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there is considerable interest in the study of equations of the form @xmath3 that describe breaking waves . in this paper we consider the equation @xmath4 with parameters @xmath5 ( not all zero ) and @xmath6 . this 4-parameter family contains several integrable equations . for @xmath7 and @xmath8 , equation reduces respectively to the camassa - holm equation @xcite @xmath9 and the degasperis - procesi equation @xcite @xmath10 while for @xmath11 , equation becomes the novikov equation @xcite @xmath12 the three equations , , are integrable in the sense of having a lax pair , a bi - hamiltonian structure , as well as hierarchies of local symmetries and local conservation laws , and they also possess peaked travelling wave solutions . in addition to these integrable equations , many other non - integrable equations that admit breaking waves are included in the 4-parameter family . for instance , there is the @xmath13-equation @xmath14 which unifies the camassa - holm and degasperis - procesi equations @xcite . there is also a modified version of the @xmath13-equation @xcite @xmath15 which includes the novikov equation . no other cases of the two equations and are known to be integrable @xcite .    an equivalent form of the 4-parameter equation is given by @xmath16 in terms of the momentum variable @xmath17 with parameters @xmath18 this parametric equation is invariant under the group of scaling transformations @xmath19 , @xmath20 , @xmath21 with @xmath22 .    in section  [ conslaws ] , we classify the low - order conservation laws of equation and show that the hamiltonians of the camassa - holm and novikov equations are admitted as local conservation laws by equation if and only if @xmath23 and @xmath24 . we consider peaked travelling waves in section  [ peakons ] and use a weak formulation of equation to show that single peakon and multi - peakon solutions are admitted if and only if @xmath23 and @xmath25 when @xmath26 . we derive the explicit equations of motion for @xmath27 peakon / anti - peakon solutions and also obtain the constants of motion inherited from the local conservation laws of equation .    in section  [ unifiedeqn ] , we combine the previous results to obtain a natural 1-parameter family of equations @xmath28 given by @xmath23 , @xmath24 , @xmath25 , @xmath26 , where a scaling transformation @xmath29 is used to put @xmath30 . since this 1-parameter family unifies the camassa - holm and novikov equations , we will refer to it as the _ gchn equation_. ( similar unified equations have been considered previously from related perspectives @xcite . ) we then discuss some general features of the dynamics of its @xmath31 peakon / anti - peakon solutions and we show that wave - breaking occurs for certain solutions describing collisions between peakons and anti - peakons in the case @xmath2 . finally , in section  [ remarks ] , we make some concluding remarks including a possible scenario for wave - breaking in the cauchy problem for weak solutions . for the 4-parameter equation , a _ local conservation law _ @xcite is a space - time divergence @xmath32 holding for all solutions @xmath33 of equation , where the _ conserved density _ @xmath34 and the _ spatial flux _ @xmath35 are functions of @xmath36 , @xmath37 , @xmath38 and derivatives of @xmath38 . the spatial integral of the conserved density @xmath34 satisfies @xmath39 and so if the flux @xmath35 vanishes at spatial infinity , then @xmath40= \int_{-\infty}^{\infty } t dx=\const\ ] ] formally yields a conserved quantity for equation . conversely , any such conserved quantity arises from a local conservation law . if the conserved quantity is purely a boundary term , then the local conservation law is called _ trivial_. this occurs when ( and only when ) the conserved density is a total @xmath37-derivative and the flux is a total @xmath36-derivative , related by @xmath41 for all solutions @xmath33 of equation , where @xmath42 is some function of @xmath36 , @xmath37 , @xmath38 and derivatives of @xmath38 . two local conservation laws are _ equivalent _ if they differ by a trivial conservation law , thereby giving the same conserved quantity up to boundary terms . the set of all conservation laws ( up to equivalence ) admitted by equation forms a vector space on which there is a natural action @xcite by the group of all lie symmetries of the equation .    