Standing and travelling waves in a parabolic-hyperbolic system

We consider a nonlinear system of partial differential equations which describes the dynamics of two types of cell densities with contact inhibition. After a change of variables the system turns out to be parabolic-hyperbolic and admits travelling wave solutions which solve a 3D dynamical system. Compared to the scalar Fisher-KPP equation, the structure of the travelling wave solutions is surprisingly rich and to unravel part of it is the aim of the present paper. In particular, we consider a parameter regime where the minimal wave velocity of the travelling wave solutions is negative. We show that there exists a branch of travelling wave solutions for any nonnegative wave velocity, which is not connected to the travelling wave solution with minimal wave velocity. The travelling wave solutions with nonnegative wave velocity are strictly positive, while the solution with minimal one is segregated in the sense that the product \begin{document}$ uv $\end{document} vanishes.


Introduction. A travelling wave solution is a special type of solution which
propagates with a constant velocity and a fixed profile. Travelling wave phenomena are ubiquitous in many fields of science. The propagation of impulse in nerve fibers and the spread of fronts in combustion and chemical reactions are examples of such waves ( [13,14]). Besides, travelling waves often characterise the large time behaviour of solutions in PDEs. Therefore, travelling waves have attracted researchers' attention.
In the present paper, we consider nonnegative solutions of the system where α, k and γ are positive constants. If v ≡ 0, u satisfies the degenerate Fisher-KPP equation. System (1) describes the growth and the motility of two types of cell densities u(x, t) and v(x, t) with contact inhibition ( [9,15,16]). A feature of system (1) is the mobility term which is not a usual diffusion. This means that each cell density moves towards less populated regions by sensing the population pressure which is described by the gradient of total cell density. In a future paper we shall extensively discuss the biological interpretation of the mathematical results obtained in this and other papers. Here, we concentrate on the mathematical novelties of the results. Setting w = u + v, the parabolic-hyperbolic character of system (1) becomes clear: Given nonnegative and bounded initial data u 0 (x) and v 0 (x), the evolution problem for (1) has a solution (u(x, t), v(x, t)), and if u 0 v 0 = 0 a.e. in R, then u(x, t)v(x, t) = 0 for all t > 0 and a.e. x ∈ R (see [1,2,4]). In other words, if u and v are initially segregated, they remain segregated for later time. The segregation property reflects the hyperbolic character of the system. Recently, a similar result for the Neumann boundary value problem in 1D was obtained by Carrillo et al( [8]).
In view of the segregation property, it is natural to ask whether or not there exist segregated TWs. In [5], it was shown that there exists a unique wave velocity c, depending on γ and the quotient k/α, for which system (1) has a unique segregated TW up to translation: for each positive α, k and γ there exist unique nonnegative functions U (z) and V (z) and c ∈ R such that u(x, t) = U (x − ct) and v(x, t) = V (x − ct) satisfy (1) Here, we fix the interface at z = 0. In addition − 1 2 < c <  it is monotonically decreasing in α and its sign (i.e. the moving direction) changes at k α = 1. We refer to [5] for further details. In Figure 2 we display typical profiles of segregated TWs.
Examples of profiles of overlapping TWs are shown in Figure 3. Numerical evidence suggests that the overlapping TWs are unique up to translation and there are no TWs if c < c. In numerical simulations, we use a shooting method to find TWs. For each simulation, only one trajectory connecting between two equilibria, which corresponds to a TW, can be numerically obtained. This suggests the uniqueness of TWs up to translation. On the other hand, if we can not capture any trajectory connecting between two equilibria numerically, this implies that there are no TWs.
In this way, we obtain numerical evidence about the nonexistence and the uniqueness of TWs in numerical simulations. So, if k > α = 1, the situation reminds that of the Fisher-KPP equation. The analogy goes even further: numerical evidence suggests that for a large class of (not necessarily segregated) initial data, the corresponding solutions of the evolution problem converge to a segregated TW, i.e. to a TW with minimal wave velocity (more precisely, the large time behaviour of the solution of the evolution problem depends critically on the decay rate of the initial function v 0 (x) at x → ∞; see [3]). A first guess could be that the result for k > α = 1 can be generalised to the case in which k > 1 and 0 < α < k, when the wave velocity c of the segregated TW is positive. However, as we discuss in a future paper, this is false for two reasons: (1) there is a parameter regime where the wave velocity of the segregated TW is not minimal and the branch of overlapping TWs has a turning point; (2) if α < 1, for either sufficiently small or sufficiently large values of γ, the branch of overlapping TWs does not connect to the segregated TW, but to a partially overlapping TW (i.e. U V vanishes on the halflines (−∞, 0) or (0, ∞)), which is connected to the segregated TW through a branch of partially overlapping TWs. So, the structure of the TWs is much richer than that of the scalar Fisher-KPP equation.
