Samuel F Cousin

Cyril Charlier

Pavel Kadeřávek

Thorsten Marquardsen

Jean-Max Tyburn

Pierre-Alain Bovier

Simone Ulzega

Thomas Speck

Dirk Wilhelm

Frank F Engelke

Werner Maas

Dimitrios Sakellariou

Geoffrey Bodenhausen

Philippe Pelupessy

Fabien Ferrage

High-Resolution Two-Field Nuclear Magnetic Resonance Spectroscopy

Introduction

Nuclear Magnetic Resonance (NMR) is routinely used to explore matter in a wide range of research fields. The development of NMR spectrometers at very high magnetic fields, 1 along with creative methodological developments, [START_REF] Ernst | Principles of Magnetic Resonance in One and Two Dimensions[END_REF][START_REF] Cavanagh | Protein NMR Spectroscopy: Principles and practice[END_REF][START_REF] Schanda | [END_REF][5][6][7] has allowed the study of molecules and materials of increasing complexity. [8][9][10][11][12][13] The sensitivity and resolution of most NMR experiments increase with the magnetic field. However, some NMR properties become less favourable at high magnetic fields. Transverse relaxation may become excessive at high magnetic fields. In particular, transverse relaxation rates increase quadratically with the magnetic field under fast chemical exchange or relaxation by chemical shift anisotropy (CSA), which is the main relaxation mechanism for carbonyl carbon-13 14,15 or phosphorus-31 16 nuclei. Fast transverse relaxation results in broad lines and inefficient coherence transfers. Increasing the magnetic field may thus lead to lower resolution and sensitivity. Radio-frequency (rf) pulses may also be too weak to allow broadband excitation, decoupling or coherence transfer. Can we design an NMR approach that benefits from the sensitivity and resolution of high magnetic fields and that permits the control and evolution of spin systems at low magnetic fields within the same experiment? Several groups have developed experiments where a sample is exposed to two or more magnetic fields to investigate magnetic-field dependent properties, such as zero 17,18 and ultra-low field 19 NMR or field-cycling experiments. [20][21][22] Fast field cycling experiments can be performed by two different approaches. The first option is to rapidly switch the electric current in an electromagnet to change the magnetic field after polarization to a lower magnetic field for relaxation. The magnetic field is then changed again to detect the NMR signal at a higher field. 20,23 Magnetic-field dependent relaxation rates have provided information on molecular motions in a host of systems, such as water, 24,25 water in solutions of contrast agents, [26][27][28] porous media, 29,30 or motions in synthetic polymers 31 or biomolecules. [32][33][34][35][36] The second approach consists in physically moving the sample in the fringe field of a highfield magnet (B0 HF ) to reach a lower field position (B0 LF ). One can also shuttle the sample between two magnets. 37 This can be achieved either by moving the entire NMR probe, 38,39 or only the sample with mechanical 22,40 or pneumatic systems. 21,36 Such shuttle systems have been employed in the frame of dynamic nuclear polarization (DNP). 41 Here, we present in detail a two-field NMR spectrometer designed to manipulate and observe spin systems in two magnetic centres with significantly different magnetic fields. 42 This system allows the measurement of properties that are more favourable at low magnetic fields, while benefiting from the sensitivity and resolution of a superconducting high-field magnet. In our prototype, a fairly homogeneous magnetic field at 0.33 T has been obtained by engineering a plateau in the stray field of a superconducting 14.1 T magnet (600 MHz proton resonance). Both magnetic centres are equipped with triple-resonance probes with a single-axis gradient. The samples can be transferred between the two fields by a pneumatic shuttle system. 36 Here, we describe the technical features of this system, and introduce a series of two-field NMR experiments, including a two-field heteronuclear singlequantum coherence experiment (2F-HSQC). In spite of the low resolution in the indirect dimension, some key features are accessible, such as multiplet patterns due to scalar couplings. Furthermore, we introduce a two-field homonuclear zeroquantum/single quantum correlation experiment (2F-INAZEQUATE) that can provide high-resolution spectra in both dimensions despite the moderate magnetic field inhomogeneities of the low-field magnetic centre.

Materials: Two-field NMR spectrometer

The two-field NMR spectrometer is based on a commercial NMR spectrometer operating at 14.1 T augmented by a series of accessories (Figure 1). (i) A pneumatic sample shuttle apparatus, 36 (ii) devices to obtain a plateau at 0.33 T in the stray field of the superconducting magnet, including rings of soft magnetic material (ferroshims), a set of roomtemperature shim coils, and strips of -metal, (iii) a low-field triple resonance probe operating at 14.1, 3.52 and 1.42 MHz for 1 H, 13 C and 15 N respectively at the 0.33 T, equipped to generate single-axis pulsed field gradients, (iv) systems for generating and amplifying rf-pulses at low frequencies. The TopSpin software was modified to perform experiments with six rf channels in two different probes. The individual components are described in detail below.

Ferroshims

Diverse systems can be used to obtain a plateau of magnetic field in the stray field of a high-field magnet. 37,43,44 Here, five rings of a soft magnetic alloy Ni48Fe52 have been combined to generate a plateau of magnetic field at 0.33 T in the stray field (Figure 2). An earlier design included only three rings. 44,45 The temperature of the ferroshim rings is regulated (see below) to enhance the stability of the magnetic field. Strips of μ-metal (called shim foils) and a set of electric room-temperature shim coils have improved the homogeneity. This combination of devices leads to a residual inhomogeneity of the magnetic field of about 10 ppm over a length of 2 cm. 36 along the axis of the bore of a 14.1 T superconducting magnet between the centre of the magnet to 0.6 m above this centre: without ferroshims (blue) and with ferroshims (red, simulated). (C) Magnetic field profile close to the low-field magnetic centre at 0.33 T.

Figure 3.

Low-field probe designed for rf pulses at 14.1, 3.52 and 1.42 MHz for 1 H, 13 C and 15 N, respectively in the 0.33 T magnetic centre.

Probes

The high-field triple-resonance probe has been described previously. 36 The low-field probe is also designed to isolate rf components from vibrations caused by the motions of the sample. The triple-resonance low-field probe consists of two coils: the inner coil has three windings with saddle-coil geometry. It is optimized for protons at 14.1 MHz. The outer coil has seven windings and a saddle coil geometry. Its resonance frequencies are optimized for both carbon-13 (3.52 MHz) and nitrogen-15 (1.42 MHz) frequencies. In addition, this probe is equipped with a coil to generate z-axis pulsed field gradients (up to 10 G/cm). The low-field probe does not comprise a deuterium coil. The homogeneity and stability of the low magnetic field can be monitored by proton NMR.

Low-field spectrometer

The system is shown in Figure 4. State-of-the-art Avance III HD NMR spectrometers can operate down to 6 MHz. An additional low-frequency signal-generating unit (SGU) that had originally been designed for mass spectroscopy has therefore been used to generate radiofrequency pulses with controlled phases and amplitudes at frequencies in the range 2.5 kHz to 6 MHz. Two low-frequency 200 W amplifiers (1020L Power Amplifier from Electronics & Innovation) are used for 13 CLF and 15 NLF channels. The linearity of these amplifiers is enhanced by correction tables for the amplitudes (but not for the phases) similar to standard practice for high-frequency amplifiers. These correction tables provide excellent linearity for nitrogen-15 and satisfactory linearity for carbon-13 (supporting information). Up to six channels can be used in any given experiment. Finally, two channels of a GREAT3/10 gradient amplifier, which was originally designed for a three-axis gradient system, are employed to generate pulsed field gradients (PFG's) along the z-axis of the low-and high-field probes.

Sample shuttle

This part was originally designed for high-resolution relaxometry. 28,36 The only modification is the tube that guides the sample shuttle in between the two probes, called shuttle guide. A plastic piston-like system is used here, where a spring ensures a tight fit of the shuttle guide between the top of the high-field probe and the bottom of the low-field probe. Typical shuttle transfer delays are 100 ms for motions between the high-and low-field centres. Such transfer delays correspond to an overpressure ca. 0.3 bar and an under-pressure ca. -0.25 Pa, that are both well below the extreme settings allowed for the system.

