an octahedral Co-Cl angle to expand from 88.
	configuration. 45 In fact, within the triphos series, [Co(triphos)Cl2] provided blue
	solutions in agreement with a four coordinate [Co( 2 -triphos)Cl2] structure.
	Nevertheless, it was isolated as a red five coordinate [Co( 3 -triphos)Cl2], in the solid
	state. 45 The increased stability of 4 with respect to [Co( 2 -triphos)Cl2] evidences the
	influence of the Si versus the C atom at the apex position of tripodal ligands. The
	control of the coordination behavior is likely due to small interplays of steric tension at
	the C-E-C (E = Si, C) angles. This finding is in agreement with the recent results
	obtained by Apfel et al. on the coordination behavior of the homologous RSi{CH2PPh2}3
	(R = Me, Ph) and triphos ligands with FeCl2. The authors indeed concluded, from
	titration experiments, that MeSi{CH2PPh2}3 binds FeCl2 more strongly than triphos in
	acetonitrile. 21, 45
	Views of complexes 4 and 4' are provided in Figure 2, respectively, and selected bond
	lengths and angles are listed in Table 1. X-ray diffraction confirmed a tetrahedral
	bond lengths are similar in 3 and 3' [1.962(4) Å for 3'], the Fe-P distances in 3' are environment for the Co 2+ ion in 4 and a distorted square pyramidal coordination
	notably shorter averaging 2.255(8) Å. This contraction is in agreement with the greater geometry in 4'. In the latter complex, the base is defined by the two halides, P2 and
	3s character in the phosphorus lone pair of a phosphinite ligand, compared to a P3 (rms deviation 0.018 Å) while P1 is at the vertex of the pyramid. The metal ion lies
	phosphine. The P-Fe-P and N-Fe-N bite angles average 92.7(8)° and 85.3(15)° in inside the pyramid, 0.4372(5) Å above the base and the sum of the ligand-Co-ligand
	3, and 91.1(18)° and 86.2(14)° in 3', whereas these values reach 88(2)° and 85(1)° in angles in the mean base is 351.14°. The distinct coordination environments in 4 and
	[Fe( 3 -MeC{CH2PPh2}3)(NCMe)3][BF4]2 and 92(2) and 84.4(7) in [Fe( 3 -4' are in line with the different electron donating properties of ligands 1 and 2. Because
	MeSi{CH2PPh2}3)(NCMe)3][BF4]2. phosphinite ligands P(OR)R2 are less electron donating than phosphines, CoCl2
	preferentially accommodates ligand 2 in a tridentate manner while  2 -coordination with

).

42

The mean Fe-P and Fe-N distances in 3

[2.308(11)

Å and 1.958(4) Å] compare well with those reported in [Fe( 3 -MeSi{CH2PPh2}3)(NCMe)3][BF4]2 [2.317(7) and 1.966(2) Å]

21

or [Fe( 3 -triphos)(NCMe)3][BF4]2 [2.27(1) and 1.953(6) Å].

44

While the mean Fe-N 69(3)° in the five-coordinate complex 4' to 121.73(4)° in the four-coordinate species 4. The P-M-P angles in the tridentate MeSi(YPPh2)3 (Y=CH2, O) ligands are unexceptional and quite similar in the iron and cobalt complexes.

Scheme 3 :

 3 Complexes 6 and 7 obtained from degradation of ligand 2

	Views of the complexes 5, 5', 6 and 7 are given in Figures 4-6, respectively, and selected bond lengths and angles are listed in Table 2. Complexes 5 and 5' comprise unexceptional and fall in the range of the four-coordinate complexes Cu( 3 -triphos)Br [2.3983(15) Å and 2.286(6) Å ], 65 Cu( 3 -N{CH2PPh2}3)Br [2.4965(13) Å and 2.304(4) Å], 65 Cu(PMePh2)3Br [2.507(2) Å and 2.282(9) Å ] 66 and [Cu( 2 -dppe)(- 1 -dppe)I] a four-coordinate Cu bond lengths [2.4137(12) and 2.285(4) Å for 5; 2.5491(3) and 2.276(2) Å for 5'] are [2.623(1) and 2.30(2) Å ].

+ ion, coordinated to three phosphorus atoms and one terminal halide ion, so that the copper environment can be viewed as distorted trigonal pyramidal, with the P atoms defining the base. The Cu-X (X = Br, I) and mean Cu-P

  the cyclic Si3O3 fragment and the Cu and Cl atoms is a three-fold axis of symmetry and the Cu atom is at a distance of 3.64 Å from the centroid of the Si3O3 ring. The Cu1-Cl1 and Cu1-P1 bond lengths of 2.3022(15) and 2.2550(8) Å, respectively, can be compared to those found in the other 4-coordinate complexes

	are notably shorter [X = Cl, Br, I ; 2.6893(4), 2.7177(8) and 2.6906(8) Å, respectively]. Such Cu(I)---Cu(I) distances vary considerably for ligand-supported Cu---Cu bonding interactions and contact distances as short as 2.35 Å have been reported. [Cu( 3 -triphos)Cl] [2.260 and 2.292(7) Å] 75 and [Cu(PPh3)3Cl] [2.347 and 2.318(1) Å];

72

Complex 7 crystallizes in the cubic space group Pa-3. The environment of the Cu + atom is best described as distorted tetrahedral with bonding to three phosphorus atoms of the novel Ph3Si3O3{OPPh2}3 ligand and one chloride anion. The line passing by the centroid of

Table 1 .