for conserved densities and fluxes depending on at most @xmath43 , a conservation law can be expressed in an equivalent form by a divergence identity @xmath44 where @xmath45 is called the _ multiplier_. this identity  is called the _ characteristic equation _ @xcite for the conserved density and flux . by balancing the highest order @xmath36-derivative terms @xmath46 on both sides of the equation , we directly find that @xmath47 and @xmath48 . then balancing the terms @xmath49 , we see that @xmath50 . hence the conserved density and the flux in the divergence identity must have the form @xmath51 its multiplier thus has the form @xmath52    in general , the differential order of a local conservation law is defined to be the smallest differential order among all equivalent conserved densities . a local conservation law is said to be of _ low order _ if the differential orders of @xmath34 and @xmath35 are both strictly less than the differential order of the equation .    consequently , conserved densities and fluxes of the form comprise all possible low - order conservation laws of equation . the problem of finding all low - order conservations then reduces to the simpler problem of finding all low - order multipliers . since equation is an evolution equation , it has no lagrangian formulation in terms of the variable @xmath38 . in this situation , the problem of finding multipliers can be understood as a kind of adjoint @xcite of the problem of finding symmetries . an _ infinitesimal symmetry _ @xcite of equation is a generator @xmath53 whose coefficient @xmath54 is given by a function of @xmath36 , @xmath37 , @xmath38 and derivatives of @xmath38 , such that the prolonged generator satisfies the invariance condition @xmath55 holding for all solutions @xmath33 of equation . the lie symmetry group of equation is generated by infinitesimal symmetries with coefficients of the form @xmath56 if @xmath54 is at most linear in @xmath57 and @xmath58 , then the resulting generator will yield a group of point transformations @xcite , whereas if @xmath54 is nonlinear in @xmath57 or @xmath58 , then a group of contact transformations @xcite will be generated . hence , all generators of lie symmetries admitted by equation are determined by the solutions of condition for @xmath59 . ( it is straightforward to solve this determining equation by maple to classify the lie symmetry group of equation , as shown in the appendix . ) the condition for determining all multipliers @xmath60 of low - order conservation laws admitted by equation consists of @xmath61 which arises from the property that the variational derivative ( euler operator ) @xmath62 annihilates an expression identically iff it is a space - time divergence @xcite . this condition can be split with respect to @xmath63 and @xmath64-derivatives of @xmath63 , yielding an equivalent overdetermined system of equations on @xmath65 . one equation in this system is given by the adjoint of the symmetry determining equation , @xmath66 holding for all solutions @xmath33 of equation . solutions @xmath65 of this equation are called _ adjoint - symmetries _ ( or _ cosymmetries _ ) @xcite . the remaining equations in the system comprise helmholtz conditions which are necessary and sufficient for @xmath65 to have the form . as a consequence , multipliers are simply adjoint - symmetries that have a certain variational form . for any solution of the multiplier determining equation , a corresponding conserved density and flux of the form can be recovered either through integration @xcite of the characteristic equation , which splits with respect to @xmath67 , @xmath63 , @xmath68 , @xmath49 into a system of equations for @xmath34 and @xmath35 , or through a homotopy integral formula @xcite , which expresses @xmath34 and @xmath35 directly in terms of @xmath69 . it is straightforward to show that @xmath34 and @xmath35 have the form of a trivial conservation law iff @xmath70 . thus there is a one - to - one correspondence between equivalence classes of non - trivial low - order conservation laws and non - zero low - order multipliers . both the camassa - holm equation and novikov equation possess low - order local conservations law given by the conserved densities @xcite @xmath71 where @xmath72 and @xmath73 , respectively , for the two equations . in addition , the camassa - holm equation itself is a low - order local conservation law having the conserved density @xmath74 all of these conserved densities are related to hamiltonian structures for the two equations @xcite . the corresponding multipliers are respectively given by @xmath75    to look for conserved densities of the same form for equation , we now classify all multipliers up to 1st - order @xmath76 as well as all 2nd - order multipliers with the specific form @xmath77 in each case it is straightforward to solve the determining equation by use of maple ( as shown in the appendix ) , which leads to the following classification result . [ class - loworderq ] ( i ) equation admits 0th - order multipliers only in the following cases : @xmath78 ( ii ) for any @xmath6 and any @xmath79 , equation admits no 1st - order multipliers . ( iii ) equation admits 2nd - order multipliers of the form only in the following cases : @xmath80    in light of the adjoint relationship between multipliers and symmetries , the classification of 0th- and 1st- order multipliers in proposition  [ class - loworderq ] is a counterpart of the classification of lie symmetries ( cf . proposition  [ class - pointsymm ] ) . next we obtain the corresponding conserved densities and fluxes for each multiplier  by first splitting the characteristic equation with respect to @xmath67 , @xmath63 , @xmath68 , @xmath49 where @xmath34 and @xmath35 have the form , and then integrating the resulting system of equations . this yields the following low - order local conservation laws for equation . [ class - lowordertx ] ( i ) the local conservation laws admitted by the wave - breaking equation with multipliers of at most 1st - order consist of three 0th - order conservation laws @xmath81 and two 1st - order conservation laws @xmath82 ( ii ) the local conservation laws admitted by the wave - breaking equation with 2nd - order multipliers of the form consist of two 2nd - order conservation laws @xmath83 in these conservation laws  , any terms of the form @xmath84 in the case @xmath85 should be replaced by @xmath86 . these conservation laws yield the following conserved integrals . we start with the conservation laws at 0th order . from @xmath87 , we have @xmath88 which is the conserved mass for equation . the conserved integral arising from @xmath89 is a weighted mass , @xmath90 interestingly , from @xmath91 we get a conserved integral which vanishes , but has a non - zero spatial flux . this type of conservation law arises because the multiplier converts equation into the form of a total @xmath37-derivative . next we look at the conservation laws at 1st order . from @xmath92 , the @xmath0 norm of @xmath33 is conserved , @xmath93 from @xmath94 , we have @xmath95 where @xmath96 is the center of mass of @xmath33 . since @xmath97 is conserved , it can be evaluated at @xmath98 , which yields the relation @xmath99 . this shows that the center of mass moves at a constant speed controlled by the @xmath0 norm of @xmath33 .    finally , we consider the conservation laws at 2nd order . from @xmath100 , we get @xmath101 this shows that the @xmath102 norm of @xmath103 is conserved if @xmath104 does not change sign or if @xmath105 is an even integer . the conserved integral arising from @xmath106 is a linear combination of the @xmath107 norms of @xmath108 as given by @xmath109 this can be written alternatively as a weighted @xmath110 norm when @xmath111 . it is interesting to note that simultaneous conservation of both the @xmath0 and the weighted @xmath110 norms requires the condition @xmath112 which holds iff @xmath113 and @xmath114 , but in this case @xmath115 . both the camassa - holm and novikov equations possess peaked travelling wave solutions @xcite , called peakons , @xmath116 where @xmath117 and @xmath72 , respectively , for the two equations . peakons have attracted much attention in the study of breaking wave equations .    in general , on @xmath118 , a peakon is a weak travelling wave solution satisfying an integral ( i.e.  weak ) formulation of a breaking wave equation . such a formulation is essential for deriving multi - peakon solutions . however , single peakons can be derived directly from the travelling wave reduction of a breaking wave equation , which will be the approach we use here . the manifest invariance of the 4-parameter equation under time - translation and space - translation symmetries implies the existence of travelling wave solutions @xmath119 where @xmath120 satisfies the ode @xmath121    for the travelling wave ode , an integral formulation is obtained through multiplying this ode by a test function @xmath122 ( which is smooth and has compact support ) and integrating over @xmath123 , leaving at most first derivatives of @xmath124 in the integral , which yields @xmath125 a weak solution of ode is a function @xmath120 that belongs to the sobolev space @xmath126 and that satisfies the integral equation for all smooth test functions @xmath127 with compact support on @xmath128 .    