In the present paper, we consider the case α > k > 1 and γ > 0.
Then, the velocity of the segregated TWs is negative: c < 0. We shall prove that there exists a branch of overlapping TWs for all c ≥ 0 (if c = 0 it is more appropriate to call it an overlapping standing wave). See  shown in Figure 5 overlapping TWs with non-monotone in U + V are also exhibited in accordance with the parameter values.
has a solution (U c (z), V c (z)), where U c , V c ∈ C 2 (R) and U c (z) > 0 and V c (z) > 0 for all z ∈ R.
In addition (U c (z), V c (z)) converges to a standing wave as c → 0 in the following sense: let (U c , V c ) satisfy Then any sequence c k → 0 has a subsequence (denoted again by c k ), such that: w c k converges uniformly in R to a function w 0 ∈ C 1 (R); w c k converges uniformly in R to w 0 , and w 0 > 0 in R; Condition (3) is a way to eliminate the translation invariance of the TWs. We conjecture that the weak standing wave is unique up to translation, but since we have not proved it, the convergence as c → 0 is stated up to subsequences.
As we shall see in Section 2 (see Remark 2.4), the standing wave has extremely fast decay properties (one could call it "almost segregated"). The fast decay is also visible in the constructed overlapping TWs: if c > 0, V c has exponential decay as z → ∞, but the exponent explodes as c → 0 and in the limit one recovers the fast decay of the standing wave (see Section 4). Again the structure is surprisingly different from that of the scalar Fisher-KPP equation: there is a continuum of wave velocity, [0, ∞), but the velocity c = 0 is not minimal, and the segregated TW with negative velocity is not connected to the branch of overlapping TWs defined by Theorem 1.1. Indeed, numerical evidence suggests that there are no other overlapping TWs for c < 0 (this means that we could not numerically capture any trajectory connecting between two equilibria) and that the pair (U c , V c ) defined by Theorem 1.1 is unique up to translation. However, the segregated TW is not isolated: in a forthcoming paper, we shall show that it can be connected with a new type of partially segregated TWs which possess at least two interfaces and negative wave velocity larger than that of the segregated TW.
We are not yet able to address the stability of the TWs analytically. In a future paper, we present numerical evidence for the stability of the segregated TW. For example, it attracts those solutions of the evolution problem for which u and v initially vanish for large values of respectively −x and x. The numerics is particularly delicate due to the fast decay rates of the standing wave.
In the paper, we only consider the case k > 1, but this is not an essential restriction since the transformation (u, v, γ, α, k, t, x) → (ũ,ṽ,γ,α,k,t,x) leaves system (1) invariant if we set The proof of Theorem 1.1 is given in sections 2 (if c = 0), 3 (if c > 0) and 4 (the limit c → 0). The proofs in section 3 are based on the phase plane analysis of a system of three ODEs, as might be expected from the parabolic-hyperbolic structure of system (1). This also explains the mathematical relevance of system (1) in the context of travelling waves: it is considerably easier to analyse than coupled systems of two parabolic equations, where one needs to analyse systems of four ODEs. Nevertheless, it leads to completely new phenomena, in particular its structure of TWs is much richer than that of the scalar Fisher-KPP equation.
2. Standing waves. In this section we look for overlapping standing waves and prove Theorem 1.1 in the case c = 0: given α > k > 1 and γ > 0, we must find a pair of smooth solutions U and V of problem (2) if c = 0. Mathematically it is convenient to set w = U + V and the introduce the variables (observe that ww can be interpreted as a flux and t as a mass fraction). We look for a solution for which w > 0 in R and use w ∈ k α , 1 as an independent variable for the functions s and t. Then s(w) and t(w) satisfy, for k α < w < 1, the system   with boundary conditions First we study the sign of s w and t w as functions of w and t, rewriting (4) as Then there exists a smooth and strictly decreasing function t γ : k α , 1 → [0, 1] such that for all w ∈ k α , 1 In addition the map γ → t γ (w) is strictly increasing for all w ∈ k α , 1 . Proof. We have that for all (w, t) ∈ k α , 1 × [0, 1] Observe that ζ 0 (t) → 1 as t → 0 and ζ ∞ (t) → k α as t → 1, but, on the other hand, One easily checks that, for all t ∈ (0, 1), the coefficient of ζ γ in the latter equation is positive if k > γ and negative if γ < k. Hence ζ γ (t) has no local maxima (minima) can change monotonicity at most one time in [0, 1]. Since ζ γ (0) = 1 and ζ γ (1) = k α , this implies that we distinguish three cases (see Figure 6): and t γ is the inverse of ζ γ restricted to the interval 0, and t γ is the inverse of ζ γ restricted to the interval γ(k−1)−(α−1)k (γ−k)(α−1) , 1 . One easily checks that t γ has all properties claimed in Lemma 2.1 for some δ = δ(γ, α, k). and for some δ = δ(γ, α, k).