Additional units

An auxiliary Bruker Smart Magnet System (BSMS) is used in order to control the homogeneity of the magnetic field at the low-field centre. This unit controls a set of 20 roomtemperature electric shims (BOSS1). There is no fieldfrequency lock at the low-field centre. A second two-channel Variable Temperature Unit (VTU) regulates independently the temperatures of the ferroshims and the low-field probe. The nitrogen used for the temperature regulation of the ferroshims is heated above room temperature so that the ferroshims remain at 300.2 ± 0.1 K. The nitrogen used to regulate the temperature of the low-field probe is cooled to about 263 K (-10° C) by a Bruker Control Unit (BCU) and heated to the desired temperature by a resistance in the low-field probe. The shuttle controller uses nitrogen gas at 5 to 6 bar and an internal vacuum pump.

Sample preparation

The sample shuttle 36 consists of four pieces of amorphous quartz glued together. The sample tube contains about 110 μL of solution; the active volume within the rf coils which cover about 20 mm is 60 μL. This study used 1 M solution of 13 C- 15 N labelled glycine in 97 % D2O, and a 1 M solution of sucrose at natural abundance in D2O.

Results and discussion

Homogeneity of the low-field magnetic centre

Obtaining a sufficiently homogeneous plateau in the stray field of a superconducting magnet is a challenge. The variation of the magnetic field along the 20 mm active length of the sample was determined from measurements of the stray field 36 to be 67 mT at 0.33 T, that is 2.03×10 5 ppm before compensation. Figure 5.B shows a typical proton spectrum recorded for a sample of sucrose in D2O at 0.33 T using both ferroshims and electric shims. The sensitivity of the experiment was enhanced by polarization at 14.1 T (Figure 5.A). The full width at half height is 10 ppm, which corresponds to a dramatic improvement of the homogeneity by a factor ~10 [START_REF] Schanda | [END_REF] . Such homogeneity was achieved reproducibly on several samples with adjustment of the shim currents limited to the main shim coils, with z, z 2 , and z 3 geometries.

Radiofrequency pulses in the low-field centre

Nutation experiments have been performed at 0.33 T for all channels (i.e., 1 H, 13 C, 15 N) in order to characterize the amplitudes and homogeneities of all radio-frequency fields in the low-field centre (Figure 6). The sensitivity was enhanced by allowing the polarization to build up in high field and by observing the signals in the high field. The sample is transferred to the low field where radiofrequency pulses with variable durations are applied. The transverse components of the magnetization are suppressed by gradients during the transfer from low to high field. The remaining longitudinal polarization after the radiofrequency pulse at low field is detected at high field after a 90° pulse. The amplitudes of the rf pulses at low field are comparable or superior to those obtained with the high-field probe. Typical pulse specifications at low field are summarized in Table 1. Thus, rf pulses with exceptional broadband capabilities can be readily obtained at low field. In addition, the homogeneity of rf pulses at low field is excellent, with I900/I180 = 80 ± 2 % for proton, I900/I180 = 98 ± 2 % for carbon-13 and I900/I180 = 96 ± 12 % for nitrogen-15, with I900 the intensity measured after a 900° pulse and /I180 the intensity measured after a 180° pulse. The nutation profiles are all imbalanced (the local minima are not as intense as the local maxima). This is likely due to the effect of longitudinal relaxation during the transfer from the lowfield centre to the high-field centre and the stabilization delay s. The imbalance is more pronounced for the nutation on the proton channel. An additional contribution to this imbalance is likely due to the fact the proton coil at low field is slightly smaller than the proton coil in the high-field centre.

Radiofrequency fields for the magnetic centre at 0.

Two-dimensional two-field correlation experiment

We introduce two-dimension experiments that provide correlations of chemical shifts recorded at different magnetic fields. A two-field heteronuclear single-quantum coherence experiment (2F-HSQC) was performed to demonstrate the principle of two-dimensional two-field (2D-2F) single-quantum correlation spectroscopy. The original HSQC experiment 46 was modified to provide a correlation between 13 C chemical shifts at low field in the indirect 1 dimension and 1 H chemical shits at high field in the direct 2 dimension. The pulse sequence is shown in Figure 7. After a recovery delay at high field, the 1 H polarisation is transferred to 13 C by a modified INEPT sequence 47 that generates two-spin order 2CzHz prior to the transfer of the sample to the low-field centre. An anti-phase coherence 2CxHz is then allowed to evolve under the chemical shift at low field. The polarization is again stored as a two-spin order 2CzHz while the sample is transferred by a reverse INEPT sequence back to the high-field centre. A proton signal is then detected under 13 C composite-pulse decoupling. The 2F-HSQC spectrum in Figure 8 was obtained with a 1 M sample of 13 C and 15 N labelled glycine in D2O. The signal intensity is satisfactory; the additional delays do not lead to significant losses of polarization for such a small molecule. The main drawback of the method is the impossibility to use the sensitivity enhancement scheme 49 . The linewidth in the indirect 1 dimension is dominated by field inhomogeneities, as expected from the low-field 1 H spectrum (Figure 5). Nevertheless, the resolution was sufficient to observe the line splitting due to the large one-bond scalar coupling 1 JCC = 53.5 Hz between 13 C and carboxyl 13 C' in glycine. This coupling constant corresponds to 15.2 ppm at 0.33 T. The inhomogeneous nature of line broadening at low field was confirmed in a low-field HSQC spectrum (Figure S1), which demonstrates that the proton and carbon-13 resonance frequencies are highly correlated.

The two-field-INAZEQUATE experiment

To obtain a high-resolution two-field correlation spectrum, the contributions of field inhomogeneities to the linewidth must be significantly reduced. We have demonstrated 42 that this could be achieved in heteronuclear spin systems with the use of zero-quantum coherences. Here we show that highresolution can also be recovered in homonuclear spin systems through the evolution of zero-quantum coherences at low field. Zero-quantum coherences have often been used to obtain high-resolution spectra in inhomogeneous magnetic fields. 42,[51][52][53] We modified the refocused incredible natural abundance double quantum transfer experiment (INADEQUATE) 54,55 to select zero-quantum coherences at low magnetic field and detect single-quantum coherences at high magnetic field. We dubbed this method 2F-INAZEQUATE (Figure 9). The 2F-INAZEQUATE and the gradient-selected refocused INADEQUATE experiments are based on the same principles, so only their differences need to be discussed. Most pulses are applied at low field, in order to attenuate off-resonance effects of 13 C pulses. The 13 C polarization is enhanced during the recovery delay by a steady-state nuclear Overhauser effect [START_REF] Neuhaus | The Nuclear Overhauser Effect in Structural and Conformational Analysis[END_REF][START_REF] Ferrage | [END_REF] obtained by saturating the proton polarisation at high field. The sample is then transferred to the low field where a zeroquantum coherence is excited before the evolution interval t1. Scalar couplings between 13 C and 15 N are refocused with a single 180° pulse on the NLF channel. The evolution under scalar couplings between 13 C and 1 H is suppressed by composite pulse proton decoupling. Bipolar pulsed field gradients (noted G1 in Figure 9) are applied at low field after