 1 Selected bond lengths (Å) and angles (°) in the complexes 3, 3' and 4, 4'

		3	3'		4		4'
	Distances
	Fe1-N1	1.963(4)	Fe1-N1	1.967(2)	Co1-Cl1	2.2318(10)	Co1-Cl1	2.2600(8)
	Fe1-N2	1.953(4)	Fe1-N2	1.958(2)	Co1-Cl2	2.2061(10)	Co1-Cl2	2.2418(8)
	Fe1-N3	1.958(3)	Fe1-N3	1.9624(18)	Co1-P1	2.3476(10)	Co1-P1	2.2755(8)
	Fe1-P1	2.3031(12)	Fe1-P1	2.2452(6)	Co1-P2	2.3489(10)	Co1-P2	2.1893(8)
	Fe1-P2	2.3224(11)	Fe1-P2	2.2532(6)	< Co1-P>	2.3483(6)	Co1-P3	2.2070(8)
	Fe1-P3	2.2972(12)	Fe1-P3	2.2656(6)	<P-C>	1.827(13)	< Co1-P>	2.22(4)
	<Fe-P>	2.308(11)	<Fe-P>	2.255(8)	<Si-C>	1.881(16)	<P-O>	1.640(4)
	<Fe-N>	1.958(4)	<Fe-N>	1.962(4)	Si1-C40	1.866(4)	<Si-O>	1.6402(2)
	<P-C>	1.833(8)	<P-O>	1.625(6)			Si1-C37	1.8244(19)
	<Si-C>	1.871(13)	<Si1-O>	1.6409(3)
	Si1-C40	1.855(4)	Si1-C37	1.816(2)

Table 3 .

 3 Crystal data and structure refinement details

	3	3'	4	4'

Table 4 .

 4 Catalytic hydroboration of CO2 with complexes 3,3', 4, 4' and 5, 5' as depicted in the equation below.

	Scheme 1.

(Co) (9-BBN)2

Acknowledgments

This work was supported financially by the CEA, CNRS, CHARMMMAT Laboratory of Excellence, and the European Research Council (ERC starting grant agreement no. 336467). T.C. thanks the Foundation Louis D.-Institut de France for its support. The authors thank Dr. Emmanuel Nicolas for fruitful discussions regarding NMR spectroscopy.


Evgueni Dinvay

AND Denys Dutykh

Henrik Kalisch

A COMPARATIVE STUDY OF BI-DIRECTIONAL WHITHAM SYSTEMS

In 1967, Whitham proposed a simplified surface water-wave model which combined the full linear dispersion relation of the full Euler equations with a weakly linear approximation. The equation he postulated which is now called the Whitham equation has recently been extended to a system of equations allowing for bi-directional propagation of surface waves. A number of different two-way systems have been put forward, and even though they are similar from a modeling point of view, these systems have very different mathematical properties.

. We use state-of-the-art numerical tools to try to understand existence and stability of solutions to the initial-value problem associated to these systems. We also put forward a new system which is Hamiltonian and semi-linear. The new system is shown to perform well both with regard to approximating the full Euler system, and with regard to well posedness properties.

Introduction

Consideration is given to the two-dimensional water-wave problem for an inviscid incompressible fluid with a free surface over an even bottom. As this problem has not been completely resolved mathematically, there is still interest in developing new simplified models which yield an approximate description of the waves at the free surface in the case when the waves have distinctive properties, such as small amplitude or large wave period. In particular, there is the Boussinesq scaling regime which gives a good approximate description of long waves of small-amplitude. Recently, there has been interest in full-dispersion model which aim to give an exact description of "linear" waves while still being weakly nonlinear, and therefore accommodating some nonlinear processes such as wave steepening. The idea of representing the linear dynamics exactly goes back to the work of Whitham [START_REF] Whitham | Variational methods and applications to water waves[END_REF] who conceived the equation (now called Whitham equation)

η t + gWη x + 3 2 c 0 H ηη x = 0, (1.1)
where W = w(-i∂ x ) = F -1 wF is a Fourier multiplier operator defined by the dispersive function

w(ξ) = tanh(Hξ) gξ , (1.2

)

Date: February 15, 2019. and c 0 = √ gH is the limiting long-wave speed, defined in terms of the undisturbed fluid depth H and the gravitational acceleration g. The Fourier transform F and inverse transform F -1 are defined in the standard way, such as for example in [START_REF] Whitham | Linear and Nonlinear Waves[END_REF]. It is clear that since the operator W reduces to the identity for very long waves (ξ → 0), the Whitham equation reduces to the inviscid Burgers equation for very long waves.

Recently, Whitham's idea has been extended to the study of systems of evolution equation which allow for bi-directional wave propagation. In particular, in [START_REF] Aceves-Sánchez | Numerical study of a nonlocal model for water-waves with variable depth[END_REF], Aceves-Sánchez, Minzoni and Panayotaros, found the Whitham system

η t = -HKu x -(ηu) x , (1.3)
u t = -gη x -uu x , (1.4)
and in [START_REF] Moldabayev | The Whitham Equation as a model for surface water waves[END_REF], it was shown how this system arises as a Hamiltonian system from the Zakharov-Craig-Sulem formulation of the water-wave problem using an exponential long-wave scaling.