to proceed we substitute a peaked travelling wave expression @xmath129 into equation and split up the integral into the intervals @xmath130 and @xmath131 . the first term in equation yields , after integration by parts , @xmath132 similarly , the second term in equation gives @xmath133 provided @xmath134 so that the boundary terms at @xmath135 vanish . the third and fourth terms in equation together yield @xmath136 when the terms  are combined , we find that equation reduces to @xmath137 this equation is satisfied for all test functions @xmath122 iff @xmath138 which determines the amplitude @xmath139 in the peakon expression . thus we obtain the following result . [ class - peakon ] the travelling wave equation admits a peakon solution only in the case @xmath140 where @xmath141 is the wave speed . the resulting peakon solution of equation is given by @xmath142 when the nonlinearity power @xmath143 is a positive integer , then the wave speed is necessarily positive , @xmath144 , if @xmath143 is even , as in the case ( @xmath145 ) of the novikov equation , while if @xmath143 is odd , the wave speed can be either positive or negative , @xmath146 , as in the case ( @xmath147 ) of the camassa - holm equation . the peakon solution satisfies equation only in the sense of a weak solution . this means @xmath33 is a distribution in @xmath148 with respect to @xmath149 for some @xmath150 and in @xmath126 with respect to @xmath151 such that it satisfies the integral equation @xmath152 for all test functions @xmath153 in @xmath154 . both the camassa - holm and novikov equations possess multi - peakon solutions @xcite which are a linear superposition of peaked travelling waves with time - dependent amplitudes and positions . the form of these solutions is given by @xmath155 where the amplitudes @xmath156 and positions @xmath157 satisfy a hamiltonian system of odes @xmath158 given in terms of the hamiltonian function @xmath159 the poisson bracket @xmath160 in this system arises from the respective hamiltonian operator formulations @xcite of these two equations and has the standard canonical form in the case of camassa - holm equation and a certain non - canonical form in the case of the novikov equation . we now investigate whether equation also admits multi - peakon solutions . it will be convenient to use the notation @xmath161 where the summation is understood to go from @xmath162 to @xmath1 . note that the @xmath37-derivatives of @xmath38 are given by @xmath163 and @xmath164 in terms of the sign function @xmath165 and the dirac delta distribution @xmath166 which has the properties @xmath167 for @xmath168 , and @xmath169 for all @xmath170 . to begin , we substitute the general multi - peakon expression into the integral equation . there are two ways we can then proceed . one way is to assume @xmath171 at a fixed @xmath172 , split up the integral over @xmath37 into corresponding intervals , and integrate by parts , similarly to the derivation of the single peakon solution . another way , which is simpler , is to employ the following result from distribution theory @xcite . let @xmath173 be a piecewise @xmath174 function having at most jump discontinuities at a finite number of points @xmath175 in @xmath128 . then , for any test function @xmath176 , @xmath177_{x_i } - \int_{-\infty}^{\infty } \psi \ns{f'}\;dx\ ] ] where @xmath178_{x_i } = f(x_i{}^+ ) - f(x_i{}^-)\ ] ] is the jump in @xmath173 at the point @xmath175 , and @xmath179 is the non - singular part of the distributional derivative of @xmath173 . we will now use this integration by parts relation to evaluate each term in the integral equation .    the first term in equation yields @xmath180_{\beta_i } - \psi(t,\beta_i)[u_{tx}]_{\beta_i } ) \ ; dt .   \end{aligned}\ ] ] from expression , we see @xmath181 so thus @xmath182 holds a.  e.  in @xmath183 . hence @xmath184 and thus we get @xmath185_{\beta_i } - \psi(t,\beta_i)[u_{tx}]_{\beta_i } ) \ ; dt .\ ] ]    next , the second term in equation gives @xmath186_{\beta_i } \ ; dt . \end{aligned}\ ] ] we now simplify the two parts of the integral involving @xmath187 . for the first part , we have @xmath188 after using relation and then integrating by parts . for the second part , since @xmath189 holds a.  e.  in @xmath183 , we have @xmath190_{\beta_i } \ ; dt   - \iint_{-\infty}^{+\infty } \psi ( u^{p-1}\ns{u_x^2})_x \ ; dx\ ; dt   \end{aligned}\ ] ] from applying the integration by parts relation . by simplifying @xmath191 a.  e.   with the use of relation , we see @xmath192 hence the integral becomes @xmath193_{\beta_i } \ ; dt . \end{aligned}\ ] ] then we have @xmath194_{\beta_i } \ ; dt   + \sum_{i}\int_{-\infty}^{+\infty } c\psi_x(t,\beta_i)u(\beta_i)^p [ u_x]_{\beta_i } \ ; dt . \end{aligned}\ ] ]    similarly , the third term in equation gives @xmath195_{\beta_i } \ ; dt .   \end{aligned}\ ] ] we can simplify the integral involving @xmath122 by the same steps used for the previous integral . this yields @xmath196 hence we then have @xmath197_{\beta_i } \ ; dt . \end{aligned}\ ] ]    finally , by combining the three terms , , with the fourth term in equation , we obtain @xmath198_{\beta_i } + c u(t,\beta_i)^p[u_x]_{\beta_i } ) \ ; dt \\&\qquad   + \sum_{i } \int_{-\infty}^{+\infty } \psi(t,\beta_i ) ( -[u_{tx}]_{\beta_i } + \tfrac{1}{2}(pc - b)u(t,\beta_i)^{p-1}[u_x^2]_{\beta_i } ) \ ; dt . \end{aligned}\ ] ] the jump terms are evaluated by @xmath199_{\beta_i }   = 2\alpha_i\beta_i ' , \quad [ u_x]_{\beta_i }   = -2\alpha_i \\ [ u_{tx}]_{\beta_i }   = \frac{d\,[u_x]_{\beta_i}}{dt } = -2\alpha_i ' \\ [ u_x^2]_{\beta_i }   = 2u_x(\beta_i)[u_x]_{\beta_i } = -4\alpha_iu_x(\beta_i)\end{gathered}\ ] ] which all follow directly from the expressions and . thus we get @xmath200 this equation is satisfied for all test functions @xmath122 iff @xmath201 which determines the amplitudes @xmath202 and the positions @xmath203 in the multi - peakon expression . thus we have established the following result . [ class - npeakon ] the integral equation admits an @xmath1-peakon solution for all @xmath204 only in the case @xmath205    from propositions  [ class - npeakon ] and  [ class - peakon ] , we have a classification of all cases for which the 4-parameter equation possesses both single peakon and multi - peakon solutions . [ class - peakons ] the 4-parameter equation admits single peakon and multi - peakon solutions iff @xmath206 in this case , a general @xmath1-peakon solution has the form , where the amplitudes @xmath156 and positions @xmath157 satisfy the system of odes @xmath207 in terms of the separations @xmath208    this result generalizes related work in ref.@xcite which established the existence of single and multi - peakon solutions for a 2-parameter equation defined by the case @xmath209 , @xmath30 of equation . ( in particular , the derivation in ref.@xcite was completely formal , whereas the steps here provide a rigorous proof applied to the more general 4-parameter equation . ) it is easy to check that the general @xmath1-peakon ode system  reduces to the well - known multi - peakon systems for the @xmath13-equation when @xmath210 , which includes the the camassa - holm equation and the degasperis - procesi equation when @xmath211 and @xmath212 , respectively , as well as for the novikov equation when @xmath11 . the ode system  for the amplitudes and positions of the @xmath1 peakons in the expression inherits constants of motion ( i.e.  time - independent quantities ) given by the conserved integrals that are admitted by equation in the case . from theorem  [ class - lowordertx ] , there are six conserved integrals  which we can consider . the first conserved integral yields @xmath213 this quantity @xmath214 is the total mass for the @xmath1-peakon solution . a weighted mass arises from the second conserved integral , @xmath215 the next conserved integral gives @xmath216 which is the @xmath0 norm of the @xmath1-peakon solution . the fourth conserved integral does not exist in the case , since @xmath217 and @xmath218 together imply that @xmath219 . last , the two conserved integrals and are nonlinear in @xmath220 which is a distribution . as a consequence , both these integrals are ill - defined for the the @xmath1-peakon solution . from theorems  [ class - lowordertx ] and  [ class - peakons ] , the low - order conservation laws  as well as the @xmath1-peakon solution expression of the camassa - holm and novikov equations are admitted simultaneously by the 4-parameter equation iff its parameters @xmath221 satisfy @xmath222 after a scaling transformation @xmath29 is used to put @xmath30 , equation reduces to the 1-parameter gchn equation presented in section  [ intro ] . the explicit system describing @xmath1-peakon solutions of the gchn equation for all @xmath26 is given by @xmath223 where @xmath156 and @xmath157 are , respectively , the amplitudes and positions appearing in the general @xmath1-peakon expression . the @xmath0 norm of the @xmath1-peakon solution provides a constant of motion @xmath224 which is determined by the initial amplitudes and initial separations . when all of the amplitudes are positive , @xmath225 , for all @xmath226 , the solution expression is a superposition of @xmath27 peakons , each of which is right moving . in this case , the constant of motion directly gives the inequality @xmath227 , @xmath228 , which implies that any collisions among the @xmath1 peakons are elastic . when all of the amplitudes are negative , @xmath229 , for all @xmath226 , the solution expression is instead a superposition of @xmath27 anti - peakons , each of which is either right moving if @xmath143 is even or left moving if @xmath143 is odd . similarly to the previous case , the constant of motion yields @xmath230 , @xmath228 , implying that any collisions among the @xmath1 anti - peakons are elastic .    