1 H HF CPD CPD CPD f 3 f 4 t 1 2 t a f 1 f 2 t 1 2 d d ' G 1 -G 1 G 2 -G 2
1 H LF 13 C HF 15 N LF PFG 15 N HF 13 C LF t a t a t a f acq the evolution period t1. These gradients will defocus all coherences with a non-zero coherence order. Only homonuclear zero-quantum coherences and populations have a non-zero expectation value after these bipolar pulsed field gradients. The delay 2a is used to convert the desired polarization into in-phase single-quantum coherences Cy and selectively refocus the desired components with a bipolar gradient pair G2. In the zero-quantum version of the experiment, G2 = 0; while in the double-quantum version of the experiment, G2 = 2G1. A 90° pulse on 13 C follows to bring the desired polarization along the z-axis for storage during the transfer to high field. At high field, a single 90° pulse is followed by 13 C signal detection. The efficiency of the excitation of zero-quantum coherences relies on the imbalance between the longitudinal 13 C polarization of the two nuclei before the first 90° pulse. Here, the difference in nuclear Overhauser enhancements between the 13 C nuclei in the explored molecule was sufficient to ensure the efficient excitation of a zero-quantum coherence. Other approaches, such as the application of selective inversion pulses, may be employed to amplify such an imbalance. Note that a purely high-field INAZEQUATE experiment provides spectra with poor quality as carbon-13 pulses are not broadband at 14.1 T. The two-field INAZEQUATE experiment provides a correlation between the single-quantum carbon-13 signals in the direct dimension and zero-quantum frequencies in the indirect dimension. The spectra were obtained with a sample of 1 M 13 C and 15 N labelled of glycine in D2O (Figure 10). The signals in the indirect dimension appear at 131.99 ppm, which is close to the difference of 130.99 ppm between the chemical shifts of 13 C  ( 13 C' ( broadening due to field inhomogeneities are largely suppressed since homonuclear zero-quantum coherences are not affected. The full line-width at half height of the 13 C'signals is 0.11 ppm in the indirect dimension (or 0.39 Hz). This resolution is significantly higher than what was achieved in heteronuclear spin systems, where the lowest full line width at half height was of 0.8 ppm 42 . This higher resolution in the twofield INAZEQUATE spectrum is likely do to a combination of factors: (1) the full cancellation of the effect of magnetic field inhomogeneities, (2) the negligible contribution of the chemical shift anisotropy to relaxation at low field and (3) the favourable relaxation properties of zero-quantum coherences in spin systems with dipolar relaxation. Two-field NMR spectroscopy offers the unique opportunity to explore nuclear spin properties at two vastly different magnetic fields. Here, we demonstrate that chemical shifts and scalar couplings can be evaluated from coherent evolution at low magnetic fields in a two-field two-dimensional correlation. This feature has been exploited in our study of a molecule prone to significant chemical exchange 42 . Further work will focus on the investigation of spin systems with more favourable relaxation properties at low field, as well as the efficient control of spin systems allowed by broadband radiofrequency pulses at low field. Currently, the main drawback of two-field NMR is the intrinsic sensitivity of our prototype, which should be improved in future developments. The ability to carry out NMR experiments at several fields will allow NMR spectroscopists to develop a new library of pulse sequences designed to perform different parts of a sequence (polarisation, chemical shift evolution, coherence transfer, etc.) at the most advantageous magnetic field.

Conclusions

We have introduced a two-field NMR spectrometer that includes two magnetic centres with different homogenous magnetic fields (14.1 T and 0.33 T) combined with a sample shuttle device. The high magnetic field is provided by a standard superconducting NMR magnet, while the second homogenous magnetic field is an engineered plateau in the stray field of the superconducting magnet, with residual inhomogeneities on the order of 10 ppm. Both magnetic centres are equipped with triple-resonance probes with z-axis pulsed field gradients. A six-channel NMR console is used to generate rf pulses. A pneumatic shuttle allows fast sample transport between the two magnetic centres. The pulse sequences for both magnetic centres are conveniently controlled by the software Topspin. We introduce pulse sequences where rf-pulses are applied and evolution under chemical shifts occurs at two fields. Correlation spectra with single-or double-quantum coherences at low field display broad lines in the indirect dimension, of the fairly crude 10 ppm homogeneity of the low magnetic field. Homonuclear correlations obtained in the 2F-INAZEQUATE experiment, by contrast, feature narrow lines in both dimensions. High-resolution in the indirect dimension is achieved by evolution of homonuclear zero-quantum coherences at low field, which is largely insensitive to line broadening due to moderate magnetic field inhomogeneities. This proof-of-concept experiment demonstrates the feasibility of high-resolution two-field NMR spectroscopy. Two-field NMR can be applied to a host of systems that benefit from low magnetic field, which opens an avenue for novel NMR methods developments.

Figure 2 .

 2 Figure 2. (A)Ferroshims designed to compensate the inhomogeneity of the magnetic field positioned with respect to the distance from the magnetic centre at 14.1 T. (B) Measured magnetic field profile36 along the axis of the bore of a 14.1 T superconducting magnet between the centre of the magnet to 0.6 m above this centre: without ferroshims (blue) and with ferroshims (red, simulated). (C) Magnetic field profile close to the low-field magnetic centre at 0.33 T.

Figure 1 .

 1 Figure 1. Schematic representation of the two-field NMR spectrometer. The subscripts LF and HF distinguish channels dedicated to the low-field centre at 0.33 T and the highfield centre at 14.1 T. The components of the 600 MHz spectrometer are shown in blue with black letters, the components designed for the low-field centre are shown in orange with white letters. Soft magnetic materials (ferroshims) are shown in red. The signal generating units (SGUs) can generate radiofrequency pulses at six frequencies. A deuterium channel allows for field-frequency locking at high field.

Figure 4 .

 4 Figure 4. Design of the two-field NMR spectrometer. Red lines indicate devices and connections added to those found in a conventional high-field spectrometer.

Figure 5 .

 5 Figure 5. (A) Pulse sequence for obtaining a proton NMR spectrum at 0.33 T. Polarization is allowed to build up at 14.1 T during a recovery delay τrec = 5 s. The sample is then transferred to low field during τHL = 80 ms. After a stabilization delay τs = 100 ms, a 90° pulse is applied and the signal is acquired at low field. The sample is transferred back to high field during the delay τLH = 80 ms if several scans need to be recorded. (B) Proton spectrum at 0.33 T for a sample of 1 M sucrose in D2O, recorded with a single scan.

Figure 6 . 1 N

 61 Figure 6. Two-field nutation experiments designed to characterize the three rf fields at low field. (A) Pulse sequence for nutation experiments at low field. The nutation angle 1 H

Figure 7 .

 7 Figure 7. Pulse sequence for the two-field heteronuclear single-quantum coherence (2F-HSQC) experiment. Proton chemical shifts at high field are correlated with carbon-13 chemical shifts at low field. Narrow (filled) and wide (open) rectangles represent 90° and 180° pulses, respectively. The stabilization delay s = 100 ms allows settling of convection currents and vibrations induced by the sample landing. The delay δ is chosen to obtain an echo for t1 = 0. The transfer of polarization is performed with  = 1/(4JCH) = 1.72 ms. The duration for sample transfer from the high-to low field is  m low-to high field is  pulses are applied along the x-axis of the rotating frame, unless otherwise indicated. The phase cycles are: 1=8{x}, 8{-x}; 2=x, -x; 3=2{x}, 2{-x}; 4= 4{x},4{-x}; 5= 8{x}, 8{-x};acq={x}, 2{-x}, {x}, {y}, 2{-y}, {y}. Composite pulse decoupling during detection was carried with a WALTZ-16 scheme 48 applied on-resonance to the 13 C  spins with an rf amplitude  C cpd/(2) = 2.08 kHz.

Figure 8 .

 8 Figure 8. 1 HHF/ 13 CLF -2F-HSQC spectrum of 1 M glycine in D2O acquired with the pulse sequence of Figure 7. The scalar coupling 1 JHC = 4.5 Hz between the carboxyl 13 C' and

Figure 9 .

 9 Figure 9. Pulse sequence for 2F-INAZEQUATE. Narrow black and wide-open rectangles represent 90° and 180° pulses, respectively. Pulses are applied along the x-axis of the rotating frame unless otherwise indicated. The delays are  = 1/4JCC = 4.7 ms, δ and δ'are chosen to obtain an echo for t1 = 0. Composite pulse decoupling was performed with a WALTZ-16 scheme 48 with rf amplitudes of 1 kHz on both HF and LF proton channels. Gradient pulses were applied along the z-axis and had duration of 1 ms and maximum amplitudes G1 = 4 G/cm and G2 = 0 G/cm. In the 2F-INADEQUATE variant, G2 = 8 G/cm. The phase cycle was 1= x, -x; 2 = x; 3 = 2{-x}, 2{x}; 4= 4{-x}, 4{x}; acq = x, -x. Frequency sign discrimination in the indirect dimension was performed with phase alternation of 2 in a TPPI manner.50

Figure 10 .