The operator K is defined by the Fourier symbol tanh(Hξ)

Hξ

, so that we have the relation HK = gW 2 . It can be seen that since the operator K reduces to the identity operator for very long waves (ξ → 0), this Whitham system reduces to the classical shallow-water system for very long waves. In the remainder of this article, we will refer to the system (1.3), (1.4) as the ASMP system.

The system (1.3), (1.4) has been studied in a number of works. In particular, it was shown in [START_REF] Ehrnström | Existence of a highest wave in a fully dispersive two-way shallow water model[END_REF] that it admits periodic traveling-wave solutions and features a highest cusped wave on the bifurcation branch. The modulational stability of its periodic traveling-wave solutions has been investigated numerically in [3], and the system has been studied numerically in the presence of an uneven bottom in [START_REF] Vargas-Magana | A Whitham-Boussinesq long-wave model for variable topography[END_REF]. Moreover, it was shown in [START_REF] Ehrnström | A conditional well-posedness result for the bidirectional Whitham equation[END_REF] that the initialvalue problem on the real line is well posed locally-in-time for data that are strictly positive and bounded away from zero.

On the other hand, the system

η t = -Hv x -(ηv) x , (1.5)
v t = -gKη x -vv x (1.6)
was put forward by Hur and Pandey in [START_REF] Hur | Modulational instability in a full-dispersion shallow water model[END_REF], and it was shown to behave somewhat more favorably than (1.3), (1.4) with regard to modulational instability and local well posedness (see also [3]. We will call this system the HP system. In the current work, it is shown how the ASMP system (1.3), (1.4) and the HP system (1.5), (1.6) can be related by an asymptotic change of variables. Using the new variables, it is also possible to obtain a Hamiltonian system which is much less sensitive to instabilities than either the ASMP or HP system. We also show that the new system yields better approximations to the full water-wave problem than any of the other bi-directional Whitham system in use so far. We also present two other Hamiltonian systems, the right-left system, where dependent variables are chosen to represent wave propagating mainly to the left or to the right, and the essentially right-going system For the sake of completeness, we also include the Matsuno system in our study since it is easily obtained using the Hamiltonian theory.

The Hamiltonian formalism

A two-dimensional water-wave problem with the gravity g and the mean depth H is under consideration. The fluid is supposed to be inviscid and incompressible with irrotational flow. The unknowns are the surface elevation η(x, t) and the velocity potential φ(x, z, t). The fluid domain is the set {(x, z) ∈ R 2 | -H < z < η(x, t)} extending to infinity in the positive and negative horizontal x-direction. Liquid motion is governed by the Euler system consisting of the Laplace's equation in this domain

φ xx + φ zz = 0 for x ∈ R, -H < z < η(x, t), (2.1)
the Neumann boundary condition at the flat bottom

φ z = 0 at z = -H, (2.2)
the kinematic condition at the free surface

η t + φ x η x -φ z = 0 for x ∈ R, z = η(x, t), (2.3)
and the Bernoulli equation

φ t + 1 2 φ 2 x + φ 2 z + gη = 0 for x ∈ R, z = η(x, t). (2.4)
The total energy of the fluid motion consists of potential and kinematic energy:

H = R η 0 gz dzdx + 1 2 R η -H |∇φ| 2 dzdx. (2.5)
It is known that the system (2.1)-(2.4) is equivalent to a certain Hamiltonian system. Indeed, with the trace Φ(x, t) = φ(x, η(x, t), t) of the potential at the free surface and the Dirichlet-Neumann operator G(η) the total energy (2.5) takes the form

H = 1 2 R gη 2 dx + 1 2 R ΦG(η)Φdx. (2.6)
We regard H(η, Φ) as a functional on a dense subspace of L 2 (R) × L 2 (R). We do not wish to specify smoothness of functions η, Φ and the exact domain of the functional H at this point, but we assume its variational derivatives lie in L 2 (R). The pair (η, Φ) represents the canonical variables for the Hamiltonian functional (2.6) with the structure map

J η,Φ = 0 1 -1 0
and so the Hamiltonian equations have the form

η t = δH δΦ , Φ t = - δH δη .
(2.7)

This evolutionary system in L 2 (R) is known to be equivalent to the Euler system (2.1)-(2.4). However, it does not simplify the problem since in general there is no explicit expression for the operator G(η).

Weakly nonlinear approximations

In this section several approximations to Hamiltonian (2.6) will be presented. Each one will give rise to a system that can be considered as an approximate model to (2.7). The analysis is mainly heuristic consisting of arguments represented in [START_REF] Craig | Hamiltonian long-wave approximations to the water-wave problem[END_REF][START_REF] Dinvay | The Whitham equation with surface tension[END_REF], for example.

Regarding the self-adjoint operator D = -i∂ x in L 2 (R) we assume that the Dirichlet-Neumann operator appearing in (2.6) may be approximated by the sum

G(η) = G 0 +G 1 (η) where G 0 (η) = D tanh(HD), G 1 (η) = DηD -G 0 ηG 0 .
Such substitution should not change the Hamiltonian significantly since the remaining terms in the truncated operator G(η) are of at least quadratic order in η and its derivatives.

