We investigated magnetic and thermodynamic properties of $S$ = 1/2 quasi-one-dimensional antiferromagnet KCuMoO$_4$(OH) through single crystalline magnetization and heat capacity measurements. At zero field, it behaves as a uniform $S$ = 1/2 Heisenberg antiferromagnet with $J$ = 238 K, and exhibits a canted antiferromagnetism below $T_mathrm{N}$ = 1.52 K. In addition, a magnetic field $H$ induces the anisotropy in magnetization and opens a gap in the spin excitation spectrum. These properties are understood in terms of an effective staggered field induced by staggered g-tensors and Dzyaloshinsky-Moriya (DM) interactions. Temperature-dependencies of the heat capacity and their field variations are consistent with those expected for quantum sine-Gordon model, indicating that spin excitations consist of soliton, anti-soliton and breather modes. From field-dependencies of the soliton mass, the staggered field normalized by the uniform field $c_mathrm{s}$ is estimated as 0.041, 0.174, and 0.030, for $H parallel a$, $b$, and $c$, respectively. Such a large variation of $c_mathrm{s}$ is understood as the combination of staggered g-tensors and DM interactions which induce the staggered field in the opposite direction for $H parallel a$ and $c$ but almost the same direction for $H parallel b$ at each Cu site.
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