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The phase diagram of the quasi-one-dimensional magnet NaCuMoO$_4$(OH) isestablished through single-crystal NMR and heat-capacity measurements. The$^{23}$Na and $^1$H NMR experiments indicate a spiral and a collinearspin-density-wave (SDW) order below and above $B_c$ = 1.5-1.8 T, respectively.Moreover, in the paramagnetic state above the SDW transition temperature, thenuclear spin-lattice relaxation rate $1/T_1$ indicates anisotropic spinfluctuations that have gapped excitations in the transverse spectrum butgapless ones in the longitudinal spectrum. These static and dynamic propertiesare well described by a theoretical model assuming quasi-one-dimensional chainswith competing ferromagnetic nearest-neighbor interactions $J_1$ andantiferromagnetic next-nearest-neighbor interactions $J_2$ ($J_1$-$J_2$chains). Because of the excellent crystal quality and good one dimensionality,NaCuMoO$_4$(OH) is a promising compound to elucidate the unique physics of thefrustrated $J_1$-$J_2$ chain.

Collinear Spin-density-wave Order and Anisotropic Spin Fluctuations in the Frustrated $J_1$--$J_2$ Chain Magnet NaCuMoO$_4$(OH)