After integration by parts (Lemma 2.1 in [START_REF] Dinvay | The Whitham equation with surface tension[END_REF]) it leads to

H = 1 2 R gη 2 + ΦG 0 Φ -η(DΦ) 2 -η(G 0 Φ) 2 dx. (3.1)
One may notice the relative advantage of this approximation immediately. Instead of integrating the system (2.1)-(2.4), the much simpler system (2.7) with Hamiltonian (3.1) is to be solved.

In works on the surface water-wave problem, it has been common to use unknowns other than the potential Φ. Here, we use the variable

u = Φ x = φ x + η x φ z = φ τ 1 + η 2
x , which is proportional to the velocity component of the fluid ϕ τ which is tangent to the surface. This change of variables transforms the Hamiltonian (3.1) to

H = 1 2 R gη 2 + u tanh HD D u + ηu 2 + η(tanh HDu) 2 dx. (3.2)
From now on, we will refer to the pair (η, u) as Boussinesq variables. Note that unlike (η, Φ) these new variables are not canonical. The corresponding structure map has the form

J η,u = 0 -∂ x -∂ x 0
and the Hamiltonian system (2.7) transforms to

η t = -∂ x δH δu , u t = -∂ x δH δη . (3.
3)

It will become clear later that it is convenient to introduce yet another change of dependent variables. We define the new velocity variable v = Ku, where the transformation K is defined by the expression

K = tanh HD HD , (3.4)
which shows that it is an invertible and bounded Fourier multiplier operator. While the physical meaning of the new velocity variable v = K∂ x Φ = i tanh(HD)Φ/H is not clear, it will be shown later that it can be used to find a new system of equations which has desirable mathematical properties. In these new variables the Hamiltonian functional H(η, v) has the form

H = 1 2 R gη 2 + HvK -1 v + η(K -1 v) 2 + η(HDv) 2 dx (3.5)
with the structure map

J η,v = 0 -K∂ x -K∂ x
0 and the Hamiltonian system (2.7) transforming to

η t = -K∂ x δH δv , v t = -K∂ x δH δη . (3.6)
In physical problems a question often arises if there is a way to split waves on right-and left-going components. One possible way of doing this splitting is to regard the linearization of the problem given in elevation-velocity variables and then change variables [START_REF] Moldabayev | The Whitham Equation as a model for surface water waves[END_REF]. Namely, regard the following transformation

r = 1 2 (η + Wu), s = 1 2 (η -Wu) (3.7)
where W is supposed to be an invertible function of the differential operator D. The inverse transformation has the form

η = r + s, u = W -1 (r -s). (3.8)
Omitting the details provided in [START_REF] Dinvay | The Whitham equation with surface tension[END_REF] we notice that to split the linearized system into two independent equations one needs to take

W = H g K = tanh HD gD .
(3.9)

The new variables r and s correspond to right-and left-going waves, respectively. Returning to the nonlinear theory we want to obtain a new Hamiltonian system with respect to unknown functions (3.7). Using the variables r and s and integrating by parts puts the Hamiltonian (3.2) into the form

H = 1 2 R 2g(r 2 + s 2 ) + (r + s)(W -1 (r -s)) 2 + (r + s)( gG 0 (r -s)) 2 dx, (3.10)
with the structure map

J r,s = -W∂ x /2 0 0 W∂ x /2,
and the Hamiltonian system (2.7) transforming to

r t = - 1 2 W∂ x δH δr , s t = 1 2 W∂ x δH δs . (3.11)
In what follows we perform a Hamiltonian perturbation analysis based on the assumption of smallness of wave gradients. Regard a wave-field with a characteristic non-dimensional wavelength λ = l/H, amplitude α = a/H and velocity β = b/ √ gH where l, a and b are typical dimensional parameters. Define the small parameter µ = 1/λ. Usually α and β are identified and regarded as functions of wave-number µ. For justification of the models derived below there is no need for this identification or concretization of the dependence α, β on µ. The meaning of the scaling is of course that η = HO(α), u = √ gHO(β)

and HD = -iH∂ x = O(µ). During our derivations, omission of higher-order terms is applied only to the Hamiltonian expressions (3.2), (3.5). The main idea is that high-order dispersive effects have little effect on the energy of the motion. Moreover, this approach guarantees that the obtained systems are Hamiltonian.

3.1. Matsuno model. The first useful system can be obtained if we take Hamiltonian (3.2) as it is and find the corresponding variational derivatives. Taking any real-valued square integrable smooth function h and using the definition

R δH δu (x)h(x)dx = d dτ τ =0 H(u + τ h, η) = = 1 2 R (HhKu + HuKh + 2ηuh + 2η(tanh HDu) tanh HDh) dx
one arrives after integration by parts to δH δu = HKu + ηutanh HD(η tanh HDu)

and in the same way to

δH δη = gη + 1 2 u 2 + 1 2 (tanh HDu) 2 .
Thus System (3.3) transforms to η t = -HKu x -(ηu) x + tanh HD(η tanh HDu) x , (3.12)

u t = -gη x -uu x -(tanh HDu) tanh HDu x (3.13)
which appeared in [START_REF] Lannes | Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation[END_REF], and is similar to the systems found in [START_REF] Choi | Nonlinear evolution equations for two-dimensional surface waves in a fluid of finite depth[END_REF] and [START_REF] Matsuno | Two-dimensional evolution of surface gravity waves on a fluid of arbitrary depth[END_REF]. It is not known so far if the system is well posed, but from a modeling point of view, it is sometimes regarded as the most exact model of all the so called bidirectional Whitham systems. Even though this system conserves the Hamiltonian (3.2), it turns out that this system is very sensitive to aliasing due to spatial discretization.