in the case when some amplitudes have opposite signs , or an amplitude changes its sign at some @xmath172 , the solution expression then describes a superposition of both peakons and anti - peakons . although the constant of motion is still non - negative , the amplitudes are no longer bounded by @xmath231 . as a consequence , wave breaking can occur in collisions , which we will now show for the case @xmath2 .      for @xmath2 , the system  describing @xmath232-peakon solutions @xmath233 takes a simple form . first , the constant of motion can be used to express the relative separation @xmath234 in terms of the two amplitudes @xmath235 and @xmath236 through the relation @xmath237 then , the equations of motion for the two positions @xmath238 and @xmath239 and the two amplitudes @xmath235 and @xmath236 are given by @xmath240 with @xmath241 and @xmath242 if another constant of motion could be found for this system , then the system could be reduced to two separated odes for the two amplitudes , plus two quadratures for the two positions , which would allow the general solution to be obtained . even without another constant of motion , it is still possible to do a qualitative analysis of all solutions by studying the phase plane @xmath243 of the coupled odes  for the amplitudes . we start from the relation , which imposes inequalities on the amplitudes , @xmath244 for a given value of @xmath245 , these two inequalities define the domain for all @xmath232-peakon solutions in the phase plane @xmath243 . the boundary of the domain corresponds to the two equalities latexmath:[\[\label{h - circ } \alpha_1 ^ 2 + \alpha_2 ^ 2 = h ,   \quad    ( \alpha_1 + \alpha_2)^2 = h , \quad    the circle comprises the equilibrium points of the amplitude odes  in the phase plane . each point on the circle is a limit of a @xmath232-peakon solution describing an asymptotic superposition of two @xmath162-peakon solutions , in which the amplitudes are constant and the positions are infinitely separated . the lines each constitute a degenerate @xmath232-peakon solution in which the two positions coincide and the sum of the two amplitudes is constant , describing a peakon solution @xmath247 in the case of the upper line , and an anti - peakon solution @xmath248 in the case of the lower line . the entire solution domain divides into four parts which are related by a reflection symmetry @xmath249 . one part of the domain is given by the points lying between the circle and the upper line in the first quadrant , which comprises all solutions describing two peakons . there is a counterpart given by the points lying between the circle and the lower line in the third quadrant , which comprises all solutions describing two anti - peakons . the two other parts of the domain comprise all solutions describing a peakon and an anti - peakon . these parts are given by the points between the segments of the upper and lower lines that lie outside of the circle . within this solution domain in the phase plane , the flow defined by the amplitude odes  depends on the nonlinearity power @xmath143 and the sign of the separation @xmath250 . we are interested in flows that describe a collision between a peakon and an anti - peakon . this condition can be used to determine @xmath251 at @xmath98 at each point in the phase plane by considering the ode @xmath252 for the separation . if @xmath253 , then the relative separation @xmath254 between the peakon and anti - peakon will be decreasing only if @xmath255 . similarly , if @xmath256 , then the relative separation @xmath254 between the peakon and anti - peakon will be decreasing only if @xmath257 . hence , a necessary condition for a collision to occur is that @xmath258 and @xmath250 have opposite signs during the flow . since @xmath259 can occur only on the upper and lower lines , which are boundaries of the domain in which solutions describe a collision between a peakon and an anti - peakon , we can impose @xmath260 at each point in the phase plane . note @xmath261 holds iff @xmath262 when @xmath143 is odd , and @xmath263 when @xmath143 is even . the points given by @xmath262 in the phase plane consist of the lines and @xmath264 , while the points given by @xmath265 consist of the lines that are perpendicular to each of those three lines . consequently , hereafter we will consider initial conditions @xmath266 and @xmath267 without loss of generality . ( note that reversing the sign in the initial condition will correspond to reflecting the flow about the line @xmath268 in the phase plane . )    under the collision condition and initial conditions , the flow then depends only on the nonlinearity power @xmath143 . the case @xmath147 , which represents the camassa - holm equation , is special , since there is another constant of motion @xmath269 which is given by the total mass . this implies that the flow simply consists of parallel lines in the phase plane . in all other cases @xmath270 , the flow is no longer given by straight lines and has a much richer structure .    ]    ]    the flows for all even powers @xmath271 are qualitatively similar to the case @xmath145 , which represents the novikov equation . a picture of the phase plane for @xmath272 is shown in fig . [ p=4-plane ] . clearly , in the second quadrant , the upper line @xmath273 is a stable asymptotic attractor for solutions describing a peakon ( @xmath274 ) and an anti - peakon ( @xmath275 ) , while the lower line @xmath276 is an unstable asymptotic attractor . in the fourth quadrant , these behaviours are reversed . the flows for all other odd powers @xmath277 are qualitatively similar to the case @xmath278 which is shown in fig . [ p=3-plane ] . in the second quadrant , both the upper and lower lines @xmath279 are stable asymptotic attractors for solutions describing a peakon ( @xmath274 ) and an anti - peakon ( @xmath275 ) . the line @xmath280 is an unstable asymptotic attractor . in the fourth quadrant , the behaviour is the same .    in all cases @xmath270 , the flow will evolve the initial amplitudes toward a stable attractor line . this evolution is shown in figs . [ p=3-collide ] and  [ p=4-collide ] for the cases @xmath278 and @xmath272 , respectively , where the initial positions of the peakon and anti - peakon are chosen to be distinctly separated . we see that the peakon and anti - peakon collide such that their peak amplitudes become closer while the slope at locations @xmath37 in between the two peaks rapidly increases ( without bound ) as relative separation between their positions decreases to zero in a finite time . this blow - up in the slope seen in figs . [ p=3-blowup ] and  [ p=4-blowup ] is an example of wave breaking . .35   ( @xmath281 , @xmath282),title="fig : " ]    .35   ( @xmath281 , @xmath282),title="fig : " ]    .35   ( @xmath281 , @xmath282),title="fig : " ]    .35   ( @xmath281 , @xmath282),title="fig : " ]    .33   ( @xmath283 , @xmath284),title="fig : " ]    .33   ( @xmath283 , @xmath284),title="fig : " ]    .33   ( @xmath283 , @xmath284),title="fig : " ]    .33   ( @xmath283 , @xmath284),title="fig : " ]    there is a qualitative explanation of why the blow - up in the slope @xmath58 between the two peaks in a collision solution occurs in a finite time . consider the asymptotic attractor solution @xmath285 corresponding to the upper line . this solution arises from the initial condition @xmath286 and @xmath287 . the amplitude odes yield @xmath288 whereby @xmath289 and @xmath290 as @xmath291 such that @xmath292 is constant for all @xmath226 . any solution having an initial condition close to @xmath286 and @xmath287 will exhibit a similar long - time behaviour for @xmath235 and @xmath236 , as a consequence of continuous dependence of solutions on initial data for the odes . since @xmath293 , the solution remains continuous and bounded at all @xmath37 for @xmath226 , whereas the slope @xmath294 has jump discontinuities at @xmath295 and @xmath296 and becomes unbounded at @xmath297 ( with @xmath298 ) as @xmath299 . the same kind of wave - breaking behaviour can be expected to occur in collisions between peakons and anti - peakons when @xmath300 .    .3 , title="fig : " ]    .3 , title="fig : " ]    .3 , title="fig : " ]    .3 , title="fig : " ]    .3 , title="fig : " ]    .3 , title="fig : " ]    .3 , title="fig : " ]    .3 , title="fig : " ] at first sight , the gchn equation seems closely analogous to the @xmath13-equation : both equations unify two integrable equations , possess @xmath1-peakon solutions , and exhibit wave breaking phenomena . however , there are important differences . firstly , the nonlinearities in the @xmath13-equation are purely quadratic , whereas the gchn equation has nonlinearities of degree @xmath301 and thereby connects two integrable equations with different nonlinearities . secondly , the @xmath0 norm of solutions @xmath33 is conserved for the @xmath13-equation only if @xmath302 , when the @xmath13-equation reduces to the camassa - holm equation . in contrast , the @xmath0 norm is conserved for the gchn equation for all @xmath6 .    in a subsequent work , we will explore further properties of the gchn equation and its multi - peakon solutions . there are numerous interesting questions . can a wave - breaking result similar to those for the camassa - holm and novikov equations be established for classical solutions ? how will the wave - breaking behaviour depend on @xmath143 ? in particular , a plausible criteria for wave - breaking is @xmath303 which generalizes the criteria known @xcite in the camassa - holm case @xmath147 and the novikov case @xmath145 . in another direction , for any @xmath143 other than these two known integrable cases @xmath147 and @xmath145 , does the equation have a hamiltonian formulation or perhaps integrability properties ? s.c . anco is supported by an nserc research grant . p.l . da silva and i.l . freire would like to thank fapesp ( scholarship n. 2012/22725 - 4 and grant n. 2014/05024 - 8 ) and capes for financial support . freire is also partially supported by cnpq ( grant n. 308941/2013 - 6 ) . the referee is thanked for remarks which have improved this paper .        