 10 Figure 10. (A) 2D-2F-INAZEQUATE spectrum of 1 M carbon-13 and nitrogen-15 labelled glycine in D2O. The signals appear at 131.99 ppm in the vertical indirect dimension. (B) 1D spectrum extracted along the vertical dimension (at 41.30 ppm in the direct dimension) of another 2D 2F-INAZEQUATE spectrum obtained with a smaller spectral width to have a higher resolution, with the same pulse sequence and the same pulse lengths. The narrow peak obtained in B is folded many times so the absolute value of the chemical shift is irrelevant. The comparison of the two-field INAZEQUATE with two-field HSQC and two-field INADEQUATE spectra (Supporting information) illustrates the gain in resolution in the indirect dimension. The two-field INAZEQUATE experiment thus provides a two-field 2D spectrum with high resolution in both dimensions.Two-field NMR spectroscopy offers the unique opportunity to explore nuclear spin properties at two vastly different magnetic fields. Here, we demonstrate that chemical shifts and scalar couplings can be evaluated from coherent evolution at low magnetic fields in a two-field two-dimensional correlation. This feature has been exploited in our study of a molecule prone to significant chemical exchange42 . Further work will focus on the investigation of spin systems with more favourable relaxation properties at low field, as well as the efficient control of spin systems allowed by broadband radiofrequency pulses at low field. Currently, the main drawback of two-field NMR is the intrinsic sensitivity of our prototype, which should be improved in future developments. The ability to carry out NMR experiments at several fields will allow NMR spectroscopists to develop a new library of pulse sequences designed to perform different parts of a sequence (polarisation, chemical shift evolution, coherence transfer, etc.) at the most advantageous magnetic field.

  33 T

	Channel	Power	90° pulse	1/2	I900/I180
		(W)	(s)	(kHz)
	1 H	0.5	10.2	24.5	80 ± 2 %
	13 C	100	11.1	22.5	98 ± 2 %
	15 N	50	20	12.5	96 ± 12 %

Acknowledgements

This work was funded by the European Research Council (ERC) under the European Community Seventh Framework Program (FP7/2007-2013), ERC Grant Agreement 279519 (2F4BIODYN), as well as from the Agence Nationale de la Recherche (ANR-11-BS07-031-01).


J E M Morales

G James

A Tonnelier

Travelling waves in a spring-block chain sliding down a slope

Keywords: Travelling front, block-spring system, excitability, piecewise-linear differential equations Propagation d'ondes, système bloc-ressort, excitabilité, équation différentielle linéaire par morceaux

Travelling waves in a spring-block chain sliding down an inclined plane are studied. For a piecewise-linear spinodal friction force, we construct analytically front waves. Pulse waves are obtained as the matching of two travelling fronts with identical speeds. Explicit formulas are obtained for the wavespeed and the wave form in the anti-continuum limit. The link with propagating phenomena in the Burridge-Knopoff model is briefly discussed.

Introduction

Spatially discrete extended systems have a wide range of applications ranging from natural sciences to engineering or social sciences. In physics, they frequently appear as ideal mass spring systems with nearest-neighbours coupling and have been used extensively to describe the dynamics of microscopic structures such as the vibration in crystals or micromechanical arrays [START_REF] Kivshar | Nonlinear surface modes in monoatomic and diatomic lattices[END_REF][START_REF] Maniadis | Mechanism of discrete breather excitation in driven micromechanical cantilever arrays[END_REF][START_REF] Rhoads | Nonlinear dynamics and its applications in micro-and nanoresonators[END_REF] or to approximate the macroscopic behaviour of deformable systems. Recent studies on soft media have driving a renewed interest in the dynamics of elastically coupled systems with a special emphasis on transition waves [START_REF] Raney | Stable propagation of mechanical signals in soft media using stored elastic energy[END_REF]. In this work we consider a spring-block system that slides down a slope due to gravity (see Fig. 1). Each block is subjected to a nonlinear friction force. This system offers a simple macroscopic model for the frictional interaction of two structures. We consider here a friction force of spinodal type (see Fig. 2 for an example). Such friction laws have been reported to induce excitable dynamics [START_REF] Cahn | Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice[END_REF] reminiscent of neural excitability [START_REF] Izhikevich | Dynamical systems in neuroscience: The geometry of excitability and bursting[END_REF][START_REF] Ermentrout | Mathematical foundations of neuroscience[END_REF], i.e., a perturbation above a certain threshold produces a large excursion in the phase space before returning to an equilibrium state. In biology, it is well documented that a large class of excitable media is able to support nonlinear solitary waves [START_REF] Murray | Mathematical biology. Biomathematics[END_REF]. It has been recently shown that excitable mechanical systems also have the capacity to induce self-sustained solitary waves [START_REF] Cartwright | Burridge-Knopoff models as elastic excitable media[END_REF][START_REF] Cartwright | Dynamics of elastic excitable media[END_REF][START_REF] Muratov | Traveling wave solutions in the Burridge-Knopoff model[END_REF]. In contrast with classical excitable media, these systems are elastic rather than diffusive. In many studies, the analysis of discrete travelling patterns heavily relies on a continuum approximation of the original model. In the spring-block model presented here, we directly tackle the discrete nature of the equations and use an idealized piecewise-linear friction force to derive exact expressions for propagating waves. This bilinearization approach has been used in a variety of contexts to study travelling waves in lattices, see e.g. [START_REF] Flytzanis | High velocity dislocation motion and interatomic force law[END_REF][START_REF] Cahn | Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice[END_REF][START_REF] Tonnelier | McKean caricature of the FitzHugh-Nagumo model: traveling pulses in a discrete diffusive medium[END_REF][START_REF] Elmer | Spatially discrete FitzHugh-Nagumo equations[END_REF][START_REF] Atkinson | Motion of a Frenkel-Kontorova dislocation in a onedimensional crystal[END_REF][START_REF] Truskinovsky | Kinetics of martensitic phase transitions: Lattice model[END_REF][START_REF] Vainchtein | The role of spinodal region in the kinetics of lattice phase transitions[END_REF][START_REF] Truskinovsky | Solitary waves in a nonintegrable fermi-pasta-ulam[END_REF][START_REF] Rosakis | New solutions for slow moving kinks in a forced frenkelkontorova chain[END_REF][START_REF] Schwetlick | Existence of dynamic phase transitions in a one-dimensional lattice model with piecewise quadratic interaction potential[END_REF]. The paper is organized as follows. We first derive in Sec. 2 the governing equations for the chain of elastically coupled blocks. Then we study in Sec. 3 the dynamical properties of an isolated block and demonstrate that a bistable behaviour exists when a spinodal friction force is considered. In Sec. 4 we perform numerical simulations of the coupled system and show that the bistability property induces travelling patterns, such as fronts and pulses. In Sec. 5 we construct the travelling fronts analytically using a piecewise-linear friction force. The anti-continuum limit is studied in Sec. 6. The link between front and pulse waves is studied in Sec. 7. We then conclude by connecting the results to a dynamical model of an earthquake fault, the so-called Burridge-Knopoff stick-slip model.