3.2. ASMP model. Simplifying the Hamiltonian through and appropriate scaling such as α = O(µ N ) and thus discarding the last integrand in (3.2), one arrives at the system

η t = -HKu x -(ηu) x , u t = -gη x -uu x .
This is the system (1.3), (1.4) mentioned in the introduction. The corresponding Hamiltonian is

H = 1 2 R gη 2 + HuKu + ηu 2 dx. (3.14)
This is also a Hamiltonian system with respect to the same Boussinesq variables η, u in the same sense as (3.3). This model started to attract attention after it appeared in [START_REF] Aceves-Sánchez | Numerical study of a nonlocal model for water-waves with variable depth[END_REF] and [START_REF] Moldabayev | The Whitham Equation as a model for surface water waves[END_REF]. The local well-posedness of the system (1.3)-(1.4) is proved [START_REF] Ehrnström | A conditional well-posedness result for the bidirectional Whitham equation[END_REF] by imposing the additional condition inf η(x, 0) > 0 on the initial surface elevation. It should be remarked that this condition may mean that the system is not useful from a physical point of view since all surface water wave models should have the property that the mean elevation be zero. However strictly positive solutions, like solitons for example, have always featured prominently in the analysis of such systems. In a recent paper by Claassen and Johnson [3] the well-posedness for more general initial data was questioned. In fact the authors showed numerically that the ASMP system is probably ill-posed in L 2 (T). However, our computations suggest to assume this is not the case in L 2 (R) and so that the system is probably well-posed on the real line. We also show that periodic discretization affects numerical computations significantly. ). This results in the Hamiltonian

H = 1 2 R gη 2 + HvK -1 v + ηv 2 dx (3.15)
with the Gâteaux derivatives

δH δv = HK -1 v + ηv, δH δη = gη + 1 2 v 2 .
Thus for the Hamiltonian (3.15), the system (3.6) has the form

η t = -Hv x -K(ηv) x , (3.16
)

v t = -gKη x -K(vv x ). (3.17)
To the best of our knowledge this system is completely new. One may notice that the nonlinear part of System (3.16)-(3.17) contains only the bounded operator K∂ x , which could mean that it is at least a locally well-posed system. Moreover we shall see later that among all bidirectional Whitham systems this is numerically the most stable one.

If one formally substitutes the operator K into the nonlinear part of (3.16)-(3.17) by unity according to the long wave approximation K = 1 + O(µ 2 ) then one arrives at the system

η t = -Hv x -(ηv) x , v t = -gKη x -vv x ,
i.e. system (1.5), (1.6) which was introduced by Hur & Pandey [START_REF] Hur | Modulational instability in a full-dispersion shallow water model[END_REF]. This system does well in the sense of numerical stability comparing with ASMP model but not as well as its Hamiltonian relative (3.16)-(3.17). Unlike the system (3.16)-(3.17) one cannot say for certain if the Hur-Pandey system is Hamiltonian with the same structure map as the original water-wave problem.

3.4.

Right-left waves model. Again simplifying the Hamiltonian (3.10) up to O(µ 2 αβ 2 ) we obtain

H = g R r 2 + s 2 + 1 2H (r + s)(r -s) 2 dx (3.18)
with the Gâteaux derivatives

δH δr = 2gr + g 2H (3r + s)(r -s), δH δs = 2gs - g 2H (3s + r)(r -s).
Hence for the Hamiltonian functional (3.18) the bi-directional Whitham system has the form

r t = -gWr x - g 4H W∂ x (3r + s)(r -s), (3.19)
s t = gWs x - g 4H W∂ x (3s + r)(r -s). (3.20)
This system is also new even though it has implicitly appeared in a recently submitted paper [START_REF] Dinvay | The Whitham equation for hydroelastic waves[END_REF], where it was not investigated further. Here we emphasize its usefulness and demonstrate that this system also outperforms the system (1.5)-(1.6) in the sense of numerical stability. Moreover, the variables r, s have a clear physical meaning and in particular initial data are easier to obtain. This means that sometimes the initial elevations r(x, 0) and s(x, 0) can be measured directly as opposed to velocity variables. We do not know if the system is well-posed. It deserves note that the symbol of the unbounded operator W∂ x behaves like a square root at infinity. This fact might be enough to obtain well posedness.

In any case, as shown later, the system has favorable numerical stability properties.

3.5. Uncoupled twin-unidirectional model. One may notice that in the system (3. [START_REF] Lannes | Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation[END_REF])-(3.20), the coupling between the dependent variables is due to the following part of Hamiltonian (3.18):

H coupling = - g 2H R rs(r + s)dx. (3.21)
This part may sometimes be neglected. Then we arrive to the Hamiltonian

H = g R r 2 + s 2 + 1 2H (r 3 + s 3 ) dx (3.22)
and the corresponding Hamiltonian system consisting of the two independent equations

r t = -gWr x - 3g 2H Wrr x , (3.23)
s t = gWs x + 3g 2H Wss x . (3.24)
The first equation is a modification of the equation proposed by Whitham [START_REF] Whitham | Variational methods and applications to water waves[END_REF][START_REF] Whitham | Linear and Nonlinear Waves[END_REF]. The second one is its analogue for left-going waves. It is not known if they are well-posed even though for a large class of similar equations the answer is affirmative [START_REF] Ehrnström | Classical well-posedness in dispersive equations with nonlinearities of mild regularity, and a composition theorem in Besov spaces[END_REF]. We shall see below that it is quite often the case that colliding waves almost do not affect each other and one may admit independence and regard basically just the equation (3.23). Up to small terms, the final result is obtained by linear superposition (3.8). Indeed in Figure 2 the dependence on time of interaction energy (3.21) for the Right-left system (3.19), (3.19) is represented. One can see that the interaction is going on for a short time and is of negligible order. This results in a small residual of solution after the interaction.