See also Figure 7 for parts (ii) and (iii) of Lemma 2.2. Proof. Let ζ γ be defined as in the proof of Lemma 2.1. By (6), which implies (7), and which implies (8). The remainder of the proof is immediate.
if the orbit (w, t ε (w)) enters the set Ω 1 for a certain value w < 1, it remains in Ω 1 and, by (7), s ε (1) > 0. Hence we have found a contradiction and (13) Then t εw (w) < 0 for w near 1 and we can follow the orbit (w, t ε (w)) backwards, i.e. for decreasing values of w: if it enters Ω 2 for a certain w ε > k α , it remains in Ω 2 for all k α < w < w ε and, by (8), . Hence w ε < k α + ε δ → k α as ε → 0, and we have proved (13). By (13), the total variation of t ε in k α , 1 is uniformly bounded. By Helly's selection theorem, and since s ε is uniformly Lipschitz continuous, there exist a sequence ε n → 0, a nonincreasing function t ∈ L ∞ k α , 1 and a Lipschitz continuous function s such that, as n → ∞, and a.e. in k α , 1 , and (t εn ) w → t w weakly * (in the sense of bounded measures on k α , 1 ). In particular s k α = s(1) = 0, and (s, t) satisfies the differential equations in (9) in the open set where s > 0. Observe that the latter set is non-empty: if s = 0 in k α , 1 , then st w = s w = 0 for a.e. w ∈ k α , 1 , but by Lemma 2.2 this is impossible. Let (a, b) ⊆ k α , 1 be an open interval in the set where s > 0 such that s(a) = s(b) = 0. Since s is Lipschitz continuous in (a, b), 1/s is not integrable near a and b, and it follows from the equation for t that In addition s w (a + ) ≥ 0 and If a > k α , it follows from (14), (15), Lemma 2.2 and the fact that t(w) is nonincreasing, that s w > 0 a.e. in some neighborhood (a − δ 1 , a), and since s(a) = 0 this yields a contradiction. Hence a = k α and, by (14) and Lemma 2.2, Reasoning similarly we prove that if b < 1, s w < 0 a.e. in some neighborhood (b, b+δ 1 ), and since s(b) = 0 this yields a contradiction. Hence b = 1 and t(1 − ) = 0.
Remark 2.4. The decay of V (z) and U (z) is extremely fast: there exist positive constants C 0 and C 1 such that We proceed formally. Approximately, as z → ∞ and w → 1, .
Hence V ≈ t → 0 faster than any power of 1 − w and Then Then 3. Travelling waves: c > 0. In this section we prove that, given α > k > 1 and γ > 0, for any velocity c > 0 there exists a one-parameter family of overlapping travelling waves, i.e. strictly positive and smooth solutions (U (z), V (z)) of problem (2). The main ingredients of the proof are the following: (i) The introduction of new variables, chosen for mathematical convenience: so U = rw and V = (1 − r)w. Observe that cR can be interpreted as a flux, r as a mass fraction, and R satisfies For overlapping TWs the functions R, r and w are smooth and satisfy 0 < r < 1, w > 0, R > 0 ( ⇔ w > −c) and the parameter found in (ii) is used as a shooting parameter.
A similar method was used in [3] in the case that α = 1 (with a different choice of variables), but if α > k > 1 the analysis is much more complex. For example, in [3] we used w as an independent variable since w(z) is monotonic if α = 1. As we shall see below, if α > k > 1 there are parameter values for which w(z) is not monotonic.