Model

Let us consider an isolated block of mass m and position x(t) that slips down a slope under gravity and subject to a velocity-dependent friction force F dx dt . The dynamical equations read

m d 2 x dt 2 + F dx dt = G (1)
where G is the tangential component of the gravity force. A steady state of (1) exists when the block achieves a constant velocity motion dx dt = V where F (V ) = G. Let us consider an infinite chain of identical blocks linearly coupled through Hookean springs of stiffness k that slips at the constant speed V over an inclined surface (see Fig. 1). The dynamical equations in a frame moving at velocity V are given by:

dy n dt = u n , m du n dt = k∆ d y n -F (V + u n ) + G, n ∈ Z (2)
where y n represents the displacement of the n th block from the steady sliding state and u n its velocity. The term ∆ d y n = y n+1 -2y n + y n-1 is the discrete Laplacian. The system may be interpreted as a variant of the Burridge-Knopoff model [START_REF] Burridge | Model and theoretical seismicity[END_REF] where the shear stress described by the local potential is replaced by a constant tangential force induced by gravity. The dynamics of system (2) is explored for three normalized non-monotonic friction laws F ε , F c and F 0 , depicted in Fig. 2A-C and given by

F ε (v) = 1 -α + N (v) v √ ε + v 2 , F c (v) = 3.2v 3 -7.2v 2 + 4.8v, (3)
F 0 (v) = v/a -αH(v -a),
where

N (v) = ε + 4 max(|v| -a, 0) 2 + α 2 max(a -|v|, 0) 2
, and H is the Heaviside step function.

For convenience, the cubic friction force F c is given for a = 1 where a is the location of the local minimum, i.e. the transition point from velocity-weakening (v < a) to velocity-strenghtening (v > a) regime. The friction function F ε describes a regularized generalized Coulomb law as ε → 0. The cubic friction force F c describes a smooth spinodal friction law similar to the one introduced in [START_REF] Cartwright | Burridge-Knopoff models as elastic excitable media[END_REF]. The piecewise linear function F 0 reduces the velocity-weakening region to a jump discontinuity. It captures some properties of spinodal friction laws and is convenient for analytical computations.

Bistable single block dynamics

For a single block, (2) reads The y-nullcline is defined by u = 0 whereas the u-nullcline is obtained by solving F (V + u) = F (V ) so that the vertical axis u = 0 always defines in the (u, y) plane the set of fixed points for an isolated block. It is easy to check that the two associated eigenvalues are given by λ 1 = -F (V ) m , λ 2 = 0 so that the equilibrium straight line is stable (but not asymptotically stable). In (4), the dynamics of the velocity u does not depend on the position y so that system (4) behaves as a one dimensional dynamical system whose bifurcation diagram is shown in Fig. 3 for the three friction laws (3) where V is taken as the bifurcation parameter. For V ∈ (a, V max ), where V max is the velocity value such that F (V max ) equals the local maximum in F and V max > a, there exist three fixed points U 1 < U 2 < U 3 = 0 whose stability is governed by the eigenvalue

dy dt = u. (4)
m du dt = -F (V + u) + F (V ). Inria Fε(v) v 1 1 -α a Fc(v) v 1 1 -α b a F0(v) v 1 1 -α a (a) (b) (c)
µ i = -F (V +Ui) m , i ∈ {1, 2,
3}, respectively. A saddle-node bifurcation occurs at V = V max and a transcritical bifurcation takes place at V = a. For V ∈ (a, V max ), the two fixed points U 1 and U 3 are stable whereas U 2 is unstable and behaves as an excitation threshold. For an initial condition below U 2 the trajectory of the system tends towards U 3 = 0, whereas for a sufficiently strong perturbation the system reaches asymptotically the state U 1 illustrating the excitable dynamics of an isolated block. Depending on the initial state, the system can switch from a neighbourhood of U 3 to U 1 and vice versa. For the cubic friction force F c (v), the threshold is given by

U 2 = - 3 2 V + 9 8 + 1 8 ∆(V ) (5)
where

∆(V ) = -48V 2 + 72V -15 1/2 (one has ∆(V ) ∈ R for V ∈ [1/4, 5/4]).
We have

U 1 = - 3 2 V + 9 8 - 1 8 ∆ (V ) ,
and V max = 5/4. For the friction force F 0 (v), the threshold is simply defined as

U 2 = a -V, (6)
the stable fixed point u 1 is given by

U 1 = -αa, (7)
and we have V max = a(1 + α). For the regularized generalized Coulomb law F ε , as ε → 0 the threshold converges to

U 2 = (a -V ) 1 + 2 α
and the stable fixed point u 1 to

U 1 = -V
and we have V max = a(1 + α 2 ). In the following we are interested in the excitability regime where the velocity of the single block has two stable steady states and we fix a V value in the interval delimited by the two bifurcation points, i.e. V ∈]a, V max [. As we will show in the sequel, the bistability property is a key feature for the existence of travelling fronts in the block-spring chain.

Travelling waves

Let us consider the block-spring slider model with the regularized generalized Coulomb law F ε . We choose parameters so that each block exhibits a bistable behaviour. The parameters of the friction law are those of Fig. 3(a). Model (2) can be rewritten in terms of velocity as

m d 2 u n dt 2 = k∆ d u n - du n dt F (V + u n ). (8)
We initialize the network by applying a suitable perturbation to the steady state U 3 = 0. A localized perturbation is applied on the first block at the left edge of the network, see Fig. 4 for more details. We consider a finite chain of blocks with free boundary conditions. For the numerical simulations, we use the adaptive Lsoda solver with a time step ∆t = 0.001 and with a minimal error tolerance of 1.5e -8. Unless stated otherwise, we take m = 0.15. We observe the existence of travelling fronts as shown in Fig. 4(a). In addition, two types of pulse solutions are observed: (i) pulse waves with expanding width and (ii) pulse waves with constant shape as plotted in Fig. 4(b) and (c), respectively. Propagating fronts (similar to the one shown in Fig. 4(a)) are the dominant pattern when the threshold is close to the resting state, i.e., for V close to a (|U 2 | 1). The speed of the propagating front increases with the coupling value k but, at the same time, the parameter range where front waves exist shrinks (without vanishing). As the stationary sliding velocity increases, a front to pulse transition occurs where the excitation spreads over the network and leads to pulses with expanding width (see Fig. 4(b)). The rate of expansion of the enlarging pulse decreases as the sliding velocity increases leading to the existence

Morales & James & Tonnelier

c) V = 1.025, k = 1 (d) V = 1.18, k = 2, (e), V = 1.025, k = 1, (f) V = 1.1, k = 1.
of a pulse with constant width as shown in Fig. 4(c). For V → V max , the threshold approaches the fixed point U 1 and a perturbation fails to produce a travelling pattern. Qualitatively similar results are obtained for the cubic friction force F c and for the piecewise linear friction force F 0 . The profiles of the travelling waves observed in Fig. 4(a,c) are shown in Fig. 5(a,b), respectively, and are compared with those obtained with the cubic law (Fig. 5(c,d)) and the piecewise-linear law (Fig. 5(e,f)). The travelling patterns for the three friction forces have similar shapes and mainly contrast in their amplitude that is determined by the distance between the two stable fixed points. A non-monotonic wave profile is observed for the travelling fronts with the existence of a dip behind the front (see Fig. 5(c,e) whereas the dip is too small to be seen in Fig. 5(a)). Interestingly, similar profiles were obtained for traveling fronts in a chain of bistable oscillators Inria [START_REF] Duanmu | Traveling wave solutions in a chain of periodically forced coupled nonlinear oscillators[END_REF]. Qualitatively, these results are not affected by the mass parameter (simulations not shown). The enlarging pulse observed in Fig. 4(b) may be seen as the superposition of two travelling fronts with two different propagating speeds (see Fig. 6). The initial front is qualitatively similar to the waveform shown in Fig. 5(a) and is followed by a travelling front that propagates in the same direction but with a lower speed and that connects the two stable states in a reversed order. The localized pulse waves shown in Fig. 5(b,d,f) are thus expected to appear when the two travelling fronts have the same speed. These observations are analytically explained in the next section for the piecewise-linear law F 0 .