The numerical approach

All the models discussed in the project are solved by treating the linear part L and the nonlinear part N separately using a split-step scheme. In other words we solve a system of the form

z t = L(z) + N (z) (4.1)
which is treated by solving the systems z t = L(z) and z t = N (z). Denote by exp(tL) an integrator of the first one and exp(tN ) an integrator of the the second one. We make use of a symplectic integrator of 6th order introduced by Yoshida [START_REF] Yoshida | Construction of higher order symplectic integrators[END_REF]. The main advantage of such an integrator is that the time step can be made relatively large which can accelerate calculations greatly. Yoshida developed his numerical scheme for separable finite Hamiltonian systems, however, it proved to be efficient also in water wave problems [START_REF] Carter | Bidirectional Whitham equations as models of waves on shallow water[END_REF]. Below we describe the method in application to the models derived. Following Yoshida a one step integrator for the whole system (4. Here we take the following set of weights w 3 = 0.784513610477560, w 2 = 0.235573213359357,

w 1 = -1.17767998417887, w 0 = 1.315186320683906.
One can notice that the integrator is symmetric. The meaning of the product is that each time step is divided into substeps. The systems z t = L(z) and z t = N (z) are solved using spectral methods. Moreover, the first one for each model can be solved exactly. For example, the linearization of the system (3.16)-(3.17) has the following solution

η(t) = cos Utη 0 -iHD sin Ut U v 0 , v(t) = -ig/H tanh HD sin Ut U η 0 + cos Utv 0 ,
with the initial data η 0 , v 0 . The operator U has the form

U = gG 0 = gD tanh HD. (4.2)
These formulas represent the integrator exp(tL) for the systems (3.16)-(3.17

η(t) = cos Utη 0 -i tanh HD sin Ut U u 0 , u(t) = -igD sin Ut U η 0 + cos Utu 0 ,
with the initial data η 0 , u 0 .

For the system (3.19)-(3.20), the operator exp(tL) is diagonal,

gW∂ x = igWD = i gG 0 sgn D,
and the linearized problem has the solution

r(t) = exp(-itU sgn D)r 0 , s(t) = exp(itU sgn D)s 0 ,
where r 0 , s 0 are initial right-and left-going waves, respectively, and U is defined by (4.2). For all models discussed here, we use the standard Runge-Kutta scheme of 4th order as the nonlinear integrator exp(tN ). It is explicit but not symplectic. One might argue that it makes the whole integrator exp(tL + tN ) not symplectic any more.

As an alternative we also ran all computations with a symplectic Euler scheme, such as described in [START_REF] Hairer | Geometric Numerical Integration[END_REF]. This scheme turns out to be explicit for most of the models discussed here. Indeed, for example, for the ASMP model (1.3)-(1.4) one step of the semi-implicit Euler method has the form

η n+1 = η n -δt∂ x (HKu n + η n+1 u n ), u n+1 = u n -δt∂ x (gη n+1 + 1 2 u 2 n )
that can be resolved with respect to η n+1 as follows. On the space l N 2 define operator

B n f = -δt∂ x (u n f ) that is bounded B n δtN max u n . Expecting uniform boundedness of solution u one can choose the time step δt = O(1/N) so that B n C < 1. Thus (1 -B n )η n+1 = η n -δt∂ x HKu n is resolved as η n+1 = (1 + B n + B 2 n + . . .)(η n -δt∂ x HKu n ).
Hence η n+1 , u n+1 are resolved via η n , u n and the scheme is explicit and symplectic at the same time.

The numerical scheme of the free-surface problem for the Euler equations is based on a time-dependent conformal mapping of the fluid domain into a strip. A complete description of the method can be found in [START_REF] Li | A Numerical Study of the Exact Evolution Equations for Surface Waves in Water of Finite Depth[END_REF]27].

Numerical experiments

The model systems described above are now characterized with respect to numerical instability due to spatial discretization. For the numerical experiments we make the problem nondimensional by setting H = 1 and g = 1. The computational domain is -L ≤ x ≤ L, with L = 70. Initial conditions are imposed by means of

η 0 (x; x 0 , a, λ) = a • sech 2 (f (x -x 0 )) -C, (5.1)
where

f (λ) = 2 λ log 1 + √ 2 , C(λ) = a 2f L (tanh f (L -x 0 ) + tanh f (L + x 0 )) .
Here C(λ) and f (λ) are chosen so that L -L η 0 (x)dx = 0, and the wave-length λ is the distance between the two points x 1 and x 2 at which η 0 (x 1 ) = η 0 (x 2 ) = a/2. Below we always take the wave-length λ = √ 5. In all problems below we are interested in time evolution from t 0 = 0 to t max = 50. In cases of collision of two waves we send them towards each other. So first of all we simulate problems that cannot be described by unidirectional models like KdV or Whitham equations. Secondly, one can see that all the models introduced are in line with the effect of quasi-elastic interaction of waves. So after collision waves behave as independent with slight tails. In all experiments below we provide initial data η(x, 0) and Φ(x, 0) for the Euler system. Initial data for the approximate models can easily be obtained by applying transformations of variables u(x, 0) = ∂ x Φ(x, 0), (3.4) and (3.7). According to (3.8) one can make quasi-right moving waves taking the surface velocity u(x, 0) = W -1 η(x, 0).

