We generalize the classical work of Adlam and Allen [Phil. Mag.3, 448 (1958)] on solitons in a cold plasma propagating perpendicular to the magnetic field to include the effects of plasma pressure. This is done by making extensive use of the properties of total momentum conservation (denoted by the term ‘momentum hodograph’, since it yields a locus in the plane of the electron and proton speeds in the direction of the wave) and the energy integral of the system as a whole. These relations elucidate the phase and integral curves of stationary flows, from which soliton solutions may be constructed. In general, only compressive solitons are permitted, and we have found an analytical expression for the critical fast Mach number as a function of the proton acoustic Mach number, which shows that it varies from its classical value of 2 (at large proton acoustic Mach numbers) to unity, where the incoming flow is proton-sonic. At the critical fast Mach number, two possible soliton-like solutions can be constructed. One is the classical compression, in which the magnetic field develops a cusp in the centre of the wave. The other is a compression in the magnetic field followed by a deep depression in the centre of the wave, which is completed by the mirror image of this signature of compression–rarefaction. This structure involves a smooth supersonic–subsonic transition in the proton flow. For Mach numbers in excess of the critical one, this kind of structure can also be constructed, but now the magnetic field is cusp-like at the points of maximum compression.