Construction of travelling fronts for the piecewise-linear friction force

A travelling front solution of (8) takes the form:

u n (t) = ϕ(n -c t) (9)
where ϕ(∞) = U 3 = 0 and ϕ(-∞) = U 1 [START_REF] Flytzanis | High velocity dislocation motion and interatomic force law[END_REF] with U 1 = 0 a stable equilibrium. The function ϕ describes the waveform, and c is the wave speed that has to be determined. Substitution of ( 9) into [START_REF] Elmer | Spatially discrete FitzHugh-Nagumo equations[END_REF] gives the advance-delay differential equation

c 2 mϕ (ξ) = k(ϕ(ξ + 1) + ϕ(ξ -1) -2ϕ(ξ)) + c d dξ F 0 (V + ϕ(ξ)) (11)
where ξ = n -ct ∈ R is the travelling wave coordinate. Front solutions connect two different stable steady states as n → ±∞. In contrast, travelling pulses tend towards the same stable equilibrium as n → ±∞. consider here the piecewise linear force F 0 and we assume that each block is in a bistable regime, i.e. we have V ∈ (a, a(1 + α)) and U 1 = -αa as in [START_REF] Duanmu | Traveling wave solutions in a chain of periodically forced coupled nonlinear oscillators[END_REF]. We assume that the travelling front solution crosses the threshold (6) for only one value of ξ. Translation invariance of travelling waves allows us to fix this value to ξ = 0 and we seek for a solution such that

   ϕ(ξ) < a -V for ξ < 0, ϕ(0) = a -V, ϕ(ξ) > a -V for ξ > 0. ( 12
)

RR n°8995

Using [START_REF] Kivshar | Nonlinear surface modes in monoatomic and diatomic lattices[END_REF] to simplify the nonlinear term F 0 (V + ϕ), system (11) takes the form

c 2 mϕ (ξ) = k [ϕ(ξ + 1) + ϕ(ξ -1) -2ϕ(ξ)] + . . . c a ϕ (ξ) -αcδ(ξ), (13)
where δ(ξ) is the Dirac delta function. Equation ( 13) is a linear non-autonomous differential equation so that one may attempt to use the Fourier transform to derive an analytic solution. However a certain amount of care is needed to correctly handle the Fourier transform of ϕ due to the nonzero boundary condition at -∞.

We look for ϕ(ξ) in the form

ϕ(ξ) = αa [ψ(ξ) + H(ξ) -1] , ψ(ξ) ∈ L 2 (R), lim ξ→±∞ ψ(ξ) = 0, (14)
where ψ(ξ) has to be determined. Equation ( 13) is re-expressed in terms of ψ(ξ) and Fourier transform is applied to determine ψ(ξ), and subsequently ϕ(ξ). Integrating ( 13), gives

c 2 mϕ = k ∧ * ϕ + c a ϕ + αc(1 -H), (15)
where ∧(ξ) = max (1 -|ξ|, 0) is tent function, and where we used for any f ∈ L 1 loc (R),

(∧ * f )(ξ) = ξ+1 ξ f (s)ds - ξ ξ-1 f (s)ds. (16)
Note that [START_REF] Murray | Mathematical biology. Biomathematics[END_REF] together with [START_REF] Flytzanis | High velocity dislocation motion and interatomic force law[END_REF] remains equivalent to the original problem ( 13)- [START_REF] Flytzanis | High velocity dislocation motion and interatomic force law[END_REF]. Injecting ( 14) into [START_REF] Murray | Mathematical biology. Biomathematics[END_REF], gives

c 2 mψ - c a ψ -k ∧ * ψ = k ∧ -c 2 mδ, ( 17
)
where we used the property ∧ * (H -1) = ∧. Taking the Fourier transform as ψ(λ) = R e -2πiλξ ψ(ξ)dξ in [START_REF] Raney | Stable propagation of mechanical signals in soft media using stored elastic energy[END_REF], we obtain

2iπλc 2 m - c a -k2iπλsinc 2 (λ) ψ(λ) = ksinc 2 (λ) -c 2 m, (18)
where we used ∧(λ) = sinc 2 (λ) with sinc(λ) = sin(πλ)/πλ. Let us introduce

K(λ) = 2iπλ c 2 m -ksinc 2 (λ) - c a -1
where one has

K(λ), K(ξ) ∈ L 2 (R) (K denotes the inverse Fourier transform of K). From d K dλ ∈ L 1 (R) and using -2iπξK = F -1 d K dλ ∈ L ∞ (R)
, one has lim ξ→±∞ K(ξ) = 0 (F -1 denotes the inverse Fourier transform). From [START_REF] Rhoads | Nonlinear dynamics and its applications in micro-and nanoresonators[END_REF], we obtain

ψ = kK * ∧ -c 2 mK. ( 19
)
Since ∧ ∈ L 1 (R) we have K * ∧ ∈ L 2 (R), and because K, ∧ ∈ L 2 (R) then K * ∧ ∈ C 0 (R) decays to zero when ξ → ±∞. Consequently ψ(ξ) given by ( 19) satisfies the properties assumed in [START_REF] Muratov | Traveling wave solutions in the Burridge-Knopoff model[END_REF], and defines a unique solution in L 2 (R). Therefore ( 14) is a solution of ( 13) with boundary Inria conditions [START_REF] Flytzanis | High velocity dislocation motion and interatomic force law[END_REF]. Regularity properties of ϕ(ξ) can be inferred from the following identity obtained from ( 14) and ( 17)

c 2 m αa ϕ = c a ψ + k ∧ * ψ + k ∧ . ( 20
)
This implies ϕ ∈ L 1 loc (R) (since ∧ * ψ ∈ L 2 (R)) and thus ones has ϕ ∈ C 0 (R). We also get from [START_REF] Murray | Mathematical biology. Biomathematics[END_REF]

that ϕ ∈ C 0 (R + ) ∩ C 0 -), hence ϕ ∈ C 1 (R + ) ∩ C 1 (R -), and thus (15) gives ϕ ∈ C 1 (R + ) ∩ C 1 (R -). We get finally ϕ ∈ C 2 (R + ) ∩ C 2 (R -) ∩ C 0 (R).
From the analytical expression of ϕ, we can derive an equation to determine the wave speed of the front. Using ( 14), we set

ϕ(ξ) + ϕ(-ξ) 2 = αa 2 [ψ(ξ) + ψ(-ξ) -1] ( 21
)
where ψ is defined by ( 19) (note that we used H(-ξ) = 1 -H(ξ) to eliminate the Heaviside function). Using the threshold condition ϕ(0) = a -V from ( 12) together with ( 21) and ( 19), we obtain that the wave speed satisfies

αac 2 m[K(0 + ) + K(0 -)] + 2(a -V ) + 1 = 0. (22)
This scalar equation allows us to compute c numerically using a Newton-type method. Computation of K is done using a Gauss-Konrod quadrature formula in a truncated interval [-10 6 , 10 6 ].

We restrict to c > 0 (the case c < 0 can be deduced by symmetry, see section 7). A plot of the resulting analytical profile ( 14) is shown in Fig. 7(a) and compared with the numerical simulation of [START_REF] Elmer | Spatially discrete FitzHugh-Nagumo equations[END_REF]. A perfect matching is realized between the two trajectories. The typical dependence of the wave speed on the stationary sliding velocity, V , and on the coupling, k, is shown in Fig. 7(b).

Anti-continuum limit

In this section the small coupling limit is explored. We consider the case c > 0 (see section 7 for the case c < 0). From ( 17) and ( 19) with k → 0, we have the leading order equation

c 2 mK - c a K -k ∧ * K = δ, (23)
where we look for a solution of the form

K = K 0 + kK 1 + O(k 2 ). (24)
Inserting ( 24) in [START_REF] Truskinovsky | Kinetics of martensitic phase transitions: Lattice model[END_REF], and equating orders of leading terms in k, we obtain

c 2 m(K 0 -νK 0 ) = δ, (25)
c 2 m(K 1 -νK 1 ) = ∧ * K 0 , (26)
where ν -1 = cam. Observe (25) has the unique bounded solution [START_REF] Muratov | Traveling wave solutions in the Burridge-Knopoff model[END_REF] where k = 0.3, V = 1.025, a = 1 and α = 0.2 (full line). The trajectory is indistinguishable from the ones obtained from the numerical simulation of the chain. The asymptotic approximation (29) obtained for k 1 also shown (dashed grey). We obtain c = 1.55 from the threshold condition (12) (the dashed line defines the threshold u 2 = a -V ). (b) Wave speed curves in the (c, k) plane obtained from [START_REF] Tonnelier | Stabilization of pulse waves through inhibition in a feedforward neural network[END_REF] for V = 1.025, 1.05, 1.075 and 1.1 (from right to left, respectively).