to classify all of the lie symmetries admitted by the 4-parameter equation , we first substitute a general coefficient function @xmath59 into the symmetry determining equation . next we eliminate @xmath63 , @xmath304 , @xmath305 through writing the equation in the solved form @xmath306 and doing the same for its differential consequences . the determining equation then splits with respect to @xmath220 , @xmath307 , @xmath49 , @xmath68 , @xmath67 , @xmath308 into a linear overdetermined system of 10 equations for @xmath309 : @xmath310 equation shows that @xmath54 is a linear function of @xmath57 and @xmath58 , and hence @xmath311 for some functions @xmath312 . after simplifying the remaining equations  , we obtain a system of 14 equations @xmath313 we solve this linear overdetermined system by the following steps . first , an integrability analysis of the system of equations  is carried out using the maple package _ rifsimp _ , which yields 9 cases . next , in each case the reduced system of equations is integrated using the maple command _ pdsolve_. last , the solutions are merged , which leads to the following classification result . [ class - pointsymm ] ( i ) for any @xmath6 and any @xmath79 , equation admits no contact symmetries . ( ii ) the point symmetries admitted by equation for all @xmath6 and all @xmath79 consist of @xmath314 ( iii ) equation admits additional point symmetries only in the following cases : @xmath315    the symmetries @xmath316 , @xmath317 , @xmath318 respectively generate one - dimensional point transformation groups consisting of time - translations @xmath319 , space - translations @xmath320 , and scalings @xmath321 , @xmath322 , with group parameter @xmath323 . the extra symmetry @xmath324 generates the one - dimensional point transformation group @xmath325 , @xmath326 , which is a galilean boost , and the other extra symmetry @xmath327 generates the one - dimensional point transformation group @xmath328 , @xmath329 , which is a non - rigid dilation . to classify all 1st - order multipliers admitted by the 4-parameter equation , we first substitute the expression into the determining equation , which splits into a linear overdetermined system of 5 equations for @xmath330 . the system contains the equations @xmath331 which yield @xmath332 after the remaining 3 equations are split with respect to @xmath57 and @xmath58 , we obtain the following system of 8 equations @xmath333 we solve this linear overdetermined system by the same three steps used in solving the symmetry system . this yields the five distinct cases presented in parts ( i ) and ( ii ) of proposition  [ class - loworderq ] .    finally , by splitting and simplifying the determining equation for second - order multipliers of the form , we obtain a linear overdetermined system of 13 equations for @xmath334 . one of the equations in this system is given by @xmath335 which yields @xmath336 the remaining 10 equations then split with respect to @xmath58 , leading to a system of 6 equations @xmath337 solving this linear overdetermined system by the same steps used in solving the multiplier system  , we obtain the two distinct cases presented in part ( iii ) of proposition  [ class - loworderq ] .                  p.l . da silva and i.l . freire , an equation unifying both camassa - holm and novikov equations , _ proceedings of 10th aims conference _ , ( 2014 ) [ accepted ] . see also p.l . da silva and i.l . freire , strict self - adjointness and shallow water models , _ arxiv:1312.3992 _ ( 2013 ) .
a 4-parameter polynomial family of equations generalizing the camassa - holm and novikov equations that describe breaking waves is introduced . 
 a classification of low - order conservation laws , peaked travelling wave solutions , and lie symmetries is presented for this family . 
 these classifications pick out a 1-parameter equation that has several interesting features : it reduces to the camassa - holm and novikov equations when the polynomial has degree two and three ; it has a conserved @xmath0 norm and it possesses @xmath1-peakon solutions , when the polynomial has any degree ; and it exhibits wave - breaking for certain solutions describing collisions between peakons and anti - peakons in the case @xmath2 .