This section is organised as follows. In subsections 3.1 and 3.2 we discuss the linearisation at z = ±∞, and in 3.3 we collect some auxiliary results. In the remaining subsections we use a shooting method to solve problem (I) in different parameter regimes. The distinction in different regimes is dictated by the linearisation problems, and often leads to different qualitative properties of the TWs. Linearisation around (1, 1, 1). We linearise the equations around (R, r, w) = (1, 1, 1):

3.1.
The eigenvalues are and, if λ 1 = λ 2 and γ(k − 1) = k(α − 1), the corresponding eigenvectors are 0)). The first components of v 1 and v 2 are smaller than the third components, Observe that since The sign of the second component of v 1 is determined by For each k > 1 and γ > 0, there exists a unique c 0 > 0 such that Since λ 1 , λ 2 < 0, there exists a local 2-dimensional stable manifold M s for the nonlinear system around (R, r, w) = (1, 1, 1), which, if λ 1 = λ 2 , is tangent to the plane orthogonal to .
By the translation invariance with respect to z, the local stable manifold M s consists of a one-parameter family of invariant C 1 -curves. M s is transversal to the invariant set {(R, r, w) ; R > 0, r = 1, w > 0}. Therefore there exists a (local) solution such that the corresponding curve C 0 lies on M s and is tangent to v 2 . This solution is determined by the equation If λ 1 = λ 2 , it follows from the fast stable manifold theorem (see the Appendix) that each invariant curve in M s is tangent to v 1 or v 2 . More precisely, exactly two incoming orbits are tangential to the eigenvector which corresponds to the most negative eigenvalue. Distinguishing the cases k(α − 1) − γ(k − 1) > 0(< 0) and λ 1 > λ 2 (< λ 2 ), this leads to 4 different pictures for the orbits near (1, 1, 1). See Figures 8, 9, 10 and 11.
Since M u is transversal to the invariant set {(R, r, w) ; R > 0, r = 0, w > 0}, there exist 2 (local) solutions which form a curve C 1 which lies on M u and is tangent to V 2 . This solution is determined by the equation If Λ 1 = Λ 2 , it follows from the fast stable manifold theorem (see the Appendix) that each invariant curve in M u is tangent to V 1 or V 2 . More precisely, exactly two outcoming orbits are tangential to the eigenvector which corresponds to the largest eigenvalue. Distinguishing the cases α(k − 1) − γ(α − 1) > 0(< 0) and Λ 1 > Λ 2 (< Λ 2 ), this leads to 4 different pictures for the orbits near ( k α , 0, k α ). See Figures 12, 13, 14 and 15.
Finally we observe that M u is transversal to the plane {R = w}. On the intersection line, with direction r + o(r) as r → 0 and so, by (21), 3.3. Preliminary remarks. We rewrite the equations for r and R of problem (I) as Below we collect some properties of the functions f and h: h(1, 1) = h( k α , 0) = 0, or H(0) = k α , H(1) = 1 ; Since H(r) is defined if the function H is well-defined and smooth in the interval [0, 1]. Since for all C ∈ R the equation H(r) = C leads to a quadratic equation in r, H changes sign at most once in [0, 1]: and Then γ > α > k and, by (18) and (22), λ 1 > λ 2 and Λ 1 < Λ 2 . See also Figures 9 and 12. We shall prove that (33) ⇒ problem (I) has a solution for which w is non monotonic.
More precisely we shall show that w has exactly one maximum with a value larger than 1 and does not possess local minima. Let H be defined by (27). By (28), (29), (31) and (33), and since γ > α(k−1) α−1 , there exists ρ 0 ∈ (0, 1) such that Let M u be the local unstable manifold around ( k α , 0, k α ). We are interested in those orbits contained in M u for which r ≥ 0. We use a parameter µ ∈ [0, 1] to label them, where µ = 0 (µ = 1) corresponds to the part of the curve C 1 for which w < k α (w > k α ). We recall that the solutions corresponding to µ = 0 and µ = 1 satisfy r = 0 and equation (23).
To characterise µ precisely, we define U δ as the intersection of M u and the sphere of radius δ > 0 and center k α , 0, k α . We fix δ so small that U δ is a closed and simple curve, which intersects each outgoing orbit contained in M u exactly once. We know already which orbits correspond to µ = 0 and µ = 1, and we denote their intersection points with U δ by x 0 and, respectively, x 1 . Let U + δ be the part of U δ where r ≥ 0, so U + δ is a curve connecting x 0 and x 1 . We denote its length by L δ . Given an outgoing orbit which intersects U + δ at a point x, we define its label µ as 1/L δ times the length of the curve in U + δ which connects x 0 and x. By construction, µ varies between 0 and 1. We represent each of these orbits by a solution (R µ (z), r µ (z), w µ (z)) which is well defined for z ≤ 0 and crosses U + δ at z = 0: (R µ (0), r µ (0), w µ (0)) ∈ U + δ for all µ ∈ [0, 1].