K 0 (ξ) = - 1 c 2 m e νξ H(-ξ), (27) RR n°8995
where K 0 ∈ L 1 (R), hence the solution of (26) reads

K 1 = K 0 * ∧ * K 0 = ∧ * K 0 * K 0 , = 1 c 2 m ∧ * K 0 + ν ∧ * K 0 * K 0 , (28)
where we used K 0 = 1 c 2 m δ + νK 0 . Using [START_REF] Rosakis | New solutions for slow moving kinks in a forced frenkelkontorova chain[END_REF] with ( 27) and (28), the approximation for ϕ up to

O(k 2 ) reads ϕ(ξ) = αa(e νξ -1)H(-ξ) + αak -c 2 mK 1 (ξ) + . . . (K 0 * ∧)(ξ) + O(k 2 ) ( 29
)
where we used the identity H(-ξ) = 1 -H(ξ). Expression (29) allows to obtain an approximation of the wave speed c for small k. From ϕ(0) = a -V and (29), we get We obtain after some calculations (see appendix A)

a -V = αak -c 2 mK 1 (0) + (K 0 * ∧)(0) + O(k 2 ), = -αkc(∧ * K 0 * K 0 )(0) + O(k 2 ), := S(c)k + O(k 2 ). (30)
S(c) = 2αma 3 - αa 2 c (2amc + 1)e -1/amc + 1 (31)
In order to approximate c, we drop O(k 2 ) terms in (30). The wave speed can be estimated from the solution of

ν -2 + (ν + 2)e -ν = V -a αma 3 k (32)
where ν -1 = acm. It can be shown that the left-hand side of (32) defines a bijective function on R that passes through the origin so that (32) admits a unique solution. Let fix the values of V and a, and look for solutions c ≈ 0 when k ≈ 0. Observing the exponential decay e -ν → 0 as c → 0, we have, from (31) and (30) the leading order approximation

ν = 2 + V -a αma 3 k (33)
for k, c → 0. Therefore we obtain the following approximation for the wave speed

c ∼ 1 2am + V -a αa 2 k (34) RR n°8995
The coexistence of this reverse front and the front satisfying ( 10)- [START_REF] Izhikevich | Dynamical systems in neuroscience: The geometry of excitability and bursting[END_REF] with the different velocity c(V ) can be used to understand the broadening of pulses reported in section ??, as well as the existence of steadily propagating pulses observed for particular sliding velocities. Indeed, we can see from Fig. 8(b) that the function V → c(V ) is decreasing (this is also clear from the leading order approximation (34)). Consequently, gluing the two above fronts to form a pulse decaying to 0 at infinity, the trailing front (at the rear of the propagating pulse) will be slower if V < Ṽ , resulting in a broadening of the pulse. This regime occurs for V ∈ (a, a(1 + α 2 )). In the critical case V = a(1 + α

2 ), we have V = Ṽ and the two fronts have identical velocities, thereby maintaining a steadily propagating pulse (this case is shown in Fig. 5(f)). Conversely, for V ∈ (a(1+ α

2 ), a(1+α)), the trailing front is faster and no pulse wave can propagate. Starting from an initial bump condition, an annihilation occurs when the trailing front reaches the leading front. In conclusion, the condition for the existence of broadening pulses reads

V < V * where V * = a α 2 + 1 . (39)
For V > V * , pulse fails to propagate whereas for V = V * a stable pulse is observed, with a width determined by the initial perturbation. In the small coupling limit, this pulse has a wave speed c ∼ 2ak according to approximation (34).

Discussion

We studied localized travelling waves in a nonlinear lattice describing a block-spring chain sliding down a slope and experiencing friction. Wave propagation was illustrated for different spinodal friction laws. For a particular range of stationary sliding velocities, the medium is made of blocks exhibiting bistabilities and supports nonlinear solitary transition waves (wave fronts). Interesting links can be made with recent results on waves in bistable lattices [START_REF] Raney | Stable propagation of mechanical signals in soft media using stored elastic energy[END_REF][START_REF] Nadkarni | Unidirectional transition waves in bistable lattices[END_REF]. For an idealized piecewise-linear friction force, we constructed analytically travelling fronts and analysed their wave speeds. In contrast with the discrete Nagumo equation, propagating fronts exist at small coupling values, i.e., propagation failure does not occur atnweak coupling strength. As already observed in a different context [START_REF] Tonnelier | Stabilization of pulse waves through inhibition in a feedforward neural network[END_REF], the travelling pulses are shaped by the concatenation of two travelling front solutions and pulse propagation failure occurs when the back wave is faster than the front wave. We determined analytically the parameter range where pulses of constant width occur, i.e., the leading front and the trailing front have the same velocity. It is worth noting that this analysis does not rely on a time scale separation and differs from the asymptotic construction of pulses done in [START_REF] Carpio | Pulse propagation in discrete systems of coupled excitable cells[END_REF]. In particular, the pulse width is not determined by the equality of the velocity of the two fronts but depends on the initial excitation. The present study is also of interest for the understanding of the dynamics of the Burridge-Knopoff model where the time evolution of the system is given by

γ ÿn = k c ∆ d y n -F (V + ẏn ) -y n . (40)
Let us define y n (t) = -F (V ) + γz n (t/γ) and k = γk c . Assuming γ 1, then, the Burridge-Knopoff model (40) can be rewritten in the fast time scale as

zn = k∆ d z n -F (V + żn ) + F (V )
that coincides with (2). Therefore, for small γ values, the front waves of (2) provide useful information on the dynamics of pulse propagation in the Burridge-Knopoff model (40). More precisely, fronts approximate the transition region from the ground state to the excited state. This is shown in Fig. 9 where the fast time scale of the Burridge-Knopoff model is accurately reproduced by model [START_REF] Burridge | Model and theoretical seismicity[END_REF].

A Computation of S(c)

We compute here the explicit expression of S(c) = -αc (∧ * K 0 * K 0 ) (0). We reexpress K 0 as K 0 (ξ) = -G(ξ) c 2 m where G(ξ) = e ν H(-ξ), hence we have

S(c) = - α c 3 m 2 (∧ * G * G) (0). (41)
We have

that is a decreasing function of ξ for ξ 0. The leading approximation of the wavefront is zero for ξ ≥ 1 and has a decreasing profile for ξ sufficiently small, therefore the wavefront is nonmonotonic and presents (at least) one dip after the front. Notice that the function occuring on the righthand side of (44) has a minimum at ξ = -1/ν that may be used to approximate the dip location.

Figure 1 :

 1 Figure 1: Mechanical representation of the block-spring slider model where m is the mass, k is the spring constant and V is the sliding velocity. The steady state corresponds to F (V ) = G with G = mg sin θ.

Figure 2 :

 2 Figure 2: Non-monotonic friction laws. (a) Coulomb-like friction force F ε , where ε = 10 -4 . (b) The cubic friction force F c (v), where b = 0.5, a = 1 and α = 0.2. (c) The piecewise linear friction force F 0 (v).

Figure 3 :

 3 Figure 3: Bifurcation diagrams of the single block model. Stationary state, u, as a function of the stationary sliding velocity, V , for (a) regularized generalized Coulomb friction force F ε (a = 1, α = 0.2, ε = 10 -4 ) (b) the cubic friction force F c and (c) the piecewise linear friction force F 0 (a = 1, α = 0.2)). Solid lines represent stable states (denoted U 1 and U 3 ) and dotted lines are for unstable states (U 2 ).