As was already said the splitting method we are making use of allows us to take relatively large time steps. So we take δt = 0.05 when the number of Fourier harmonics is either N = 512 or N = 1024. This choice is dictated by the stiffness of the ASMP model (1.3)-(1.4) since the scheme becomes unstable for large N and might need filtering due to the probable ill-posedness of the model. In comparative experiments, on the other hand, we do not want to use any filtration.

Experiment 5.1 (A). Consider a collision of two approaching positive waves. Let a = 0.2 and x 0 = 20. Impose initial surface η(x, 0) = η 0 (x; x 0 ) + η 0 (x; -x 0 ) and initial potential

Φ(x, 0) = - x 0 W -1 η 0 (ξ; x 0 )dξ + x 0 W -1 η 0 (ξ; -x 0 )dξ.
All approximate systems in Experiment (A) are solved on the grid with N = 1024.

Experiment 5.2 (B). Consider a collision of a trough and a convex wave. Let a = 0.1 and x 0 = 20. Impose initial surface η(x, 0) = η 0 (x; x 0 )η 0 (x; -x 0 ) and initial potential

Φ(x, 0) = - x 0 W -1 η 0 (ξ; x 0 )dξ - x 0 W -1 η 0 (ξ; -x 0 )dξ.
All approximate systems in Experiment (B) are solved on the grid with N = 512.

Experiment 5.3 (C). Consider a collision of two troughs. Let a = 0.1 and x 0 = 20. Impose initial surface η(x, 0) = -η 0 (x; x 0 )η 0 (x; -x 0 )

and initial potential

Φ(x, 0) = x 0 W -1 η 0 (ξ; x 0 )dξ - x 0 W -1 η 0 (ξ; -x 0 )dξ.
All approximate systems in Experiment (C) are solved on the grid with N = 512.

Experiment 5.4 (E1-E3). Consider the evolution of waves with the initial surface elevation η(x, 0) = η 0 (x;

x 0 = 0)
where a = 0.3 and x 0 = 0. Impose firstly (E1) initial potential

Φ(x, 0) = x 0 W -1 η 0 (ξ)dξ, than secondly (E2) initial potential Φ(x, 0) = x 0 η 0 (ξ)dξ,
and finely (E3) initial potential Φ(x, 0) = 0.

All approximate systems in Experiments (E1-E3) are solved on the grid with N = 1024. Note that the initial potential of Experiment (E2) creates only approximately a right-going wave according to the linear long wave theory. Anyway neither the conditions of Experiment (E1) or of Experiment (E2) induce completely one way propagation as numerical results shows. Surprisingly, initial potentials of the type as in Experiment (E3) lead to better correspondence between approximate models and the Euler system then initial potentials of the type as in Experiment (E2). And moreover, of the type as in Experiment (E2) lead to the better correspondence then of the type as in Experiment (E1). We believe it is mainly a technical feature since the initial error of evaluation surface potential via W -1 and integration normally increases with the time. In order to quantitatively compare the accuracy of each approximate model we calculate the differences between Euler solutions and solutions of each system correspondingly. These

η initial = η(x, 0) 2 dx.
Here η Euler (x, t) is the solution for the Euler system and η model (x, t) corresponds either to ASMP, Hur-Pandey, Hamiltonian Hur-Pandey or Right-Left system. The corresponding results are represented in Table 2. As was stated above some models work better in the sense of numerical stability. There were many discussions about ill-posedness of ASMP model [3]. In the next experiment we provide an example with initial data satisfying the condition for local well posedness. One can see that the initial data is lifted over the real axis so the mean value is approximately The black curve is the approximate solution of the full Euler system at t = 50. The color coding is as follows: purple -Hamiltonian HP system; red -right-left system; blue -ASMP system; green -HP system. 0.35. It is known from Ehrnström, Pei, Wang [START_REF] Ehrnström | A conditional well-posedness result for the bidirectional Whitham equation[END_REF] that we are in a locally well posed situation, however, the obtained solution seems very unstable as one can see in Figure 8. This experiment was repeated with different time integrators, including the symplectic first-order Euler method described in Section 4. The results were always the same, pointing to doubts about the long-time well posedness of the ASMP system. In order to systematize our experiments regarding the well posedness and stability of the Whitham systems, we used the following initial data: The black curve is the approximate solution of the full Euler system at t = 50. The color coding is as follows: purple -Hamiltonian HP system; red -right-left system; blue -ASMP system; green -HP system.