Proof. If µ ∈ A, it follows from (38) and the equation for R µ that h(w µ , r µ ) ≤ 0 at z = z * µ . Hence, by (27) and (34), We claim that lim sup Arguing by contradiction we assume that z * µ k → ∞ as µ k → µ − 1 for some sequence {µ k } ⊂ A 0 . By (37) and (44), w µ k ≤ w µ k (z * µ k ) ≤ H(ρ 0 ) in (−∞, z * µ k ), and it easily follows from the equations in problem (I) that (R µ1 , r µ1 , w µ1 ) is a solution of problem (I) in (−∞, ∞) which converges to (1, 1, 1) as z → ∞. Since R µ k > w µ k and w µ k < 1 in (−∞, z * µ k ), such behaviour is not compatible with the linearisation around (1, 1, 1) (see also Figure 9; here we have used that, by the fast stable manifold theorem, the solutions in M s which we consider, are tangent to v 1 as z → ∞) and we have found a contradiction.
Since µ 2 ∈ A 0 , Lemma 3.5 and its proof imply that The discussion of Case 1 is completed by the following result.
Remark 3.7. In a forthcoming paper on homoclinic TWs we shall use the following observation, which follows easily from the above construction and the stable manifold theorem. If µ < µ 2 (so µ ∈ A 0 \ B) and µ 2 − µ is sufficiently small, then w µ in decreasing in (z * µ , z µ ) and crosses the plane w = 1 at some z ∈ (z * µ , z µ ). In particular, the solution orbit will be close to that of the part of the one-dimensional unstable manifold of (1, 1, 1) for which R < w, i.e. the orbit defined by dR dw = w 2 (1−w) c 2 (w−R) for w < 1 and R(1) = 1 (the convergence to this orbit, as µ → µ − 2 , is not uniform: the solutions remain for ever larger values of z near the equiibrium point (1, 1, 1) before following the unstable manifold).
In addition w is either increasing or non-monotonic. In the latter case w has exactly one maximum with a value larger than 1 and it does not possess local minima. We do not know whether both cases really occur.
To prove our claim we follow the procedure in the previous section. The difference is that now λ 1 < λ 2 instead of λ 1 > λ 2 (see Figure 8). In section 3.4.1 we only used the latter inequality to prove (45). In this case we are not able to prove (45) and distinguish two cases: if (45) is satisfied, we follow the proof in section 3.4.1 and obtain a solution for which w is non-monotonic; if (45) is not satisfied, it follows from what is written just below (45) that problem (I) has a solution for which w is monotonic.
In the present version the remark also applies to the cases 3, 4 and 5 discussed below.
Until now we have assumed that γ > k(k−1) 2 αc 2 + k − 1(> 0). In this section we show that the existence and monotonicity results for the solution of problem (I) which we have proved in sections 3.4.1 and 3.4.2 remain valid if 0 < γ < k(k−1) 2 problem (I) has a solution for which w is non-monotonic with exactly one maximum large than 1, and if problem (I) has a solution for which w is either increasing or non-monotonic with exactly one maximum large than 1. Condition (50) implies that Λ 1 > Λ 2 instead of Λ 1 < Λ 2 , a condition which concerns the local structure of the solutions near ( k α , 0, k α ), as z → −∞. Below we indicate where we have previously used the sign of Λ 1 − Λ 2 , and how we can adapt the various proofs.

3.4.4.
A remark on exceptional cases. There are exceptional wave velocities c for which the strict inequalities in the previous sections are not satisfied (for example, it may happen that λ 1 = λ 2 ). Such velocities, say c 0 , are isolated, and it is possible to approximate them by a sequence of velocities for which we already know that problem (I) has a solution. For a different parameter regime, in [3] we have indicated how to prove that the corresponding TWs converge to a solution of problem (I) with velocity c 0 (of course one should control the translation invariance in doing so). Therefore we omit the proofs in the present. Similarly, we shall not pay attention to exceptional velocities in the cases which will be treated below.
In this section we prove that (51) ⇒ problem (I) has a solution for which w is increasing.
It remains to prove the following result.