Figure 4 :

 4 Figure 4: Numerical simulations of equation (8) with the regularized Coulomb friction force F ε with the same parameters as in Fig. 3. We display spatiotemporal plots of the velocity variable u n of (a) a travelling front (k = 0.5 and V = 1.01), (b) a broadening pulse (k = 1 and V = 1.025), (c) a steadily propagating pulse solution (k = 1 and V = 1.046). An initial perturbation u 0 (0) = -10 is applied on the first block of the chain. Computations are performed for m = 0.15.

Figure 5 :

 5 Figure 5: Plots of the velocity waveforms u n (t) of the block-spring model in the travelling wave coordinate ξ = n -ct. The wave profiles in (a,b) are obtained with the regularized generalized Coulomb law F ε and correspond to the travelling waves shown in Fig. 4(a,c), respectively. Plots (c) and (d) represent the wave profiles obtained with the cubic friction force, F c . Plots (e) and (f) represent the wave profiles obtained with the piecewise linear friction law F 0 . The wave speed is (a) c = 1.95, (b) c = 2.21, (c) c = 3.06, (d) c = 3.16, (e) c = 3.16 (f) c = 1.45. For the piecewise linear law, we use a = 1 and α = 0.2. Other parameters are those of Fig. 4 for (a-b) and we take (c) V = 1.025, k = 1 (d) V = 1.18, k = 2, (e), V = 1.025, k = 1, (f) V = 1.1, k = 1.

Figure 6 : 2 .

 62 Figure 6: Plot of the velocity waveforms u n (t) of the block-spring model. The two localized pulses correspond to a snapshot of the travelling pattern shown in Fig. 4(b) at two different locations (n = 25, n = 50). The initial front propagates at speed c = 2.45 and the rear front at c = 2. Other parameters are those of Fig. 4(b)

Figure 7 :

 7 Figure 7: (a) Travelling front solution computed from the explicit formula[START_REF] Muratov | Traveling wave solutions in the Burridge-Knopoff model[END_REF] where k = 0.3, V = 1.025, a = 1 and α = 0.2 (full line). The trajectory is indistinguishable from the ones obtained from the numerical simulation of the chain. The asymptotic approximation (29) obtained for k 1 also shown (dashed grey). We obtain c = 1.55 from the threshold condition (12) (the dashed line defines the threshold u 2 = a -V ). (b) Wave speed curves in the (c, k) plane obtained from[START_REF] Tonnelier | Stabilization of pulse waves through inhibition in a feedforward neural network[END_REF] for V = 1.025, 1.05, 1.075 and 1.1 (from right to left, respectively).

Figure 8 :

 8 Figure 8: (a) Speed curves of the travelling front solution in the (c, k) plane for V = 1.0025, 1.005, 1.0075 and 1.01 (from right to left, respectively). Curve solutions (c, k) computed with (34) (dashed grey) accurately describes the exact curves (c, k) computed with (22) (black continuous) in the limit c → 0. (b) A zoom of the dashed square region in panel (a) is shown. Parameter values are α = 0.2 and a = 1.

ξFigure 9 :

 9 Figure 9: Comparison of a front solution of (2) (solid line) with a pulse supported by the Burridge-Knopoff model (dotted line). Computations are performed for the cubic friction law F c and the following parameters : γ = 0.05, k c = 10, V = 1.025.

1 0( 1 3 = e -ν ν 3 ( 2 + ν) + - 2 + ν ν 3 - 2 + ν ν 3 = 2 )

 1133223232 )G(-s -τ )dτ = se -νs H(s) with s > 0, therefore(∧ * G * G) (0) = R ∧(τ )(G * G)(-τ )dτ = -τ )(τ e -ντ )dτ = ν + e -ν ν -2 + 2e -ν ν with ν = (cam) -1 . We further calculate e -ν ν 3 (2 + ν) + -ν -2 2ν -1 -(2ν -1 + 1)e -ν -1(42)Inserting (42) into (41), givesS(c) = αν -2 c 3 m 2 2ν -1 -1 -(2ν -1 + 1)e -ν ,(43)and (31) follows.InriaB Monotonicity of the approximated frontFrom (28) and (29) one has the following asymptotic approximation of the wavefromϕ(ξ) = αa(e νξ -1)H(-ξ) -αck ∧ * K 0 * K 0 (ξ) + O(kWe note ϕ 1 = ∧ * K 0 * K 0 and we calculate ϕ 1 = 0 for ξ ≥ 1 andϕ 1 (ξ) = 1 m 2 c 4 ν 3 (2 + ν(1 -ξ)e ν(ξ-1) + . . . +(2 -ν(1 + ξ)e ν(ξ+1) + (2νξ -4)e νξfor ξ ≤ -1. For ξ 0 we obtain the following approximationϕ 1 (ξ) ∼ -2(cosh(ν) -1) m 2 c 4 ν 2 ξe νξ .Therefore the travelling front takes the leading formϕ(ξ) ∼ -αa + 2α(cosh(ν) -1)k m 2 c 3 ν 2 ξe νξas ξ 0. Using c ∼ αa 2 k V -a we have ϕ(ξ) ∼ -αa + 2(V -a)(cosh(ν) -1)ξe νξ

RR n°8995

where the leading order approximation reads c ∼ αa 2 k V -a . Formula (34) was derived under the assumption that c is small for k small, and one can easily check that c given by (34) satisfies c → 0 as k → 0. To evaluate the accuracy of the asymptotic approximation (34), we compare in Fig. 8 the (c,k) curves obtained from [START_REF] Tonnelier | Stabilization of pulse waves through inhibition in a feedforward neural network[END_REF] with those computed from (34) for different sliding velocities V . The asymptotic approximation (29) of the waveform is compared with the exact solution (see Fig. 7(a)). A good matching between the two wave profiles is found. Monotonicity analysis of the approximated waveform (29) shows that the velocity profile is nonmonotonic, i.e. a dip always exists behind the front (see appendix B).

Reverse travelling fronts and pulses

In the previous section we have constructed travelling fronts connecting the two stable equilibria U 1 = -αa (when n → -∞) and 0 (when n → +∞). In this analysis we have restricted our attention to travelling fronts with positive velocity c(V ) (for now we consider the dependency of front velocity in V and discard the other parameters). Using symmetry arguments, we show in the sequel the existence of travelling fronts with negative velocity satisfying the same boundary conditions. We also deduce the existence of travelling fronts with positive velocity satisfying reverse boundary conditions (u n → -αa when n → +∞ and u n → 0 when n → +∞).

Let us start with some symmetry considerations. Consider the advance-delay equation ( 11)

This problem admits the invariance

Moreover, the piecewise-linear friction force F 0 is antisymmetric about v = a, i.e. we have

As a consequence, one can readily check that (11)-( 35) is invariant by the one-parameter family of transformations

where λ ∈ R is arbitrary. Now let us use the above invariances in order to obtain reverse travelling fronts. We define ζ = -α a -ϕ, so that ζ and ϕ connect stable equilibria in reverse order at infinity. Applying invariance (37) for U 3 = 0 and λ = α a = -U 1 , it follows that ϕ is a solution of ( 11 [START_REF] Izhikevich | Dynamical systems in neuroscience: The geometry of excitability and bursting[END_REF] with sliding velocity V admits the front solution φ = -α a-ζ, satisfying the boundary conditions [START_REF] Flytzanis | High velocity dislocation motion and interatomic force law[END_REF] and having a negative velocity -c(a (2+α)-V ). Consequently, the search of front solutions of ( 10)- [START_REF] Izhikevich | Dynamical systems in neuroscience: The geometry of excitability and bursting[END_REF] can be reduced to the case c > 0 examined in section 5, since all fronts with c < 0 can be deduced by symmetry.

Furthermore, ϕ(ξ) = φ(-ξ) = -α a-ζ(ξ) defines another solution of [START_REF] Izhikevich | Dynamical systems in neuroscience: The geometry of excitability and bursting[END_REF] with sliding velocity V . This front has a positive velocity c( Ṽ ) = c(a (2 + α) -V ) and satisfies the reverse boundary conditions ϕ(-∞) = 0, ϕ(+∞) = -αa.