Problems with the HP system (1.5)-(1.6) may occur if an initial trough is deep enough. In the example shown on Figure 9 we have to filter half of the high Fourier modes to make computations stable. The resulting noisy solution continues its propagation and one can notice that all the oscillations happen around some reasonable mean curve that can be obtained easily by solving either the system (3.16)-(3.17 Experiment 5.7. Suppose a = 0.2 and x 0 = 0. Solve Matsuno System (3.12)-(3.13) with initial surface η(x, 0) = η 0 (x) and initial velocity u(x, 0) = KW -1 η(x, 0). We take the time step δt = 0.1 and the number of grid points N = 512.

One might think that a numerical method conserving the total energy could remove the instabilities in the solution. Unfortunately this is not the case. We applied a simple projection method [START_REF] Hairer | Geometric Numerical Integration[END_REF] to obtain a conservative method. With this method, energy was indeed conserved, and we managed to get a constant instead of the time-varying energy shown in Figure 11. However, the solutions itself remained noisy such as in Figure 10, and the computational cost is substantially higher than in the nonconservative method.

1

 1

3. 3 .

 3 Hamiltonian version of the Hur-Pandey model. Regarding the Hamiltonian (3.5) given in the new variables defined above, one may discard the last integral in the expression and simplify the next one staying in the same framework of accuracy up to O(µ 2 αβ 2

1 ) 2 and d 1 =

 121 is approximated by the product exp[δt(L + N ))] = exp(c 1 δtL) exp(d 1 δtN ) exp(c 2 δtL) • . . . • exp(d 7 δtN ) exp(c 8 δtL) where δt is the time step and c i , d i are constants given by c 1 = c 8 = w 3 /2, c 2 = c 7 = (w 3 + w 2 )/2, c 3 = c 6 = (w 2 + w 1 )/2, c 4 = c 5 = (w 1 + w 0 )/d 7 = w 3 , d 2 = d 6 = w 2 , d 3 = d 5 = w 1 , d 4 = w 0 .

  ) and (1.5)-(1.6) since the linear part L is the same for those two.For the systems (1.3)-(1.4) and (3.12)-(3.13) the integrator exp(tL) has the form

  4) are black, of the ASMP system (1.3)-(1.4) are blue, of the Hur-Pandey system (1.5)-(1.6) are green, of the Hamiltonian Hur-Pandey system (3.16)-(3.17) are purple, and of the right-left system (3.19)-(3.20) are red.

Figure 6 .

 6 Figure 6. Experiment (E2). The thin grey curve represents the initial data.

Experiment 5. 5 .Figure 7 .

 57 Figure 7. Experiment (E3). The thin grey curve represents the initial data.

Figure 8 .

 8 Figure 8. Approximate solution of the ASMP Whitham system with initial data satisfying the condition inf η 0 > 0.

Figure 9 .Figure 10 .

 910 Figure 9. Left panel: De-aliased solution of the HP system with N = 512 and time step = 0.05. Snapshot is taken at t = 25. Right panel: The same for Hamiltonian version of the HP system and the Right-Left system.

Figure 11 .

 11 Figure 11. Total energy of the self-stabilized solution of Matsuno system as a function of time t, with N = 512 and time step δt = 0.1.

Table 1 .

 1 Hamiltonians H for various systems, evaluated at t = 50. In all presented figures initial elevation profiles are marked by grey lines. Solutions of the Euler system (2.1)-(2.

	0.20
	0.15
	0.10
	0.05
	0.00
	0.05
	0.10
		80	60	40		20	0	20	40	60	80
	Figure 1. Experiment (A). The thin grey curve represents the initial data. The
	black curve is the approximate solution of the full Euler system at t = 50. The
	color coding is as follows: purple -Hamiltonian HP system; red -right-left system;
	blue -ASMP system; green -HP system.
	0.16						0.002
	0.14
	0.12						0.000
	0.10
							0.002
	0.08
	0.06
							0.004
	0.04
	0.02						0.006
	0.00
	0.02						0.008
	0	10	20	30	40	50	0	10	20	30	40	50
	Figure 2. Left panel: Development of the Hamiltonian for total initial energy
	H = 0.1420, and the coupling term H coupling for Experiment (A). Right panel: close-up of the graph of H coupling .

Table 2 .

 2 Errors E, evaluated at t = 50.

	0.06
	0.04
	0.02
	0.00
	0.02
	0.04
	0.06
	0.08
	0.10
	80	60	40	20	0	20		40	60	80
	Figure 4. Experiment (C). The thin grey curve represents the initial data. The
	black curve is the approximate solution of the full Euler system at t = 50. The
	color coding is as follows: purple -Hamiltonian HP system; red -right-left system;
	blue -ASMP system; green -HP system.
		Experiment	A	B	C	E1	E2	E3
		ASMP		0.488 0.109 0.149 0.883 0.768 0.153
		Hur-Pandey	0.253 0.085 0.126 0.339 0.315 0.082
		Hamiltonian HP 0.167 0.130 0.106 0.231 0.207 0.061
		Right-Left		0.167 0.089 0.128 0.240 0.218 0.048
	errors are measured in the integral L 2 -norm normalized by initial condition as follows
				E =	η Euler -η model η initial

Figure

5

. Experiment (E1). The thin grey curve represents the initial data. The black curve is the approximate solution of the full Euler system at t = 50. The color coding is as follows: purple -Hamiltonian HP system; red -right-left system; blue -ASMP system; green -HP system. where η Eulerη model = max t (η Euler (x, t)η model (x, t)) 2 dx and

Acknowledgments

This research was supported in part by the Research Council of Norway through grants 213474/F20 and 239033/F20.