3.6.3. Case 8: if γ < α(k−1) α−1 , λ 1 < λ 2 , we proceed as in section 3.4.3: the existence and monotonicity results for the solution of problem (I) considered in cases 6 and 7 remain valid if More precisely: if problem (I) has a solution for which w is non-monotonic with exactly one minimum smaller than k α (see Figures 10 and 14), and if problem (I) has a solution for which w is either increasing or non-monotonic with exactly one minimum smaller than k α (see Figures 10 and 15).
4. Convergence to standing waves as c → 0. We briefly discuss the behaviour of the constructed overlapping TWs (U c , V c ) for small values of c. Let 0 < c ≤ 1 and set w c = U c + V c . To eliminate translation invariance we require that w c (0) = 1 2 (1 + k α ). It easily follows from the equation for w which is obtained by adding the equations for U and V in (2) and the uniform Lipschitz continuity of w c that |w c | < C for some constant C which does not depend on c. In addition it follows from their construction that U c (and so V c = w c − U c ) are uniformly bounded in BV (R). Therefore any sequence c k → 0 has a subsequence, which we denote again by c k , such that: -w c k converges uniformly in R to a function w 0 ∈ C 1 (R); -w c k converges uniformly in R to w 0 ; -U c k converges in L 2 loc (R) to a function U 0 ∈ BV (R); -V c k converges in L 2 loc (R) to V 0 = w 0 − U 0 . Let ζ ∈ C 2 (R) have compact support. Since So (U 0 , V 0 ) is a (weak) standing wave.
It also follows from the construction in the previous section that, for fixed α, k and γ, w c > 0 in R for sufficiently small c > 0, so w 0 ≥ 0 in R. By the strong maximum principle, w 0 > 0 in R, in particular k α < w 0 < 1 in R.
Finally we consider the decay rates of the constructed overlapping TWs for small values of c > 0. Setting r c = U c /w c , it follows from the linearisation of the equation for r c around ( k α , 0, k α ) that so, for fixed but small c, U c has exponential decay as z → −∞ but the exponent explodes as c → 0. Similarly, kc z as z → ∞.
These decay rates are not suitable to pass to the limit c → 0. Formally, a more precise result can be obtained as follows. For small values of c > 0 the decay rates as z → ±∞ are determined by λ 2 and Λ 2 . We focus here on the decay as z → −∞. Since Λ 2 → γα k as c → 0 + , it follows from a straightforward calculation that for sufficiently small c w c ≈ γα k (w c − k α ) as z → −∞.
On the other hand r c satisfies, as a function of w, the equation Integrating we obtain that, in the limit c → 0, U c and w c satisfy the relation − log U c ≈ k(k − 1) γα(w c − k α ) as z → −∞.
Together with (57) this result is compatible with the decay rate (17) for the standing wave. A similar procedure leads to a limiting (as c → 0) decay rate for V c (z) as z → ∞ which is compatible with (16). 5. Concluding remarks. When in a healthy tissue environment, abnormal tissue (which is slightly different from healthy tissue) appears as a result of mutation, it is biologically and medically relevant to know how the abnormal tissue expands. To address this question, mathematical models are proposed and analysed. In this paper, we consider a system which describes between normal and abnormal cell densities, say u and v, respectively. Since TWs often describe the large time behaviour of solutions, it is important to consider TWs to answer the above question. Particularly, we focus on TWs of (1) in the parameter regime where α > k > 1 and γ > 0. It is known ( [5]) that segregated TWs, which satisfy U V = 0 in R, have negative wave velocities c < 0. In the present paper we discuss the existence of overlapping TWs satisfying U, V > 0 in R and c ≥ 0. If c = 0, (2) possesses standing waves with extremely fast decay tails. To our best knowledge, this type of TWs has not been reported so far, and it seems that this feature comes form a parabolic-hyperbolic nature of (2). If c > 0, the overlapping TWs can be either monotonic or nonmonotonic in U + V according to the parameter values. But then a subsequent question arises spontaneously: is there any TW for c ∈ (c, 0)? Or is the segregated TWs with negative wave velocity isolated? In numerical tests we could not find any overlapping TWs for c ∈ (c, 0). However, in a forthcoming paper we will prove the existence of TWs for c ∈ (c, 0) which are neither segregated nor overlapping: we call them partially overlapping TWs.
Though we are gradually unraveling TWs of (2), the global structure of TWs is not yet completely understood. This will be the subject of future work.
From (58) and (59) we obtain the one-dimensional differential manifold S fast of class C 2 , and the proof is complete.